Skip to content

Advertisement

  • Research
  • Open Access

Mathematical modeling of malaria transmission global dynamics: taking into account the immature stages of the vectors

Advances in Difference Equations20182018:220

https://doi.org/10.1186/s13662-018-1671-2

  • Received: 23 December 2017
  • Accepted: 8 June 2018
  • Published:

Abstract

In this paper we present a mathematical model of malaria transmission. The model is an autonomous system, constructed by considering two models: a model of vector population and a model of virus transmission. The threshold dynamics of each model is determined and a relation between them established. Furthermore, the Lyapunov principle is applied to study the stability of equilibrium points. The common basic reproduction number has been determined using the next generation matrix and its implication for malaria management analyzed. Hence, we show that if the threshold dynamics quantities are less than unity, the mosquitoes population disappears leading to malaria disappearance; but if they are greater than unity, mosquitoes population persists and malaria also.

Finally, numerical simulations are carried out to support our mathematical results.

Keywords

  • Mosquitoes
  • Malaria transmission
  • Thresholds dynamics
  • Stability
  • Lyapunov principle

1 Introduction

The burden of infectious diseases goes beyond the individual but extends to collectives including families, communities, countries and the whole world. The impact is both social and economic as it keeps children away from school and adults away from work. Most malaria-related mortality and a large fraction of malaria cases occur in sub-Saharan Africa, where transmission is very intense. Moreover, in endemic regions, children under five, pregnant women, and non-immune adults are most at risk of mortality due to malaria. For instance, the World Health Organization estimated that there were 214 million malaria cases in 2015, resulting in about 438 thousand deaths. Costs for treatment are often very expensive for patients driving already poor families into ruin. The countrywide economic loss due to disease is immense, cementing poverty and underdevelopment particularly in low income countries [29, 30].

Although mathematical models are an abstract simplification of the reality; they can still capture the main features of the system and are more amenable to experimentation or analysis. As such, mathematical models can therefore be used to describe or predict the outcomes of epidemics or pandemics providing information that is crucial in informing public health intervention policies. This allows policy makers to optimize the use of their limited resources. Concerning the mathematical modeling of malaria, significant results have been established in the recent years since the first model introduced by Ross [28]. Ross defended that keeping the mosquitoes population under a certain threshold can lead to malaria eradication. Some years later, Macdonald [22] improved the model of Ross showing that reducing the number of mosquitoes effectively has some effects on epidemiology of malaria in areas of intense transmission. Before the role of anopheles in the spread of malaria was known, efforts to control the disease were sporadic, infrequent and insignificant. Furthermore, Aron and May [2], added various features of malaria to the model of Macdonald, such that an incubation period in the mosquito, super-infection and a period of immunity in human beings. Besides, the inclusion of acquired immunity proposed by Dietz et al. [12] was a major point of malaria modeling.

Other reviews on mathematical modeling in malaria include work by Ngwa et al. [24] and Chitnis et al. [7, 9]. Indeed, in the model proposed by Ngwa and Shu, human hosts follow an SEIRS-like pattern and vector hosts follow the SEI pattern due to their short life cycle. In [35], a similar model is described by Yang, but with only one class for humans. Humans move from the susceptible, to the exposed class at some probability when they come into contact with an infectious mosquito, and then to the infectious class, as in conventional SEIRS models. However, infectious people can then recover with, or without, a gain in immunity; and either return to the susceptible class, or move to the recovered class. Moreover, Chitnis et al. extended the model of Ngwa and Shu by assuming that, although individuals in the recovered class are immune, in the sense that they do not suffer from serious illness and do not contract clinical malaria, they still have low levels of Plasmodium in their blood stream and can infect the susceptible mosquitoes. This is one of the main features which makes a distinction between malaria and many other vector-borne diseases.

In addition, based on the susceptibility, the exposedness and the infectivity of human hosts, Ducrot et al. [13] have developed two species malaria model in which we find two host types in the human population: non-immune and semi-immune. In fact, the non-immune is supposed to be more vulnerable to malaria than the semi-immune because it has never acquired immunity against the disease. Meanwhile, the semi-immune has at least once acquired immunity in his life.

In the study of all these models, we remark that the mosquito life cycle is ignored. Generally, the authors consider a constant recruitment rate in the vector population. However, recent work has shown that some of the factors, as the age structure of mosquitoes population and climate effects, are very important for a better understanding of malaria transmission global dynamics, [5, 8, 13, 14, 23, 25, 31]. Indeed, mosquitoes undergo complete metamorphosis going through four distinct stages of development during a lifetime: egg, larva, pupa, and adult. While it is appropriate to assume that only adult mosquito are involved in the malaria transmission, the dynamics of the juvenile stages (larvae and pupae) has significant effects on the dynamics of the mosquitoes population, and then the malaria transmission global dynamics. Motivated by this work, and using the malaria model in [13] as our baseline model, we include the four distinct metamorphic stages of mosquito to formulate a mosquito-stage-structured autonomous model of malaria spread in a more general setting.

The paper is organized as follows. In Sect. 2, we present a vector age-structured model. For this model we established a threshold parameter r. Using this threshold and the Lyapunov theory, we established the local stability and the global stability of the equilibria. Section 3 concerns the malaria transmission model. Its mathematical analysis is done focusing on the boundedness, the positivity, local stability, and global stability of the equilibria. We also found two threshold parameters, respectively, denoted by r and \(r_{0}\), that determine the global dynamical behavior of malaria in an area. Section 4 is devoted to numerical simulations. A conclusion finishes the paper.

2 Vectors population growth dynamics

2.1 Mathematical formulation of the model

In this section, we formulate a model for the mosquitoes population growth basing on their life cycle. There are four main stages in the vector life cycle. The first three stages namely egg, larva and pupa are both aquatics, while the adult stage is aerial. Moreover, the eggs, the larvae and the pupae respond differently to the control measures. For instance, chemical interventions on the breeding sites has impact on the larvae and pupae population, but not on the eggs. For all these reasons, to provide some acceptable strategies to stop mosquitoes population proliferation, it would be fair to dissociate the aquatic stage in the modeling of the mosquitoes growth dynamics. Here, we propose a mathematical model of female anopheles population global behavior as in [1, 23] taking into account the four stages of mosquito. Thus, following the four different stages of the mosquito growth dynamics, we, respectively, denote
  • \(E(t)\): the number of eggs at the moment t;

  • \(L(t)\): the number of larvae at the moment t;

  • \(P(t)\): the number of pupae at the moment t;

  • \(A(t)\): the number of females at the moment t.

Let us consider the following positive transfer parameters:
  • b: the intrinsic egg-laying rate;

  • \(s_{E}\), \(s_{L}\), \(s_{P}\): respectively, the rates of transfer from eggs to larvae, from larvae to pupae, and from pupae stage to females.

  • \(d_{E}\), \(d_{L}\), \(d_{P}\), \(f_{m}\): respectively, the natural death rate of eggs, larvae, pupae, and females.

(H1):: 

We assume that the number of eggs is proportional to the number of females.

Using the hypothesis (H1) and analyzing the above diagram (Fig. 1), we obtain the following system:
$$ \textstyle\begin{cases} E'(t)=bA(t)-(s_{E}+d_{E})E(t), \\ L'(t) =s_{E}E(t)-(s_{L}+d_{L})L(t), \\ P'(t) =s_{L}L(t)-(s_{P}+d_{P})P(t), \\ A'(t)=s_{P}P(t)-f_{m}A(t). \end{cases} $$
(1)
In this model, we only take into account the phenomena of growth and death of the different stages.
Figure 1
Figure 1

An age-structured model for vector population growth dynamics: E for eggs, L for larvae, P for pupae and A for adult females

However, when we consider the difficulties due to the availability of spaces, foods and the oviposition habitat selection, the following assumptions can be made:
(H2):: 
The growth of eggs depends on the availability of the nutrients. It also depends on the availability of the space because, the oviposition habitat selection is made taking into account the possibility of development of larvae and pupae. Indeed, before laying eggs the adult mosquitoes make sure that the immature stages can develop relatively unmolested. This hypothesis leads to the following logistic growth model:
$$ \textstyle\begin{cases} E'(t)=bA(t) ( 1-\frac{E(t)}{K_{E}} ) -(s_{E}+d_{E})E(t), \\ L'(t) =s_{E}E(t) ( 1-\frac{L(t)}{K_{L}} ) -(s_{L}+d_{L})L(t), \\ P'(t) =s_{L}L(t) ( 1-\frac{P(t)}{K_{P}} ) -(s_{P}+d_{P})P(t), \\ A'(t)=s_{P}P(t)-f_{m}A(t). \end{cases} $$
(2)

2.2 Mathematical analysis of the vector model

2.2.1 Existence and uniqueness of solutions

Theorem 2.1

For any initial condition \((t_{0},X_{0})\in \mathbb{R}_{+}\times \mathbb{R}_{+}^{4}\), the system (2) admits a unique positive maximal solution.

Proof

The model (2) is described by a system of first order autonomous linear differential equations. It can be rewritten as follows:
$$ X'(t)=\mathcal{F}_{1}\bigl(X(t)\bigr), $$
where
$$ X(t)= \begin{pmatrix} E(t), L(t), P(t), A(t) \end{pmatrix}^{T} $$
and \(\mathcal{F}_{1}\) is \(\mathcal{C}^{\infty }\) of \(\mathbb{R}^{4}\) into \(\mathbb{R}^{4}\) and defined by
$$ \mathcal{F}_{1}(X)= \begin{pmatrix} f_{1}(x_{1},x_{2},x_{3},x_{4}) \\ f_{2}(x_{1},x_{2},x_{3},x_{4}) \\ f_{3}(x_{1},x_{2},x_{3},x_{4}) \\ f_{4}(x_{1},x_{2},x_{3},x_{4}) \end{pmatrix} = \begin{pmatrix} bx_{4} ( 1-\frac{x_{1}}{K_{E}} ) -(s_{E}+d_{E})x_{1} \\ s_{E}x_{1} ( 1-\frac{x_{2}}{K_{L}} ) -(s_{L}+d_{L})x_{2} \\ s_{L}x_{2} ( 1-\frac{x_{3}}{K_{P}} ) -(s_{P}+d_{P})x_{3} \\ s_{P}x_{3}-f_{m}x_{4} \end{pmatrix} $$
with \(X=(x_{1},x_{2},x_{3},x_{4})\in \mathbb{R}^{4}\). Since \(\mathcal{F}_{1}\) is \(\mathcal{C}^{\infty }\) and then \(\mathcal{C} ^{1}\), it is locally lipschitzian on \(\mathbb{R}^{4}\). Then we deduce the existence and the uniqueness of the maximal solution of the Cauchy problem associated to the system (2) with the initial condition \((t_{0},X_{0})\in \mathbb{R}_{+}\times \mathbb{R}^{4}\). In addition, the solution is \(\mathcal{C}^{\infty }\) because \(\mathcal{F}_{1}\) is \(\mathcal{C}^{\infty }\).
Now, we establish the non-negativity of the solutions. For this purpose, we proceed by absurd. Let us assume that there exists \(\bar{t}_{1}> t _{0}\) such that \(\forall t>\bar{t}_{1}, X(t)\notin \mathbb{R}_{+}^{4}\). Consider
$$\begin{aligned} t_{1}=\inf \bigl\lbrace t| X(t)\notin \mathbb{R}_{+}^{4}\bigr\rbrace , \end{aligned}$$
(3)
this means that \(\forall t\in \mathbb{R}_{+}, t_{0}\leq t < t_{1}, X(t) \in \mathbb{R}_{+}^{4}\).
Consequently, there exists \(\epsilon > 0\) such that
$$\begin{aligned} \forall t_{1}\leq t < t_{1}+\epsilon ,\quad X(t)\notin \mathbb{R}_{+} ^{4}. \end{aligned}$$
(4)
Since \(X^{*}=(0,0,0,0)\) is a steady equilibrium, the uniqueness of the solutions implies \(X(t_{1})\neq (0,0,0,0)\).
For \(t=t_{1}\), 12 cases are possible.
  1. (i)
    Let us consider the case \(X(t_{1})=(0,L(t_{1}),P(t_{1}),A(t _{1}))\), where \(L(t_{1})\), \(P(t_{1})\), \(A(t_{1})\) are positive. Since \(A(t_{1})>0\) and \(E(t_{1})=0\), from the first equation of the system (2) we have
    $$ E'(t_{1})=bA(t_{1})>0. $$
    A first order limited development of \(E(t)\) in the neighborhood of \(t_{1}\) is given by
    $$ E(t)=E'(t_{1}) (t-t_{1})+o(t-t_{1}), \quad t\rightarrow t_{1}. $$
    Thus, there exists \(\bar{\epsilon }>0\) such that, for all \(t\in [t_{1},t_{1}+\bar{\epsilon }]\), we have \(E(t)>0\). Besides, by continuity of solutions, there exists \(\bar{\bar{\epsilon }}>0\) such that, for all \(t\in [t_{1},t_{1}+\bar{\bar{ \epsilon }}]\), \(L(t)>0\), \(P(t)>0\), \(A(t)>0\). Thus,
    $$ t\in \bigl[t_{1},t_{1}+\inf \lbrace \bar{\epsilon },\bar{ \bar{\epsilon }} \rbrace \bigr], \quad \text{and} \quad X(t)\in \mathbb{R}_{+}^{4}. $$

    This result is a contradiction with the definition of \(t_{1}\) given in (3).

     
  2. (ii)

    Let us consider \(X(t_{1})=(0,0,0,A(t_{1}))\) with \(A(t_{1})>0\).

    We previously show that there exists \(\bar{\epsilon }>0\) such that, for all \(t\in [t_{1},t_{1}+\bar{\epsilon }]\), \(L(t)>0\). Besides, \(P(t_{1})=0\), \(P'(t_{1})=0\), \(P''(t_{1})=0\) and \(P'''(t_{1})=s_{L}L''(t_{1})\). Since \(L''(t_{1})=s_{E}E'(t_{1})=s_{E}bA(t _{1})>0\) for \(A(t_{1})>0\),
    $$ P'''(t_{1})=s_{E}s_{L}bA(t_{1})>0. $$
    Therefore, a three order limited development of \(P(t)\) about \(t_{1}\) is written as follows:
    $$ P(t)=P'''(t_{1}) \frac{(t-t_{1})^{3}}{6}+o\bigl((t-t_{1})^{3}\bigr), \quad t \rightarrow t_{1}. $$
    We deduce that there exists \(\bar{\bar{\epsilon }}>0\) such that, for all \(t\in [t_{1},t_{1}+\bar{\bar{\epsilon }}]\), \(P(t)>0\) and therefore, since \(A(t_{1})>0\), there exists \(\epsilon _{1}>0\) such that, for all \(t\in [t_{1},t_{1}+\epsilon_{1}]\), \(X(t)\in \mathbb{R}_{+}^{4}\). This is a contradiction with the condition (4).
     
A similar proof can easily be given for the ten other cases, which are \((E(t_{1}),L(t_{1}),P(t_{1}),0)\), \((E(t_{1}),L(t_{1}),0,0)\), \((E(t_{1}),0,0,0)\), \((E(t _{1}),0,P(t_{1}),0)\), \((E(t_{1}),0,0,A(t_{1}))\), \((0,L(t_{1}),P(t_{1}),0)\), \((0, L(t_{1}),0, A(t _{1}))\), \((0,0,P(t_{1}),0)\), \((0,L(t_{1}),0,0)\), \((0,0,P(t_{1}),A(t_{1}))\) where \(E(t_{1})\), \(L(t_{1})\), \(P(t_{1})\) and \(A(t_{1})\) are, respectively, positive. □
We investigate the asymptotic behavior of orbits starting in the non-negative cone,
$$ \mathbb{R}_{+}^{4}= \bigl\lbrace (x,y,z,w)\in \mathbb{R}^{4}|x\geq 0, y\geq 0, z\geq 0, w\geq 0 \bigr\rbrace . $$
Let us also consider the positive cone denoted by
$$ \mathbb{R}_{+}^{*4}= \bigl\lbrace (x,y,z,w)\in \mathbb{R}^{4}|x> 0, y>0, z> 0, w> 0 \bigr\rbrace . $$

2.2.2 Equilibrium states

Let us consider the following threshold parameter, called the mosquito reproduction number:
$$ r= \biggl( \frac{b}{s_{E}+d_{E}} \biggr) \biggl( \frac{s_{E}}{s_{L}+d _{L}} \biggr) \biggl( \frac{s_{L}}{ s_{P}+d_{P}} \biggr) \biggl( \frac{ s _{P}}{f_{m}} \biggr) . $$

Proposition 2.1

The system (2) always has a mosquito-free equilibrium
$$ X_{0}^{*}= ( 0,0,0,0 ) . $$
Moreover,
  • if \(r\leq 1\), then system (2) has no other equilibrium,

  • if \(r>1\), there is a unique non-trivial equilibrium,
    $$ X^{*}= \biggl( 1-\frac{1}{r} \biggr) \biggl( \frac{K_{E}}{\chi_{E}},\frac{K _{L}}{\chi_{L}}, \frac{K_{P}}{\chi_{P}}, \frac{s_{P}}{f_{m}} \frac{K _{P}}{\chi_{P}} \biggr) = \bigl( E^{*}, L^{*}, P^{*}, A^{*} \bigr), $$
    where
    $$\begin{aligned} &\chi_{E}= \biggl( 1-\frac{1}{r} \biggr) +\frac{f_{m}(s_{E}+d_{E})\chi _{P}}{bs_{P}K_{P}}, \\ &\chi_{L}= \biggl( 1-\frac{1}{r} \biggr) \biggl( 1+ \frac{(s_{L}+d_{L})K_{L}}{s _{E}K_{E}} \biggr) +\frac{(s_{E}+d_{E})(s_{L}+d_{L})K_{L} \chi_{P}}{bs _{E}s_{P} K_{P}}, \end{aligned}$$
    and
    $$ \chi_{P}=1+\frac{s_{E}K_{E}K_{P}(s_{P}+d_{P})+(s_{L}+d_{L})(s_{P}+d _{P})K_{L}K_{P}}{s_{E}s_{L}K_{E}K_{L}}. $$

Proof

Setting all the equations of system (2) to zero, we easily obtain the above results. □

Lemma 2.1

The set
$$ \Delta = \biggl\lbrace (x,y,z,w)\in \mathbb{R}^{4} | 0\leq x\leq K_{E}, 0\leq y\leq K_{L}, 0\leq z\leq K_{P}, 0 \leq w\leq \frac{s_{P}}{f_{m}}K_{P} \biggr\rbrace $$
is positively invariant by the system (2).

Proof

Let us consider \((t_{0},X_{0}=(E_{0},L_{0},P_{0},A_{0}))\in \mathbb{R}_{+}\times \mathbb{R}_{+}^{4}\) and \(([t_{0},T_{\max }],X=(E,L,P,A))\) a maximal solution of Cauchy problem associated to (2) with the initial condition \((t_{0},X_{0})\), \((T_{\max }\in [t_{0},+\infty ])\).

Let us consider \(t_{1}\in [t_{0},T_{\max }]\). We must show that
  • if \(E(t_{1})\leq K_{E}\) then, for all \(t_{1}\leq t\leq T_{\max }\), \(E(t)\leq K_{E}\),

  • if \(L(t_{1})\leq K_{L}\) then, for all \(t_{1}\leq t\leq T_{\max }\), \(L(t)\leq K_{L}\),

  • if \(P(t_{1})\leq K_{P}\) then, for all \(t_{1}\leq t\leq T_{\max }\), \(P(t)\leq K_{P}\),

  • if \(A(t_{1})\leq \frac{s_{L}}{f_{m}}K_{P}\) then, for all \(t_{1} \leq t\leq T_{\max }\), \(A(t)\leq \frac{s_{L}}{f_{m}}K_{P}\).

  1. 1.

    Let us show that, for all \(t\in [t_{0},T_{\max }]\), \(E(t)\leq K _{E}\).

    Assume that there exists \(\epsilon_{1}>0\) such that \(t_{1}\leq t_{1}+ \epsilon_{1}< T_{\max }\) and \(E(t_{1}+\epsilon_{1})>K_{E}\). We choose \(t_{1}^{*}=\inf \lbrace t \geq t_{1} | E(t)>K_{E}\rbrace \). Since \(E(t_{1}^{*})=K_{E}\), a first order limited development of \(E(t)\) in the neighborhood of \(t_{1}^{*}\) is given by
    $$ E(t)=K_{E}+E'\bigl(t_{1}^{*}\bigr) \bigl(t-t_{1}^{*}\bigr)+o\bigl(t-t_{1}^{*} \bigr), \quad t\rightarrow t_{1}^{*}. $$
    Besides, using the first equation of system (2), and replacing \(E(t_{1}^{*})\) by \(K_{E}\), we obtain
    $$ E'\bigl(t_{1}^{*}\bigr)=-(s_{E}+d_{E})K_{E}< 0. $$
    So, there exists ϵ̄ such that, for all \(t_{1}^{*} \leq t< t_{1}^{*}+\bar{\epsilon }\), \(E(t)< K_{E}\), this is absurd because of the hypothesis on \(t_{1}^{*}\). We deduce that, for all \(t\in [t_{0},T_{\max }]\), \(E(t)\leq K_{E}\).
     
  2. 2.

    Now, we want to show that, for all \(t\in [t_{0},T_{\max }]\), \(L(t)\leq K_{L}\).

    Suppose that there exists \(\epsilon_{1}\) such that, for all \(t_{1}\leq t_{1}+\epsilon < T_{\max }\) and, \(L(t_{1}+\epsilon_{1})> K_{L}\).

    Let us assume that \(t_{1}^{*}=\inf \lbrace t\geq t_{1}| L(t)> K_{L} \rbrace \). We have \(L(t_{1}^{*})=K_{L}\), so a first order limited development of \(L(t)\) about \(t_{1}^{*}\) is given by
    $$ L(t)= K_{L}+L'\bigl(t_{1}^{*}\bigr) \bigl(t-t_{1}^{*}\bigr)+o\bigl(t-t_{1}^{*} \bigr), \quad t\rightarrow t_{1}^{*}. $$
    From the second equation of system (2), by replacing \(L(t_{1}^{*})\) by \(K_{L}\) it follows that
    $$ L'\bigl(t_{1}^{*}\bigr)=-(s_{L}+d_{L})K_{L}. $$
    This result implies that \(L'(t_{1}^{*})<0\). So, there exists \(\bar{\epsilon }>0\) such that, for all \(t_{1}^{*}\leq t< t_{1}^{*}+\bar{ \epsilon }\), \(L(t)< K_{L}\). This contradicts the hypothesis. Thus, there exists for all \(t\in [t_{0},T_{\max }]\), \(L(t)\leq K_{L}\).
     
  3. 3.

    Let us also show that, for all \(t\in [t_{0},T_{\max }]\), \(P(t)\leq K_{P}\).

    We suppose that there exists \(\epsilon_{1}>0\) such that \(t_{1}\leq t _{1}+\epsilon_{1}< T_{\max }\) and \(P(t_{1}+\epsilon_{1})>K_{P}\). Set \(t_{1}^{*}=\inf \lbrace t\geq t_{1}| P(t)>K_{P} \rbrace \). As \(P(t_{1}^{*})=K_{P}\), a first order limited development of \(P(t)\) in the neighborhood of \(t_{1}^{*}\) is given by
    $$ P(t)=K_{P}+P'\bigl(t_{1}^{*}\bigr) \bigl(t-t_{1}^{*}\bigr)+o\bigl(t-t_{1}^{*} \bigr), \quad t\rightarrow t_{1}^{*}. $$
    Furthermore, from the third equation of system (2), we obtain by substituting \(P(t_{1}^{*})\) by \(K_{P}\)
    $$ P'\bigl(t_{1}^{*}\bigr)=-(s_{P}+d_{P})K_{P}< 0. $$
    Then there exists \(\bar{\epsilon }>0\) such that, for all \(t_{1}^{*} \leq t< t_{1}^{*}+\bar{\epsilon }\), \(P(t)< K_{P}\), which is absurd. We deduce that, for all \(t\in [t_{0},T_{\max }]\), \(P(t)\leq K_{P}\).
     
  4. 4.

    Finally, we show that, for all \(t\in [t_{0},T_{\max }]\), \(A(t)\leq \frac{s_{P}}{f_{m}}K_{P}\).

    We suppose the existence of \(\epsilon_{1}>0\) such that
    $$ t_{1}\leq t_{1}+\epsilon_{1}< T_{\max } \quad \text{and} \quad A(t_{1}+\epsilon_{1})> \frac{s_{P}}{f_{m}}K_{P}. $$
    Considering \(t_{1}^{*}=\inf \lbrace t\geq t_{1}, A(t)>\frac{s _{P}}{f_{m}}K_{P} \rbrace \), we have \(A(t_{1}^{*})=\frac{s_{P}}{f _{m}}K_{P}\).
    Since \(P(t_{1}^{*})=K_{P}\), it then follows that
    $$ A'\bigl(t_{1}^{*}\bigr)=s_{P}P \bigl(t_{1}^{*}\bigr)-f_{m}A\bigl(t_{1}^{*} \bigr)=s_{P}K_{P}-f_{m} \times \frac{s_{P}}{f_{m}}K_{P}=0, $$
    namely \(A'(t_{1}^{*})=0\). Hence,
    $$\begin{aligned} A''\bigl(t_{1}^{*}\bigr) =& s_{P}P'\bigl(t_{1}^{*} \bigr)-f_{m}A'\bigl(t_{1}^{*}\bigr) \\ =& s_{P}P'\bigl(t_{1}^{*}\bigr) \\ =& -s_{P}(s_{P}+d_{P})K_{P}< 0. \end{aligned}$$
    A second order limited development of \(A(t)\) about \(t_{1}^{*}\) yields
    $$ A(t)=\frac{s_{P}}{f_{m}}K_{P}+A'\bigl(t_{1}^{*} \bigr) \bigl(t-t_{1}^{*}\bigr)+A'' \bigl(t_{1} ^{*}\bigr)\frac{(t-t_{1}^{*})^{2}}{2}+o\bigl( \bigl(t-t_{1}^{*}\bigr)^{2}\bigr), \quad t \rightarrow t_{1}^{*}. $$
    In this case, there exists \(\bar{\epsilon }>0\) such that, for all \(t_{1}^{*}< t\leq t_{1}^{*}+\bar{\epsilon }, A(t)<\frac{s_{P}}{f_{m}}K _{P}\). This is absurd.

    In conclusion, for all \(t\in [t_{0},T_{\max }]\), \(A(t)\leq \frac{s _{P}}{f_{m}}K_{P}\).

     
 □

Proposition 2.2

Δ is attractive for the system (2).

Proof

Let us consider \((t_{0},X_{0}=(E_{0},L_{0},P_{0},A_{0})\in \mathbb{R} _{+}\times \mathbb{R}_{+}^{4}\setminus \Delta \) and \(([t_{0},T_{ \max }],X=(E,L,P,A))\) a global solution of Cauchy problem associated to (2) with the initial condition \((t_{0},X_{0})\).

Lemma 2.1 shows that Δ is invariant. It remains to show that there exists t such that \(X(t)\in \Delta \). We will proceed by showing the contrary to be absurd.
  • We suppose that, for all \(t\in [t_{0},+\infty [\), \(E(t)>K_{E}\). From the first equation of the system (2), we have \(E'(t)=bA(t) ( 1-\frac{E(t)}{K _{E}} ) -(s_{E}+d_{E})E(t)\). Then \(bA(t) ( 1-\frac{E(t)}{K _{E}} ) <0\), and it follows that \(E'(t)< -(s_{E}+d_{E})K_{E}\).

    Integrating from \(t_{0}\) to t, we obtain
    $$ \int_{t_{0}}^{t}E'(t)\,dt\leq - \int_{t_{0}}^{t}(s_{E}+d_{E})K_{E} \,dt, \quad \forall t\geq t_{0}. $$
    Consequently, \(E(t)\leq E(t_{0})-(s_{E}+d_{E})K_{E}(t-t_{0})\), \(t\leq t_{0}\).
    Posing \(t_{1}=t_{0}+\frac{E_{0}(t-K_{E}}{(s_{E}+d_{E})K_{E}}\), then
    $$\begin{aligned} E(t_{1}) \leq & E_{0}-(s_{E}+d_{E})K_{E} \times \biggl( t_{0}+\frac{E _{0}-K_{E}}{(s_{E}+d_{E})K_{E}}-t_{0} \biggr) \\ \leq & E_{0}-(E_{0}-K_{E}) \\ \leq & K_{E}, \end{aligned}$$
    which is a contradiction. So, for all \(t>t_{1}, E(t)\leq K_{E}\).
  • If \(L(t_{1})\leq K_{L}\), then the solution \(L(t)\) is defined in Δ, which is invariant. If not, suppose that, for any \(t\in [t_{1},+\infty [\), with \(t_{1}\) previously defined, thus \(L(t)>K_{L}\). Then, for all \(t\in [t_{1},+\infty [\), and thanks to the second equation of the system (2), we have \(L'(t)<-(s_{L}+d _{L})K_{L}\).

    By integrating between \(t_{1}\) and t and using the principle of comparison, we obtain for all \(t\in [t_{1},+\infty [\): \(L(t)\leq L(t _{1})-(s_{L}+d_{L})K_{L}(t-t_{1})\).

    Considering \(t_{2}=t_{1}+\frac{(L(t_{1})-K_{L})}{(s_{L}+d_{L})K_{L}}\), then
    $$\begin{aligned} L(t_{2}) \leq & L(t_{1})-(s_{L}+d_{L})K_{L} \biggl( t_{1}+\frac{(L(t _{1})-K_{L})}{(s_{L}+d_{L})K_{L}}-t_{1} \biggr) \\ \leq & L(t_{1})-(s_{L}+d_{L})K_{L} \times \frac{(L(t_{1})-K_{L})}{(s _{L}+d_{L})K_{L}} \\ \leq & L(t_{1})-\bigl(L(t_{1})-K_{L}\bigr) \\ \leq & K_{L}. \end{aligned}$$
    Therefore, there exists \(t_{2}>t_{1}\) such that \(L(t_{2})\leq K_{L}\), which is a contradiction. So, \(L(t)\leq K_{L}\).
  • If \(P(t)\leq K_{P}\), then the solution \(P(t)\) is defined in Δ, which is invariant. If not, suppose that, for all \(t\in [t_{2},+\infty [\), \(P(t)>K_{P}\).

    From the third equation of the system (2), we have, for all \(t\in [t_{2},+\infty [\), \(P'(t)=s_{L}L(t) ( 1-\frac{P(t)}{K_{P}} ) -(s_{P}+d_{P})P(t)\) and as \(( 1-\frac{P(t)}{K _{P}} ) <0\), then \(P'(t)<-(s_{P}+d_{P})K_{P}\).

    Integrating from \(t_{3}\) and t yields
    $$ P(t)\leq P(t_{3})-(s_{P}+d_{P})K_{P}(t-t_{3}), \quad t\leq t_{3} . $$
    Setting \(t_{3}=t_{2}+\frac{P(t_{2})-K_{P}}{(s_{P}+d_{P})K_{P}}\), then
    $$\begin{aligned} P(t_{3}) \leq & P(t_{2})-(s_{P}+d_{P})K_{P} \times \biggl( t_{2}+\frac{P(t _{2})-K_{P}}{(s_{P}+d_{P})K_{P}}-t_{2} \biggr) \\ \leq & P(t_{2})-\bigl(P(t_{2})-K_{P}\bigr) \\ \leq & K_{P}. \end{aligned}$$
    Hence, there exists \(t_{3}>t_{2}\) such that \(P(t_{3})\leq K_{P}\), which is a contradiction. Then, \(\forall t\in [t_{3},+\infty [, P(t)\leq K _{P}\).
  • If \(A(t)\leq \frac{s_{P}}{f_{m}}K_{P}\), the solution \(A(t)\) is defined in Δ, which is invariant. On the other hand, suppose that, for all \(t\in [t_{3},+\infty [\), \(A(t)>\frac{s_{P}}{f_{m}}K_{P}\). Hence, from the last equation of the system (2), for all \(t\in [t _{3},+\infty [\), \(A(t)\leq A(t_{3})-c(t-t_{3})\).

    Considering \(t_{4}=t_{3}+ \frac{A(t_{3})-\frac{s_{P}}{f_{m}}K_{P}}{c}\), it then follows that
    $$ A(t_{4}) \leq A(t_{3})-c \biggl( t_{3}+ \frac{A(t_{3})-\frac{s_{P}}{f _{m}}K_{P}}{c}-t_{3} \biggr) \leq \frac{s_{P}}{f_{m}}K_{P}. $$
    This is a contradiction.

    So, for all \(t\geq \max (t_{1},t_{2},t_{3},t_{4})\), we have \((E(t),L(t),P(t),A(t))\in \Delta \).

 □

Corollary 2.1

Let \((t_{0},X_{0}=(E_{0},L_{0},P_{0},A_{0}))\in \mathbb{R}_{+}\times \mathbb{R}_{+}^{4}\). The maximum solution of the problem of Cauchy relative to the system (2) and associated with the initial condition is global.

Proof

Let us consider \((t,X=(E,L,P,A))\in \mathbb{R}_{+}\times \mathbb{R} _{+}^{4}\), the maximal solution of the Cauchy problem relative to the system (2) and associated with the initial condition \((t_{0},X_{0})\). By Proposition 2.2 and Lemma 2.1, we know that this solution is bounded. Thus, it is global. □

2.2.3 Stability of equilibrium

Theorem 2.2

The equilibrium \(X_{0}^{*}=(0,0,0,0)\) is locally asymptotically stable if and only if \(r<1\).

Proof

The local stability of the equilibrium \(X_{0}^{*}=(0,0,0,0)\) is given by the Jacobian matrix \(D\mathcal{F}_{1}(X_{0}^{*})\) of the system evaluated at this point. We have
$$\begin{aligned} D\mathcal{F}_{1}\bigl(X_{0}^{*}\bigr) = \begin{pmatrix} -(s_{E}+d_{E}) & 0 & 0 & b \\ s_{E} & -(s_{L}+d_{L}) & 0 & 0 \\ 0 & s_{L} & -(s_{P}+d_{P}) & 0 \\ 0 & 0 & s_{P} & -f_{m} \end{pmatrix} . \end{aligned}$$
The matrix \(D\mathcal{F}_{1}(X_{0}^{*})\) can be rewritten as \(D\mathcal{F}_{1}(X_{0}^{*})=M+N\) with M a positive matrix defined by
$$ M= \begin{pmatrix} 0 & 0 & 0 & b \\ s_{E} & 0 & 0 & 0 \\ 0 & s_{L} & 0 & 0 \\ 0 & 0 & s_{P} & 0 \end{pmatrix} $$
and N is a diagonal matrix defined as follows:
$$ N= \begin{pmatrix} -(s_{E}+d_{E}) & 0 & 0 & 0 \\ 0 & -(s_{L}+d_{L}) & 0 & 0 \\ 0 & 0 & -(s_{P}+d_{P}) & 0 \\ 0 & 0 & 0 & -f_{m} \end{pmatrix} . $$
Consequently,
$$ P=-MN^{-1}= \begin{pmatrix} 0 & 0 & 0 & \frac{b}{f_{m}} \\ \frac{s_{E}}{s_{E}+d_{E}} & 0 & 0 & 0 \\ 0 & \frac{s_{L}}{s_{L}+d_{L}} & 0 & 0 \\ 0 & 0 & \frac{s_{P}}{s_{P}+d_{P}} & 0 \end{pmatrix} . $$
The characteristic polynomial is given by \(X^{4}-r\) and \(\rho (P)= \sqrt[4]{r}\). From Varga’s theorem, \(D\mathcal{F}_{1}(X_{0}^{*})\) is asymptotically stable if and only if \(r<1\). □

Theorem 2.3

The equilibrium
$$ X^{*}= \biggl( 1-\frac{1}{r} \biggr) \biggl( \frac{K_{E}}{\chi_{E}}, \frac{K _{L}}{\chi_{L}}, \frac{K_{P}}{\chi_{P}}, \frac{s_{P}}{f_{m}} \frac{K _{P}}{\chi_{P}} \biggr) $$
is locally asymptotically stable if and only if \(r>1\).

Proof

For the proof we evaluate the Jacobian matrix of \(\mathcal{F}_{1}\) at the endemic equilibrium point \(X^{*}\). It can be written as \(D\mathcal{F}_{1}(X^{*})=M+N\) where
$$ M= \begin{pmatrix} 0 & 0 & 0 & B \\ C & 0 & 0 & 0 \\ 0 & E & 0 & 0 \\ 0 & 0 & G & 0 \end{pmatrix} $$
with
$$\begin{aligned} &B=b \biggl( 1-\frac{1}{\chi_{E}} \biggl( 1-\frac{1}{r} \biggr) \biggr) , \qquad C=s_{E} \biggl( 1-\frac{1}{\chi_{L}} \biggl( 1- \frac{1}{r} \biggr) \biggr) , \\ &E= s_{L} \biggl( 1-\frac{1}{ \chi_{P}} \biggl( 1-\frac{1}{r} \biggr) \biggr) , \qquad G=s_{P}. \end{aligned}$$
Also we have
$$ N= \begin{pmatrix} -A & 0 & 0 & 0 \\ 0 & -D & 0 & 0 \\ 0 & 0 & -U & 0 \\ 0 & 0 & 0 & -H \end{pmatrix}, $$
where A, D, U and H are, respectively, given by
$$\begin{aligned} &A=\frac{bs_{P} K_{P}}{f_{m} \chi_{P} K_{E}} \biggl( 1-\frac{1}{r} \biggr) +(s_{E}+d_{E}), \qquad D=\frac{s_{E} K_{E}}{K_{L} \chi_{P}} \biggl( 1-\frac{1}{r} \biggr) +(s_{L}+d_{L}), \\ &U=\frac{S_{L} K_{L}}{K_{P} \chi_{L}} \biggl( 1-\frac{1}{r} \biggr) +(s _{P}+d_{P}), \qquad H=f_{m}. \end{aligned}$$
The matrix M is positive if and only if \(r>1\). On the other hand, the diagonal matrix N is invertible and its eigenvalues are all negative if and only if \(r>1\). Thus,
$$ -MN^{-1}= \begin{pmatrix} 0 & 0 & 0 & \alpha_{1} \\ \alpha_{2} & 0 & 0 & 0 \\ 0 & \alpha_{3} & 0 & 0 \\ 0 & 0 & \alpha_{4} & 0 \end{pmatrix} $$
with
$$\begin{aligned} &\alpha_{1}=\frac{b ( 1-\frac{1}{\chi_{E}} ( 1-\frac{1}{r} ) ) }{f_{m}}, \qquad \alpha_{2}= \frac{s_{E} ( 1-\frac{1}{\chi_{L}} ( 1- \frac{1}{r} ) ) }{\frac{bs_{P} K_{P}}{f_{m} \chi_{P} K _{E}} ( 1-\frac{1}{r} ) +(s_{E}+d_{E})}, \\ &\alpha_{3}=\frac{s_{L} ( 1-\frac{1}{ \chi_{P}} ( 1- \frac{1}{r} ) ) }{\frac{s_{E} K_{E}}{K_{L} \chi_{P}} ( 1-\frac{1}{r} ) +(s_{L}+d_{L})}, \qquad \alpha_{4}= \frac{s_{P}}{\frac{s_{L} K_{L}}{K_{P} \chi_{L}} ( 1- \frac{1}{r} ) +(s_{P}+d_{P})}. \end{aligned}$$

The characteristic polynomial of the matrix \(-MN^{-1}\) is \(X^{4}- \alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\), and the spectral radius is given by \(\rho (-MN^{-1})=\sqrt[4]{\alpha _{1}\alpha_{2}\alpha_{3}\alpha_{4}}\).

Since \(r>1\), the quantity \(\alpha_{1} \alpha_{2} \alpha_{3} \alpha _{4}\) is less than unity. Consequently, \(\rho (-MN^{-1})<1\), and then the endemic equilibrium is locally asymptotically stable. □

Theorem 2.4

The trivial equilibrium is globally asymptotically stable if and only if \(r\leq 1\).

Proof

Consider the following Lyapunov function [1, 18]:
$$ \vartheta (t)=s_{L} s_{P} \bigl( s_{E} E(t)+(s_{E}+d_{E})L(t) \bigr) +(s _{E}+d_{E}) (s_{L}+d_{L}) \bigl( s_{P} P(t)+(s_{P}+d_{P})A(t) \bigr). $$
By calculating the derivative of ϑ, we have
$$\begin{aligned} \vartheta '(t) =& s_{L} s_{P} \bigl( s_{E} E'(t)+(s_{E}+d_{E})L'(t) \bigr) +(s_{E}+d_{E}) (s_{L}+d_{L}) \bigl( s_{P} P'(t)+(s_{P}+d_{P})A'(t) \bigr) \\ =& A(t) \bigl( bs_{E}s_{L}s_{P}-f_{m}(s_{E}+d_{E}) (s_{L}+d_{L}) (s_{P}+d _{P}) \bigr) \\ &{}- \biggl( s_{E}s_{L}s_{P} \frac{E(t)L(t)}{K_{L}}+s_{L}s_{P}(s_{E}+d _{E}) (s_{L}+d_{L})\frac{L(t)P(t)}{K_{P}}+bs_{E}s_{L}s_{P} \frac{A(t)E(t)}{K _{E}} \biggr) \\ =& \frac{A(t)}{f_{m}(s_{E}+d_{E})(s_{L}+d_{L})(s_{P}+d_{P})} ( r-1 ) \\ &{}- \biggl( s_{E}s_{L}s_{P} \frac{E(t)L(t)}{K_{L}}+s_{L}s_{P}(s_{E}+d _{E}) (s_{L}+d_{L})\frac{L(t)P(t)}{K_{P}}+bs_{E}s_{L}s_{P} \frac{A(t)E(t)}{K _{E}} \biggr) . \end{aligned}$$
Since \(r\leq 1\), it then follows that \(\vartheta '(t)\leq 0\). Thanks to LaSalle’s invariance principle, the trivial equilibrium, \(X_{0}^{*}=(0,0,0,0)\) is globally asymptotically stable. □

Theorem 2.5

The non-trivial equilibrium is globally asymptotically stable if and only if \(r>1\).

Proof

Suppose that the rate of growth is greater than 1 and \(X^{*}=(E^{*},L ^{*},P^{*},A^{*})=(x^{*},y^{*},z^{*},w^{*})\).

Consider the Lyapunov function \(V_{1}:\mathbb{R}^{4}\rightarrow \mathbb{R}\) defined by
$$ V_{1}(x,y,z,w)=\frac{1}{2} \bigl( a_{1} \bigl(x-x^{*}\bigr)^{2}+a_{2} \bigl(y-y^{*} \bigr)^{2}+a _{3} \bigl(z-z^{*}\bigr)^{2}+a_{4} \bigl(w-w^{*}\bigr)^{2} \bigr) $$
with \(a=(a_{1},a_{2},a_{3},a_{4})^{t} \in \mathbb{R}_{+}^{*4}\) a positive constant vector.

Since \(r>1\), \(x^{*}\), \(y^{*}\), \(z^{*}\) and \(w^{*}\) are also positive.

It is clear that \(V_{1}(X^{*})=0\) and \(\forall (x,y,z,w)\in \mathbb{R}_{+}^{4} \setminus \lbrace X^{*}\rbrace \), \(V_{1}(x,y,z,w)>0\). Thus, the function \(V_{1}\) is well defined and the orbital derivative of \(V_{1}\) along the solution of the system (2) is
$$\begin{aligned} V_{1}'(x,y,z,w) =& a_{1} \bigl(x-x^{*} \bigr) \biggl( b t \biggl( 1- \frac{x}{K_{E}} \biggr) -(s_{E}+d_{E})x \biggr) \\ &{}+ a_{2} \bigl(y-y^{*}\bigr) \biggl( s_{E} x \biggl( 1-\frac{y}{K_{L}} \biggr) -(s _{L}+d_{L})y \biggr) \\ &{}+ a_{3} \bigl(z-z^{*}\bigr) \biggl( s_{L} y \biggl( 1-\frac{z}{K_{P}} \biggr) -(s _{P}+d_{P})z \biggr) \\ &{}+ a_{4} \bigl(w-w^{*}\bigr) (s_{P} z-f_{m} w). \end{aligned}$$
Let us adopt the following notations:
$$ \tilde{x}=x-x^{*}, \qquad\tilde{y}=y-y^{*}, \qquad \tilde{z}=z-z^{*}, \qquad\tilde{w}=w-w^{*},\qquad \tilde{X}=( \tilde{x},\tilde{y}, \tilde{z},\tilde{w})^{T} . $$
Then
$$\begin{aligned} \dot{V}_{1}(x,y,z,w) =& \tilde{X}^{T} \begin{pmatrix} -a_{1} (s_{E}+d_{E}) & 0 & 0 & a_{1} b ( 1-\frac{x^{*}}{K_{E}} ) \\ a_{2} s_{E} ( 1-\frac{y^{*}}{K_{L}} ) & -a_{2} (s_{L}+d_{L}) & 0 & 0 \\ 0 & a_{3} s_{L} ( 1-\frac{z^{*}}{K_{P}} ) & -a_{3} (s_{P}+d _{P}) & 0 \\ 0 & 0 & a_{4} s_{P} & -a_{4} f_{m} \end{pmatrix} \tilde{X} \\ &{}- \biggl( a_{1} \frac{b}{K_{E}} x^{*2} w+a_{2}\frac{s_{E}}{K_{L}} y ^{*2} x+ a_{3} \frac{s_{L}}{K_{P}} z^{*2} y \biggr) . \end{aligned}$$
Consider \(A_{1}=-D+R_{1}\) with
$$\begin{aligned} &D= \begin{pmatrix} a_{1} (s_{E}+d_{E}) & 0 & 0 & 0 \\ 0 & a_{2} (s_{L}+d_{L}) & 0 & 0 \\ 0 & 0 & a_{3} (s_{P}+d_{P}) & 0 \\ 0 & 0 & 0 & a_{4} f_{m} \end{pmatrix} ; \\ &R_{1}= \begin{pmatrix} 0 & 0 & 0 & a_{1} b ( 1-\frac{x^{*}}{K_{E}} ) \\ a_{2} s_{E} ( 1-\frac{y^{*}}{K_{L}} ) & 0 & 0 & 0 \\ 0 & a_{3} s_{L} ( 1-\frac{z^{*}}{K_{P}} ) & 0 & 0 \\ 0 & 0 & a_{4} s_{P} & 0 \end{pmatrix}. \end{aligned}$$
Considering the scalar product of \(\mathbb{R}^{4}\), the orbital derivative of the function \(V_{1}\) can be rewritten in the following form:
$$ V_{1}'(x,y,z,w)= \langle A_{1}\tilde{X}, \tilde{X} \rangle - \biggl( a_{1} \frac{b}{K_{E}} \tilde{x}^{2} w+a_{2}\frac{s_{E}}{K _{L}} \tilde{y}^{2} x+ a_{3}\frac{s_{L}}{K_{P}} \tilde{z}^{2} y \biggr) . $$
Let us introduce the following symmetric matrix:
$$\begin{aligned} S_{1} =& -D+\frac{1}{2} \bigl( R_{1}^{T}+R_{1} \bigr) \\ =& \begin{pmatrix} -a_{1}(s_{E}+d_{E}) & \frac{a_{2} s_{E}}{2} ( 1-\frac{y^{*}}{K _{L}} ) & 0 & \frac{a_{1} b}{2} ( 1-\frac{x^{*}}{K_{E}} ) \\ \frac{a_{2} s_{E}}{2} ( 1-\frac{y^{*}}{K_{L}} ) & -a_{2} (s _{L}+d_{L}) & \frac{a_{3} s_{L}}{2} ( 1-\frac{z^{*}}{K_{P}} ) & 0 \\ 0 & \frac{a_{3} s_{L}}{2} ( 1-\frac{z^{*}}{K_{P}} ) & -a _{3} (s_{P}+d_{P}) & \frac{a_{4} s_{P}}{2} \\ \frac{a_{1} b}{2} ( 1-\frac{x^{*}}{K_{E}} ) & 0 & \frac{a _{4} s_{P}}{2} & -a_{4} f_{m} \end{pmatrix} . \end{aligned}$$
Using the properties of equilibrium states, we get
$$\begin{aligned} &\biggl( 1-\frac{x^{*}}{K_{E}} \biggr) =\frac{s_{E}+d_{E}}{b}\times \frac{x ^{*}}{w^{*}}; \\ & \biggl( 1-\frac{y^{*}}{K_{L}} \biggr) =\frac{s_{L}+d_{L}}{s_{E}} \times \frac{y^{*}}{x^{*}}; \\ & \biggl( 1-\frac{z^{*}}{K_{P}} \biggr) = \frac{s_{P}+d_{P}}{s_{L}} \times \frac{z^{*}}{y^{*}}. \end{aligned}$$
Further, the matrix \(S_{1}\) becomes
$$ S_{1}= \begin{pmatrix} -a_{1}(s_{E}+d_{E}) & \frac{a_{2}}{2} ( s_{L}+d_{L} ) \frac{y ^{*}}{x^{*}} & 0 & \frac{a_{1}}{2} ( s_{E}+d_{E} ) \frac{x ^{*}}{w^{*}} \\ \frac{a_{2}}{2} ( s_{L}+d_{L} ) \frac{y^{*}}{x^{*}} & -a_{2} (s_{L}+d_{L}) & \frac{a_{3}}{2} ( s_{P}+d_{P} ) \frac{z^{*}}{y ^{*}} & 0 \\ 0 & \frac{a_{3}}{2} ( s_{P}+d_{P} ) \frac{z^{*}}{y^{*}} & -a _{3} (s_{P}+d_{P}) & \frac{a_{4} s_{P}}{2} \\ \frac{a_{1}}{2} ( s_{E}+d_{E} ) \frac{x^{*}}{w^{*}} & 0 & \frac{a _{4} s_{P}}{2} & -a_{4} f_{m} \end{pmatrix} , $$
and we obtain \(\langle A_{1}\tilde{X},\tilde{X} \rangle = \langle S_{1}\tilde{X},\tilde{X} \rangle \)
The characteristic polynomial is \(P=X^{4}+\gamma_{1} X^{3}+\gamma _{2} X^{2}+\gamma_{3} X+\gamma_{4} \) where
$$\begin{aligned} &\gamma_{1} = a_{1}(s_{E}+d_{E})+a_{2}(s_{L}+d_{L})+a_{3}(s_{P}+d _{P})+a_{4}f_{m}; \\ & \begin{aligned} \gamma_{2} ={}& a_{1} a_{2} (s_{E}+d_{E}) (s_{L}+d_{L})+a_{3} a_{4} (s _{P}+d_{P})f_{m}+ \bigl( a_{1} (s_{E}+d_{E})+a_{2} (s_{L}+d_{L}) \bigr)\\ &{} + \bigl( a_{3} (s_{P}+d_{P})+a_{4} f_{m} \bigr) ; \end{aligned} \\ & \begin{aligned} \gamma_{3} ={}& \bigl( a_{4} a_{3}(s_{P}+d_{P})f_{m} \bigl( a_{1}(s_{E}+d _{E})+a_{2}(s_{L}+d_{L}) \bigr) +a_{1} a_{2}(s_{E}+d_{E}) (s_{L}+d_{L})\\ &{}\times \bigl( a_{3}(s_{P}+d_{P})+a_{4}f_{m} \bigr) \bigr) ; \end{aligned} \\ & \begin{aligned} \gamma_{4} ={}&\frac{1}{4} \biggl( a_{1} a_{3} (s_{E}+d_{E}) \biggl( \frac{x ^{*}}{w^{*}} \biggr) (s_{P}+d_{P}) \biggl( \frac{z^{*}}{y^{*}} \biggr) -a _{2} a_{4} (s_{L}+d_{L})f_{m} \biggl( \frac{y^{*}}{x^{*}} \biggr) \biggr) ^{2} \\ &{}+a_{1} a_{2} a_{3} a_{4} f_{m}(s_{E}+d_{E}) (s_{L}+d_{L}) (s_{P}+d _{P}). \end{aligned} \end{aligned}$$
The global stability of non-trivial equilibrium can be investigated by applying the Routh–Hurwitz criterion on the characteristic polynomial.
The relevant Routh–Hurwitz determinants are
$$ \textstyle\begin{cases} \Delta_{1} = \gamma_{1}>0, \\ \Delta_{2} = \gamma_{1} \gamma_{2}-\gamma_{3}>0, \\ \Delta_{3} = \gamma_{3} \Delta_{2}-\gamma_{1}^{2} \gamma_{4}>0, \\ \Delta_{4} = \gamma_{4} \Delta_{3}>0. \end{cases} $$
It is clear that \(\Delta_{1}=\gamma_{1}>0\) and
$$\begin{aligned} \Delta_{2}={}&\gamma_{1}^{2}+ 2a_{1}^{2} a_{2}(s_{E}+d_{E})^{2}(s_{L}+d _{L})+a_{1} a_{2}^{2}(s_{E}+d_{E})^{2}(s_{L}+d_{L})^{2}\\ &{}+a_{3}^{2} a _{4} (s_{P}+d_{P})^{2} f_{m} +a_{3} a_{4}^{2} f_{m}^{2}>0. \end{aligned}$$
Let us assume that
$$\begin{aligned} &\alpha =2a_{1}^{2} a_{2}(s_{E}+d_{E})^{2}(s_{L}+d_{L})+a_{1} a_{2} ^{2}(s_{E}+d_{E})^{2}(s_{L}+d_{L})^{2}+a_{3}^{2} a_{4} (s_{P}+d_{P})^{2} f_{m} +a_{3} a_{4}^{2} f_{m}^{2}, \\ &\Delta_{3} = \gamma_{3} \Delta_{2}- \gamma_{1}^{2} \gamma_{4} =\gamma _{1}^{2} (\gamma_{3}-\gamma_{4})+ \alpha \gamma_{3}>0. \end{aligned}$$
Finally, since \(\gamma_{4}>0\), \(\Delta_{4} = \gamma_{4} \Delta_{3}>0\). □

Remark 2.1

The above results show that a vector control strategy that brings and maintains the threshold quantity r, to a value less than unity will lead to the effective control of mosquitoes population growth for the community. In other words, the requirement \(r\leq 1\) is necessary and sufficient for the effective control of the mosquitoes’ population growth [10, 23].

3 Malaria transmission dynamics model

3.1 Malaria transmission mechanism

Malaria is transmitted to humans by the female of a mosquito of the genus anopheles [24, 28]. There are four species of parasites, which are Plasmodium falciparum, Plasmodium vivax, Plasmodium malariae, and more recently Plasmodium knowlesi. However, the most pathogenic is Plasmodium malariae, but it remains a rare case. Plasmodium falciparum is a frequent case. It causes the most serious illness and is the most widespread in the tropics [27]. Mosquito-to-human malaria transmission occurs when sporozoites from the salivary gland of the mosquito are injected into the skin during blood-feeding. Parasites then pass to the liver where they replicate, each sporozoite yielding many thousands of merozoites which go on to cause patent infection. The biology of the four species of Plasmodium is generally similar and consists of two distinct phases: a sexual stage at the mosquito host and an asexual stage at the human host. The asexual phase consists of at least three forms: sporozoites, merozoites, and trophozoites. During the asexual stage, some of the parasites become gametocytes and then when a mosquito bites an infected human, it ingests the gametocytes. Hence, the parasite continues its development and invades the salivary glands of the mosquito ending the cycle [13, 27].

3.2 Model formulation of malaria transmission

In this section, we give a brief description of the different stages of our model of malaria parasite transmission. In order to derive our model, we divide the human population into two major types. The first type, called non-immune, is divided into three sub-classes, and the second type, called semi-immune, is divided in four classes. Tables 16 give a description of all these classes.
Table 1

Parameters for human hosts

Notation

Description

\(S_{e}\)

class of non-immune susceptible

\(S_{a}\)

class of semi-immune susceptible

\(E_{e}\)

class of non-immune latent

\(E_{a}\)

class of semi-immune latent

\(I_{e}\)

class of non-immune infectious

\(I_{a}\)

class of semi-immune infectious

\(R_{a}\)

class of immune semi-immune

Table 2

Parameters for vectors hosts

Notation

Description

\(S_{m}\)

class of susceptible mosquitoes

\(E_{m}\)

class of latent mosquitoes

\(I_{m}\)

class of infectious mosquitoes

Table 3

Parameters common to non-immune and semi-immune

Parameters

Biological description

\(\Lambda _{h}\)

constant recruitment rate of human (it also includes births)

p

probability for a new recruit to be non-immune

1 − p

probability for a new recruit to be semi-immune

Table 4

Parameters for semi-immune hosts

Parameters

Biological description

\(\nu _{a}\)

rate of passage from latent semi-immune to infectious

\(\alpha _{a}\)

rate of passage infectious semi-immune to immune

\(\beta _{a}\)

rate of loss of immune immunity

Table 5

Contact parameters between non-immune, semi-immune and mosquito

Parameters

Biological description

\(n_{a}\)

average number of bites per mosquito per unit of time

\(c_{me}\)

probability that an infectious mosquito bite on a susceptible non-immune transfers the infection to the non-immune

\(c_{ma}\)

probability that an infectious mosquito bite on a susceptible semi-immune transfers the infection to the semi-immune

\(c_{em}\)

probability that a bite from a susceptible mosquito on an infectious non-immune transfers the infection to the mosquito

\(c_{am}\)

probability that a bite from a susceptible mosquito on an infectious semi-immune transfers the infection to the mosquito

\(\tilde{c}_{am}\)

probability that a bite from a susceptible mosquito on an infectious immune transfers the infection to the mosquito

Table 6

Parameters for the non-immune hosts

Parameters

Biological description

\(\nu _{e}\)

rate of passage from latent non-immune to infectious

\(\alpha _{e}\)

rate of passage infectious non-immune to immune

\(\gamma _{e}\)

mortality rate due to malaria on non-immune

\(\gamma _{a}\)

mortality rate due to malaria on semi-immune

The total human population at each instant t is given by
$$\begin{aligned} N_{h}(t)=S_{e}(t)+E_{e}(t)+I_{e}(t)+S_{a}(t)+E_{a}(t)+I_{a}(t)+R_{a}(t) . \end{aligned}$$
(5)
The total vector population at each instant t is given by
$$\begin{aligned} A(t)=S_{m}(t)+E_{m}(t)+I_{m}(t). \end{aligned}$$
(6)
We make the following useful assumptions.
(H3):: 

We assume that an immigrant is either non-immune or semi-immune.

(H4):: 

We assume that the only mode of transmission is the mosquito bites.

(H5):: 

It is assumed that an individual who arrives newly in our study area has a probability p of being non-immune and a probability \(1-p\) of being semi-immune.

(H6):: 

We assume that all recruits are susceptible.

(H7):: 

We assume that the natural mortality rate \(f_{h}\) (resp. \(f_{m}\)) is constant.

(H8):: 

The probabilities \(c_{me}\), \(c_{ma}\), \(c_{em}\), \(c_{am}\), \(\tilde{c}_{am}\), the parameters \(\nu_{e}\), \(\nu_{a}\), \(\nu_{m}\), \(\alpha_{e}\), \(\alpha_{a}\), \(\beta_{a}\) and \(n_{a}\) are and the induced mortality rates \(\gamma_{e}\) and \(\gamma_{a}\) are non-negative.

The forces of infections from mosquitoes to non-immune and semi-immune are, respectively, defined by
$$\begin{aligned} &k_{e} = c_{me}n_{a} \frac{I_{m}}{N_{h}}, \\ &k_{a} = c_{ma}n_{a} \frac{I_{m}}{N_{h}}, \end{aligned}$$
and the force of infection from human to mosquitoes is
$$\begin{aligned} k_{m} =& c_{am}n_{a} \frac{I_{a}}{N_{h}}+ c_{em}n_{a} \frac{I_{e}}{N_{h}}+\tilde{c}_{am}n_{a} \frac{R_{a}}{N_{h}}. \end{aligned}$$
(H9):: 

Let us assume that \(0<\nu_{e}\leq k_{e}\), \(0<\nu_{a}\leq k_{a}\) and \(0<\nu_{m}\leq k_{m}\).

Taking into account all these above hypotheses, the inter-host dynamics has been illustrated as in [13].
Thus, the overall dynamics of the spread of the disease is reflected in a diagram. From Fig. 2, by making the balance in each compartment, we obtain the following system of ordinary differential equations:
$$\begin{aligned} \textstyle\begin{cases} \textstyle\begin{cases} E'(t)=bA(t) ( 1-\frac{E(t)}{K_{E}} ) -(s_{E}+d_{E})E(t), \\ L'(t) =s_{E}E(t) ( 1-\frac{L(t)}{K_{L}} ) -(s_{L}+d_{L})L(t), &\qquad \mbox{(S1)} \\ P'(t) =s_{L}L(t) ( 1-\frac{P(t)}{K_{P}} ) -(s_{P}+d_{P})P(t), \end{cases}\displaystyle \\ \textstyle\begin{cases} \textstyle\begin{cases} S_{e}'(t)=p\Lambda_{h}-(f_{h}+k_{e})S_{e}(t), \\ E_{e}'(t)=k_{e} S_{e}(t)-(f_{h}+\nu_{e})E_{e}(t), \\ I_{e}'(t)=\nu_{e} E_{e}(t)-(f_{h}+\gamma_{e}+\alpha_{e})I_{e}(t), \end{cases}\displaystyle \\ \textstyle\begin{cases} S_{a}'(t) =(1-p)\Lambda_{h}+\beta_{a} R_{a}(t) - (f_{h}+k_{a})S_{a}(t), \\ E_{a}'(t) = k_{a} S_{a}(t)- (f_{h}+\nu_{a})E_{a}(t), \\ I_{a}'(t) = \nu_{a} E_{a}(t)-(f_{h}+\gamma_{a}+\alpha_{a})I_{a}(t), &\qquad \mbox{(S2)} \\ R_{a}'(t) = \alpha_{e} I_{e}(t) +\alpha_{a} I_{a}(t)-(f_{h} +\beta _{a})R_{a}(t), \end{cases}\displaystyle \\ \textstyle\begin{cases} S_{m}'(t) = s_{P}P(t) - (f_{m}+k_{m})S_{m}(t), \\ E_{m}'(t) = k_{m} S_{m}(t) - (f_{m} +\nu_{m})E_{m}(t), \\ I_{m}'(t) = \nu_{m} E_{m}(t) - f_{m} I_{m}(t). \end{cases}\displaystyle \end{cases}\displaystyle \end{cases}\displaystyle \end{aligned}$$
(7)
In system (7), the subsystem (S1) represents the vectors population growth global dynamics and subsystem (S2) represents the virus transmission. Indeed, (S1) is coupled with (S2) through the variable \(P(t)\). However, to analyze the system (7) in a decoupled form, we can use the principle of limiting system [16, 36].
Figure 2
Figure 2

A schematic of the mathematical model for malaria transmission involving the human host susceptibility, exposedness and infectivity with variable non-immune, semi-immune and mosquitoes population. The dashed arrows indicate the direction of the infection and the solid arrows represent the transition from one class to another

From Eq. (6), we obtain
$$\begin{aligned} S_{m}(t)=A(t)-E_{m}(t)-I_{m}(t), \end{aligned}$$
(8)
and then, replacing the expression of \(S_{m}(t)\) given by Eq. (8) into the system (7), it is easy to see that the system (7) is equivalent to the following one:
$$\begin{aligned} \textstyle\begin{cases} \textstyle\begin{cases} E'(t)=bA(t) ( 1-\frac{E(t)}{K_{E}} ) -(s_{E}+d_{E})E(t), \\ L'(t) =s_{E}E(t) ( 1-\frac{L(t)}{K_{L}} ) -(s_{L}+d_{L})L(t), &\qquad \mbox{(S1)} \\ P'(t) =s_{L}L(t) ( 1-\frac{P(t)}{K_{P}} ) -(s_{P}+d_{P})P(t), \\ A'(t)=s_{P} P(t)-f_{m} A(t), \end{cases}\displaystyle \\ \textstyle\begin{cases} \textstyle\begin{cases} S_{e}'(t)=p\Lambda_{h}-(f_{h}+k_{e})S_{e}(t), \\ E_{e}'(t)=k_{e} S_{e}(t)-(f_{h}+\nu_{e})E_{e}(t), \\ I_{e}'(t)=\nu_{e} E_{e}(t)-(f_{h}+\gamma_{e}+\alpha_{e})I_{e}(t), \end{cases}\displaystyle \\ \textstyle\begin{cases} S_{a}'(t) =(1-p)\Lambda_{h}+\beta_{a} R_{a}(t) - (f_{h}+k_{a})S_{a}(t), \\ E_{a}'(t) = k_{a} S_{a}(t)- (f_{h}+\nu_{a})E_{a}(t), \\ I_{a}'(t) = \nu_{a} E_{a}(t)-(f_{h}+\gamma_{a}+\alpha_{a})I_{a}(t), &\qquad \mbox{(S2)} \\ R_{a}'(t) = \alpha_{e} I_{e}(t) +\alpha_{a} I_{a}(t)-(f_{h} +\beta _{a})R_{a}(t), \end{cases}\displaystyle \\ \textstyle\begin{cases} E_{m}'(t) = k_{m} (A(t)-E_{m}(t)-I_{m}(t)) - (f_{m} +\nu_{m})E_{m}(t), \\ I_{m}'(t) = \nu_{m} E_{m}(t) - f_{m} I_{m}(t). \end{cases}\displaystyle \end{cases}\displaystyle \end{cases}\displaystyle \end{aligned}$$
(9)
At any time \(t\geq 0\), the total size of the humans population and adult mosquitoes population are, respectively, given by the following equations:
$$\begin{aligned} &N_{h}'(t) =\Lambda_{h}-f_{h}N_{h}- \gamma_{e}I_{e}-\gamma_{a}I_{a} , \end{aligned}$$
(10)
$$\begin{aligned} &A'(t) = s_{P}P-f_{m}A(t) . \end{aligned}$$
(11)

Remark 3.1

The previous results indicate that the mosquito population will die out if the vector threshold r is less than or equal to unity, while the mosquito population will eventually stabilize at a positive equilibrium \((E^{*},L^{*},P^{*},A^{*})\) if the vector threshold r is greater than unity.

From the system (9), we obtain the following limit system:
$$\begin{aligned} \textstyle\begin{cases} \textstyle\begin{cases} \textstyle\begin{cases} S_{e}'(t)=p\Lambda_{h}-(f_{h}+k_{e})S_{e}(t), \\ E_{e}'(t)=k_{e} S_{e}(t)-(f_{h}+\nu_{e})E_{e}(t), \\ I_{e}'(t)=\nu_{e} E_{e}(t)-(f_{h}+\gamma_{e}+\alpha_{e})I_{e}(t), \end{cases}\displaystyle \\ \textstyle\begin{cases} S_{a}'(t) =(1-p)\Lambda_{h}+\beta_{a} R_{a}(t) - (f_{h}+k_{a})S_{a}(t), \\ E_{a}'(t) = k_{a} S_{a}(t)- (f_{h}+\nu_{a})E_{a}(t), \\ I_{a}(t) = \nu_{a} E_{a}(t)-(f_{h}+\gamma_{a}+\alpha_{a})I_{a}(t), \\ R_{a}'(t) = \alpha_{e} I_{e}(t) +\alpha_{a} I_{a}(t)-(f_{h} +\beta _{a})R_{a}(t), \end{cases}\displaystyle \\ \textstyle\begin{cases} E_{m}'(t) = k_{m} (A^{*}-E_{m}(t)-I_{m}(t)) - (f_{m} +\nu_{m})E_{m}(t), \\ I_{m}'(t) = \nu_{m} E_{m}(t) - f_{m} I_{m}(t). \end{cases}\displaystyle \end{cases}\displaystyle \end{cases}\displaystyle \end{aligned}$$
(12)

3.3 Mathematical analysis of malaria transmission model

In this part of the paper, we focus on the study of the system (12) under the influence of the mosquito growth rate [11, 13, 15, 23].

Note that the system (12) can be represented as follows:
$$ X'(t)=\mathcal{F}_{2}\bigl(X(t)\bigr) $$
where
$$ X(t)=(S_{e},E_{e},I_{e},S_{a},E_{a},I_{a},R_{a},E_{m},I_{m})^{T} $$
and
$$ \mathcal{F}_{2}(X)= \begin{pmatrix} p\Lambda_{h}- (f_{h}+k_{e})S_{e} \\ k_{e} S_{e}-(f_{h}+\nu_{e})E_{e} \\ \nu_{e} E_{e}- (f_{h}+\gamma_{e} +\alpha_{e})I_{e} \\ (1-p)\Lambda_{h}+\beta_{a} R_{a} - (f_{h}+k_{a})S_{a} \\ k_{a} S_{a}- (f_{h}+\nu_{a})E_{a} \\ \nu_{a} E_{a}-(f_{h}+\gamma_{a}+\alpha_{a})I_{a} \\ \alpha_{e} I_{e} +\alpha_{a} I_{a}-(f_{h} +\beta_{a})R_{a} \\ k_{m} (A^{*}-E_{m}-I_{m}) - (f_{m} +\nu_{m})E_{m} \\ \nu_{m} E_{m} - f_{m} I_{m} \end{pmatrix} . $$

3.3.1 Existence and positivity of solutions

Lemma 3.1

For any initial conditions, the system (12) has a unique positive solution for all \(t\geq 0\). Further, the domain \(\Delta '= \Omega_{h} \times \Omega_{m} \subset \mathbb{R}_{+}^{9}\) where
$$ \Omega_{h}= \biggl\lbrace (S_{e},E_{e},I_{e},S_{a},E_{a},I_{a},R_{a}) | 0\leq N_{h} \leq \frac{\Lambda_{h}}{f_{h}} \biggr\rbrace $$
and
$$ \Omega_{m}= \biggl\lbrace (E_{m},I_{m})| 0\leq E_{m}+I_{m} \leq \frac{s _{P}}{f_{m}}K_{P} \biggr\rbrace $$
is positively invariant and attracts all the positive orbits of \(\mathbb{R}_{+}\).

Proof

Since \(\mathcal{F}_{2}\) is \(\mathcal{C}^{1}\), it is locally Lipschitzian on \(\mathbb{R}^{9}\), we deduce the existence and uniqueness of the solution to the Cauchy problem associated with the subsystem relative to the initial condition \((t_{0},X_{0})\in \mathbb{R}\times \mathbb{R}^{9}\). Since \(\mathcal{F}_{2}\) is \(\mathcal{C}^{\infty }\), we deduce that the solution is also \(\mathcal{C}^{\infty }\).

Now, assuming that there is no disease induced death rate, (10) becomes
$$\begin{aligned} N_{h}'(t)=\Lambda_{h}-f_{h}N_{h}. \end{aligned}$$
(13)
Let us assume that \(N_{h}'(t)\leq 0\) and \(N_{m}'(t)\leq 0\).

It follows that \(N_{h}\leq \frac{\Lambda_{h}}{f_{h}}\), \(A \leq \frac{s _{P}}{f_{m}}K_{P}\).

Then, as in [6, 19], the following inequalities hold:
$$ 0 \leq N_{h}\leq \frac{\Lambda_{h}}{f_{h}}, \qquad 0 \leq A \leq \frac{s_{P}}{f_{m}}K_{P}. $$
Therefore, Eqs. (11) and (13), respectively, become
$$ N_{h}'(t)\leq \Lambda_{h}-f_{h}N_{h} $$
and
$$ A'(t)\leq \Lambda_{m}-f_{m}A(t). $$
Using the variation of constant method, between t and \(t_{0}\), we have the following solutions:
$$\begin{aligned} &N_{h}(t)=\frac{\Lambda_{h}}{f_{h}}+ \biggl( N_{h}^{0}- \frac{\Lambda _{h}}{f_{h}} \biggr) e^{-f_{h}(t-t_{0})}, \\ &A(t)=\frac{S_{P} K_{P}}{f_{m}}+ \biggl( A_{0}-\frac{\Lambda_{m}}{f _{m}} \biggr) e^{-f_{m}(t-t_{0})}. \end{aligned}$$
From the theorem of comparison, it follows that
$$ N_{h}(t)\leq \frac{\Lambda_{h}}{f_{h}}+ \biggl( N_{h}^{0}- \frac{ \Lambda_{h}}{f_{h}} \biggr) e^{-f_{h}(t-t_{0})} $$
and
$$ A(t)\leq \frac{S_{P} K_{P}}{f_{m}}+ \biggl( A_{0}-\frac{\Lambda_{m}}{f _{m}} \biggr) e^{-f_{m}(t-t_{0})}. $$
So, the total size of the humans population \(N_{h}(t) \rightarrow \frac{ \Lambda_{h}}{f_{h}}\) as \(t\rightarrow \infty \). Similarly, the total size of the mosquitoes population \(A \rightarrow \frac{S_{P} K_{P}}{f _{m}}\) as \(t\rightarrow \infty \).

It implies that the set \(\Delta '\) is bounded and we deduce the global existence of the solutions in \([0,+\infty [\).

However, assuming that \(N_{h}(t)> \frac{\Lambda_{h}}{f_{h}}\) (respectively, \(A(t)>\frac{s_{P}}{f_{m}}K_{P}\)), we obtain \(N_{h}'(t)<\Lambda_{h}-f_{h}\times \frac{\Lambda_{h}}{f_{h}}\), namely, \(N_{h}'(t)<0\).

In this case, the two hosts size would be decreasing. Since the domain \(\Delta '\) is compact, all the solutions remain there. □

3.3.2 Basic reproduction number

By linearizing the system in the neighborhood of the trivial equilibrium point \(DFE_{0}\), we obtain the following linear differential system: \(X'(t)=BX(t) \) where B denotes the Jacobian matrix of the function \(\mathcal{F}_{1}\) at the equilibrium point \(DFE_{0}\), and it is defined as follows:
$$ B= \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix}, $$
where the matrices \(B_{11}\), \(B_{12}\), \(B_{21}\) and \(B_{22}\) are, respectively, given by
$$ B_{11}= \begin{pmatrix} -B^{*} & 0 & 0 & 0 & 0 & 0 & c_{me}n_{a}\frac{S_{e}^{*}}{N_{h}^{*}} \\ \nu_{e} & -C^{*} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -D^{*} & 0 & 0 & 0 & c_{ma}n_{a}\frac{S_{a}^{*}}{N_{h}^{*}} \\ 0 & 0 & \nu_{a} & -E^{*} & 0 & 0 & 0 \\ 0 & \alpha_{e} & 0 & \alpha_{a} & -G^{*} & 0 & 0 \\ 0 & c_{em}n_{a}\frac{A^{*}}{N_{h}^{*}} & 0 & c_{am}n_{a}\frac{A^{*}}{N _{h}^{*}} & \tilde{c}_{am}n_{a}\frac{A^{*}}{N_{h}^{*}} & -H^{*} & 0 \\ 0 & 0 & 0 & 0 & 0 & \nu_{m} & -f_{m} \end{pmatrix} $$
with
$$\begin{aligned} &B^{*}=f_{h}+\nu_{e}, \qquad C^{*}=f_{h}+\gamma_{e}+\alpha_{e}, \qquad D^{*}=f_{h}+\nu_{a}, \\ &E^{*}=f_{h}+\gamma_{a}+\alpha_{a}, \qquad G^{*}=f_{h}+\beta_{a}, \qquad H^{*}=f_{h}+\nu_{m}, \\ &B_{21}= \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & -c_{me}n_{a}\frac{S_{e}^{*}}{N_{h}^{*}} \\ 0 & 0 & 0 & 0 & \beta_{a} & 0 & -c_{ma}n_{a}\frac{S_{a}^{*}}{N_{h} ^{*}} \\ 0 & -c_{em}n_{a}\frac{A^{*}}{N_{h}^{*}} & -\tilde{c}_{am}n_{a}\frac{A ^{*}}{N_{h}^{*}} & -c_{am}n_{a}\frac{A^{*}}{N_{h}^{*}} & 0 & 0 & 0 \end{pmatrix} , \\ &B_{22}= \begin{pmatrix} -f_{h} & 0 & 0 \\ 0 & -f_{h} & 0 \\ 0 & 0 & -f_{m} \end{pmatrix} ; \qquad B_{12}= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} . \end{aligned}$$
The sub-matrix \(B_{11}\) is called transmission matrix and it is Metzler stable. It can be decomposed as \(B_{11}=F+V\) where
$$ F= \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & c_{me}n_{a}\frac{S_{e}^{*}}{N_{h}^{*}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c_{ma}n_{a}\frac{S_{a}^{*}}{N_{h}^{*}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & c_{em}n_{a}\frac{A^{*}}{N_{h}^{*}} & 0 & c_{am}n_{a}\frac{A^{*}}{N _{h}^{*}} & \tilde{c}_{am}n_{a}\frac{A^{*}}{N_{h}^{*}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} $$
and
$$ V= \begin{pmatrix} -B^{*} & 0 & 0 & 0 & 0 & 0 & 0 \\ \nu_{e} & -C^{*} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -D^{*} & 0 & 0 & 0 & 0 \\ 0 & 0 & \nu_{a} & -E^{*} & 0 & 0 & 0 \\ 0 & \alpha_{e} & 0 & \alpha_{a} & -G^{*} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -H^{*} & 0 \\ 0 & 0 & 0 & 0 & 0 & \nu_{m} & -f_{m} \end{pmatrix} . $$
The inverse of the matrix V is
$$ V^{-1}= \begin{pmatrix} -\frac{1}{B^{*}} & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{\nu_{e}}{B^{*} C^{*}} & -\frac{1}{C^{*}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{1}{D^{*}} & 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{\nu_{a}}{D^{*} E^{*}} & -\frac{1}{E^{*}} & 0 & 0 & 0 \\ 0 & 0 & -\frac{\alpha_{a} \nu_{a}}{D^{*} E^{*} G^{*}} & -\frac{ \alpha_{a}}{E^{*} G^{*}} & -\frac{1}{G^{*}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -\frac{1}{H^{*}} & 0 \\ 0 & 0 & 0 & 0 & 0 & -\frac{\nu_{m}}{f_{m} H^{*}} & -\frac{1}{f_{m}} \end{pmatrix} . $$

Definition 3.1

The matrix
$$ -FV^{-1}= \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & K_{17} \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{27} \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{37} \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{47} \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{57} \\ 0 & K_{62} & 0 & K_{64} & K_{65} & 0 & 0 \\ 0 & K_{72} & 0 & K_{74} & K_{75} & 0 & 0 \end{pmatrix} $$
is called the next generation matrix. The non-zero coefficients are given by
$$\begin{aligned} &K_{17}=\frac{1}{B^{*}}\times c_{me}n_{a} \frac{S_{e}^{*}}{N_{h}^{*}}, \qquad K_{27}=\frac{\nu_{e}}{B^{*} C^{*}}\times c_{me}n_{a} \frac{S_{e} ^{*}}{N_{h}^{*}}, \qquad K_{37}= \frac{1}{D^{*}}\times c_{ma}n_{a} \frac{S_{a}^{*}}{N_{h}^{*}}, \\ &K_{47}=\frac{\nu_{a}}{D^{*} E^{*}}\times c_{ma}n_{a} \frac{S_{a} ^{*}}{N_{h}^{*}},\qquad K_{57}= \frac{\alpha_{a} \nu_{a}}{D^{*} E^{*} G^{*}}\times c_{ma}n_{a} \frac{S _{a}^{*}}{N_{h}^{*}}, \qquad K_{62}= \frac{1}{H^{*}}\times c_{em}n_{a} \frac{A^{*}}{N_{h}^{*}}, \\ &K_{64}= \frac{1}{H^{*}}\times c_{am}n_{a} \frac{A^{*}}{N_{h}^{*}}, \qquad K_{65}=\frac{1}{H^{*}}\times \tilde{c}_{am}n_{a} \frac{A^{*}}{N_{h} ^{*}}, \qquad K_{72} =\frac{\nu_{m}}{H^{*}}\times \frac{1}{f_{m}}\times c_{em}n _{a} \frac{A^{*}}{N_{h}^{*}}, \\ &K_{74}=\frac{\nu_{m}}{H^{*}}\times \frac{1}{f_{m}}\times c_{am}n _{a} \frac{A^{*}}{N_{h}^{*}}, \qquad K_{75}= \frac{\nu_{m}}{H^{*}}\times \frac{1}{f_{m}}\times \tilde{c} _{am}n_{a} \frac{A^{*}}{N_{h}^{*}}. \end{aligned}$$

Proposition 3.1

The basic reproduction number is
$$\begin{aligned} \mathcal{R}_{0}=\rho \bigl(-FV^{-1}\bigr)= \sqrt{K_{27}K_{72}+K_{47}K_{74}+K _{57}K_{75}} . \end{aligned}$$
(14)

Proof

According to the mathematical sense, the basic reproduction number is the spectral radius of next generation matrix. For this purpose, the above expression of \(\mathcal{R}_{0}\) is obtained just by using this definition and making some calculations. □

Proposition 3.2

The system (12) has a unique disease-free equilibrium given by
$$ DFE_{0}= \biggl( \frac{p}{f_{h}}\Lambda_{h},0,0, \frac{1-p}{f_{h}} \Lambda_{h},0,0,0,0,0 \biggr) . $$

Proof

When we consider the case where there are any mosquitoes and infected humans, we have \(E_{m}^{*}=I_{m}^{*}=0\) and \(E_{e}^{*}=I_{e}^{*}=E _{a}^{*}=I_{a}^{*}=R_{a}^{*}=0\); then the trivial equilibrium is \(X_{0}^{*}=(S_{e}^{*},0,0,S_{a}^{*},0,0,0,0,0,0)\). Furthermore, without infected classes, the forces of infection are equal to zero, and then \(S_{e}^{*}\) and \(S_{a}^{*}\) are obtained solving the following equations:
$$\begin{aligned} &\begin{aligned} S_{e}'(t)=0 &\quad \Leftrightarrow \quad p\Lambda_{h} -f_{h}S_{e}^{*}=0 \\ &\quad \Leftrightarrow\quad S_{e}^{*}=\frac{p}{f_{h}} \Lambda_{h}, \end{aligned} \\ &\begin{aligned} S_{a}'(t)=0 &\quad \Leftrightarrow\quad (1-p) \Lambda_{h} -f_{h}S_{a}^{*}=0 \\ &\quad \Leftrightarrow\quad S_{a}^{*}=\frac{1-p}{f_{h}} \Lambda_{h}, \end{aligned} \\ &N_{h}^{*}=S_{e}^{*}+S_{a}^{*}= \frac{\Lambda_{h}}{f_{h}}. \end{aligned}$$
Finally, \(DFE_{0}= ( \frac{p}{f_{h}}\Lambda_{h},0,0,\frac{1-p}{f _{h}}\Lambda_{h},0,0,0,0,0 ) \). □

3.3.3 Stability of equilibrium states

Theorem 3.1

The disease-free equilibrium \(DFE_{0}\) is locally asymptotically stable if \(\mathcal{R}_{0}\) is less than unity and instable if \(\mathcal{R} _{0}\) is greater than unity.

Proof

Indeed, the disease-free equilibrium \(DFE_{0}\) is stable when the spectral radius of the next generation matrix is less than unity. As \(\rho (-FV^{-}1)=\mathcal{R}_{0}\), we deduce the result. □

Let us consider the following region:
$$\begin{aligned} \Delta^{*}= \bigl\lbrace (S_{e},E_{e},I_{e},S_{a},E_{a},I _{a},R_{a},E_{m},I_{m})\in \mathbb{R}_{+}^{9} | \textstyle\begin{array}{llll} 0< S_{e} \leq S_{e}^{*},\ldots, 0< I_{m}\leq I_{m}^{*} \end{array}\displaystyle \bigr\rbrace . \end{aligned}$$
(15)
It is clear that \(\Delta^{*} \subset \Delta \). Since Δ is positively invariant, we deduce that \(\Delta^{*}\) is positively invariant.

Theorem 3.2

If \(\mathcal{R}_{0}<1\), then the DFE \(X_{1}^{*}\) is globally asymptotically stable in \(\Delta^{*}\).

Proof

Let us consider the following Lyapunov function:
$$ \mathcal{L}(t)=f_{1} E_{e}+f_{2} I_{e} +f_{3} E_{a} + f_{4} I_{a} +f _{5} R_{a} + f_{6} E_{m} +f_{7} I_{m}, $$
where
$$\begin{aligned} &\chi = \frac{k_{e}}{B^{*}}\times \frac{\nu_{e}}{C^{*}}\times \frac{k _{a}}{D^{*}} \times \frac{\nu_{a}}{E^{*}}\times \frac{\alpha_{e}+ \alpha_{a}}{G^{*}}\times \frac{k_{m}}{H^{*}+k_{m}} \times \frac{\nu _{m}}{f_{m}}, \\ &f_{1}=\chi \mathcal{R}_{0}, \qquad f_{2}= \frac{B^{*}}{k_{e}} f_{1}, \qquad f_{3}=\frac{C^{*}}{\nu_{e}}f_{2}, \qquad f_{4}=\frac{D^{*}}{k_{a}} f_{3}, \\ &f_{5}=0, \qquad f_{6}=\frac{E^{*}}{\nu_{a}}\times \frac{G^{*}}{\alpha_{e}+\alpha _{a}} f_{4}, \qquad f_{7}=\frac{f_{m}}{\nu_{m}}f_{6}, \end{aligned}$$
with Lyapunov derivative given by
$$\begin{aligned} &\mathcal{L}'(t)=f_{1} E_{e}'+f_{2} I_{e}' +f_{3} E_{a}' + f_{4} I_{a}' +f_{5} R_{a}' + f_{6} E_{m}' +f_{7} I_{m}', \\ & \begin{aligned} \mathcal{L}'(t) ={}&f_{1} \bigl( k_{e} S_{e}-B^{*} E_{e} \bigr) +f_{2} \bigl( \nu_{e} E_{e}-C^{*} I_{e} \bigr) +f_{3} \bigl( k_{a} S_{a}-D^{*} E _{a} \bigr) \\ &{}+f_{4} \bigl( \nu_{a} E_{a}-E^{*} I_{a} \bigr) + f_{6} \bigl( k_{m} A ^{*}-k_{m} I_{m}-\bigl(H^{*}+k_{m} \bigr) E_{m} \bigr) \\ &{}+ f_{7} ( \nu_{m} E_{m}-f_{m} I_{m} ) \\ ={}&k_{e} \chi S_{e} ( \mathcal{R}_{0}-1 ) + ( f_{2}\nu_{e} E_{e}-k_{e} f_{2} E_{e} ) + ( f_{4} \nu_{a} E_{a}-k_{a} f_{4} E _{a} ) \\ &{}+ \biggl( \frac{f_{m}}{\nu_{m}}f_{6} E_{m}-f_{6} \bigl(H^{*}+k_{m}\bigr)E_{m} \biggr) + \bigl( k_{e} \chi S_{e} -f_{2} C^{*} I_{e} \bigr) \\ &{}+ \bigl( f_{3} k_{a} S_{a}-f_{4} E^{*} I_{a} \bigr) + \bigl( f_{6} k_{m} A^{*}-f_{6} k_{m} I_{m}-f_{7} f_{m} I_{m} \bigr) \\ \leq{} & k_{e} \chi S_{e} ( \mathcal{R}_{0}-1 ) +\mathcal{R} _{0}\chi_{1} E_{e} \biggl( \frac{\nu_{e}}{k_{e}}-1 \biggr) +\mathcal{R} _{0} \chi_{2} E_{a} \biggl( \frac{\nu_{a}}{k_{a}}-1 \biggr) \\ &{}+\mathcal{R}_{0} \chi_{3} E_{m} \biggl( \frac{f_{m}}{\nu_{m}(H^{*}+k _{m})}-1 \biggr) + 2 \biggl( \frac{k_{e}^{2}}{\mathcal{R}_{0} B^{*} C ^{*}}-1 \biggr) C^{*} f_{2} \max \bigl(S_{e}^{*},I_{e}^{*} \bigr) \\ &{}+2 \biggl( \frac{k_{a}^{2}}{D^{*}}-1 \biggr) f_{4} \max \bigl(S_{a}^{*},I _{a}^{*} \bigr)-2f_{7} f_{m} \max \bigl(A^{*},I_{m}^{*} \bigr) \\ \leq {}& k_{e}\chi S_{e} ( \mathcal{R}_{0}-1 ) + \chi_{1} \mathcal{R}_{0} E_{e} \biggl( \frac{\nu_{e}}{k_{e}}-1 \biggr) +\chi_{2} \mathcal{R}_{0} E_{a} \biggl( \frac{\nu_{a}}{k_{a}}-1 \biggr) \\ &{}+\mathcal{R}_{0} \chi_{3} E_{m} \biggl( \frac{f_{m}}{\nu_{m}(H^{*}+k _{m})}-1 \biggr), \end{aligned} \end{aligned}$$
where
$$\begin{aligned} &\chi_{1} = k_{e} \frac{\nu_{e}}{C^{*}}\times \frac{k_{a}}{D^{*}} \times \frac{\nu_{a}}{E^{*}}\times \frac{\alpha_{e}+\alpha_{a}}{G ^{*}}\times \frac{k_{m}}{H^{*}+k_{m}}\times \frac{\nu_{m}}{f_{m}} , \\ &\chi_{2} = k_{a} \frac{\nu_{a}}{E^{*}}\times \frac{\alpha_{e}+ \alpha_{a}}{G^{*}}\times \frac{k_{m}}{H^{*}+k_{m}}\times \frac{\nu _{m}}{f_{m}}, \\ &\chi_{3} = k_{m} \frac{\nu_{m}}{f_{m}} . \end{aligned}$$
So, if \(\mathcal{R}_{0}<1\), then \(\mathcal{L}'(t)\leq 0 \). □

Theorem 3.3

If \(\mathcal{R}_{0}>1\) then the system (12) has a unique endemic equilibrium.

Proof

Let be \(X^{**}\) the non-trivial equilibrium; the components \(S_{e}^{**}\), \(E_{e}^{**}\), \(I_{e}^{**}\), \(S_{a}^{**}\), \(E_{a}^{**}\), \(I_{a}^{**}\), \(R_{a}^{**}\), \(E_{m}^{**}\), \(I_{m}^{**}\) of \(X^{**}\) are obtained setting all the equations of the system (12) to zero. Hence,
$$\begin{aligned} &S_{e}^{**}=\frac{p}{f_{h}+k_{e}^{**}}\Lambda_{h}, \qquad E_{e}^{**}=\frac{p k_{e}^{**}}{(f_{h}+k_{e}) B^{*}} \Lambda_{h}, \qquad I_{e}^{**}=\frac{p k_{e}^{**} \nu_{e}}{(f_{h}+k_{e} ^{**})B^{*} C^{*}}\Lambda_{h}, \\ &S_{a}^{**}=\frac{ ( p k_{e}^{**} \alpha_{e} \nu_{e}-(1-p)(f_{h}+k _{e}^{**})B^{*} C^{*} G^{*} ) D^{*} E^{*}}{ ( (f_{h}+k_{a}^{**})D ^{*} E^{*} G^{*}-k_{a}^{**} \alpha_{a} \nu_{a} \beta_{a} ) (f_{h}+k _{e}^{**})B^{*} C^{*}}, \\ &E_{a}^{**}=\frac{ ( p k_{e}^{**} \alpha_{e} \nu_{e}-(1-p)(f_{h}+k _{e}^{**})B^{*} C^{*} G^{*} ) k_{a}^{**} E^{*}}{ ( (f_{h}+k _{a}^{**})D^{*} E^{*} G^{*}-k_{a}^{**} \alpha_{a} \nu_{a} \beta_{a} ) (f _{h}+k_{e}^{**})B^{*} A^{*}}, \\ &I_{a}^{**}=\frac{ ( p k_{e}^{**} \alpha_{e} \nu_{e}-(1-p)(f_{h}+k _{e}^{**})B^{*} C^{*} G^{*} ) k_{a}^{**} \nu_{a}}{ ( (f_{h}+k _{a}^{**})D^{*} E^{*} G^{*}-k_{a}^{**} \alpha_{a} \nu_{a} \beta_{a} ) (f _{h}+k_{e}^{**})B^{*} C^{*}}, \\ &R_{a}^{**}=\frac{ ( p k_{e}^{**}\alpha_{e} \nu_{e}-(1-p)(f_{h}+k _{e}^{**})B^{*} C^{*} G^{*} ) (f_{h}+k_{a}^{**})D^{*} E^{*}}{ ( (f_{h}+k_{a}^{**})D^{*} E^{*} G^{*}-k_{a}^{**} \alpha_{a} \nu _{a} \beta_{a} ) (f_{h}+k_{e}^{**})\beta_{a} B^{*} C^{*}}-\frac{1-p}{ \beta_{a}} \Lambda_{h}, \\ &E_{m}^{**}= \biggl( 1-\frac{1}{r} \biggr) \frac{k_{m}^{**} s_{P} K_{P}}{(f _{m}+1)(f_{m}+\nu_{m})\chi_{P} }, \qquad I_{m}^{**}= \biggl( 1- \frac{1}{r} \biggr) \frac{k_{m}^{**} \nu_{m} s_{P} K_{P}}{f_{m}(f_{m}+1)(f_{m}+\nu_{m})\chi_{P} H^{*}}. \end{aligned}$$
 □

Theorem 3.4

If \(\mathcal{R}_{0}>1\), then the system (12) has a unique endemic equilibrium which is globally asymptotically stable in the following set:
$$ \Delta^{**}= \bigl\lbrace (S_{e},E_{e},I_{e},S_{a},E_{a},I_{a},R_{a},E _{m},I_{m})\in \mathbb{R}_{+}^{9} | \textstyle\begin{array}{llll} 0< S_{e} \leq S_{e}^{**},\ldots, 0< I_{m}\leq I_{m}^{**} \end{array}\displaystyle \bigr\rbrace . $$

Proof

Since \(\mathcal{R}_{0}>1\), the endemic equilibrium exists. Now, let us consider the following Lyapunov function [3]:
$$ \mathcal{V}(t)=\mathcal{V}_{1}(t)+\mathcal{V}_{2}(t), $$
where
$$ \mathcal{V}_{1}(t)=\frac{1}{2} \bigl( V_{se}(t)+V_{ee}(t)+ V_{ie}(t)+V _{sa}(t)+V_{ea}(t)+V_{ia}(t)+V_{ra}(t) \bigr) ^{2} $$
and
$$ \mathcal{V}_{2}(t)=\frac{1}{2} \bigl( V_{em}(t)+V_{im}(t) \bigr) ^{2} $$
with
$$\begin{aligned} &V_{se}(t)=S_{e}(t)-S_{e}^{**}, \qquad V_{ee}(t)=E_{e}(t)-E_{e}^{**}, \qquad V_{ie}(t)=I_{e}(t)-I_{e}^{**}, \\ &V_{sa}(t)=S_{a}(t)-S_{a}^{**}, \qquad V_{ea}(t)=E_{a}(t)-E_{a}^{**}, \qquad V_{ia}(t)=I_{a}(t)-I_{a}^{**}, \\ &V_{ra}(t)=R_{a}(t)-R_{a}^{**}, \qquad V_{em}(t)=E_{m}(t)-E_{m}^{**}, \qquad V_{im}(t)=I_{m}(t)-I_{m}^{**}. \end{aligned}$$
The Lyapunov function constructed above guarantees that it attains the minimum value at the endemic equilibrium \((S_{e}^{**},E_{e}^{**},I_{e}^{**},S_{a}^{**},E_{a}^{**},I_{a}^{**},R_{a}^{**},E_{m}^{**},I_{m}^{**})\).
The Lyapunov derivative of this function is given by
$$ \mathcal{V}'(t)=\mathcal{V}_{1}'(t)+ \mathcal{V}_{2}'(t) $$
with
$$ \mathcal{V}_{1}'(t)= \bigl( V_{se}(t)+V_{ee}(t)+ V_{ie}(t)+V_{sa}(t)+V _{ea}(t)+V_{ia}(t)+V_{ra}(t) \bigr) \times N_{h}'(t) $$
and
$$ \mathcal{V}_{2}'(t)= \bigl( V_{em}(t)+V_{im}(t) \bigr) \times \bigl( E _{m}'(t)+I_{m}'(t) \bigr) . $$
Consequently,
$$\begin{aligned} \mathcal{V}'(t) =& \bigl( V_{se}(t)+V_{ee}(t)+ V_{ie}(t)+V_{sa}(t)+V_{ea}(t)+V_{ia}(t)+V_{ra}(t) \bigr) \times N_{h}'(t) \\ &{}+ \bigl( V_{em}(t)+V_{im}(t) \bigr) \times \bigl( E_{m}'(t)+I_{m}'(t) \bigr) . \end{aligned}$$
It is easy to see that \(\mathcal{V}'(t)=0\) if and only if \(S_{e}=S_{e}^{**}\), \(E_{e}=E_{e}^{**}\), \(I_{e}=I_{e}^{**}\), \(S_{a}=S_{a}^{**}\), \(E_{a}=E_{a}^{**}\), \(R_{a}=R_{a}^{**}\), \(I_{a}=I_{a}^{**}\), \(E_{m}=E_{m}^{**}\), \(I_{m}=I_{m}^{**}\).

Now, let us show that \(\mathcal{V}'(t)\leq 0\).

Using the expression of \(N_{h}'(t)\) from (10) we obtain
$$\begin{aligned} \Lambda_{h}=f_{h} \bigl( S_{e}^{**}+E_{e}^{**}+I_{e}^{**}+S_{a}^{**}+E _{a}^{**}+I_{a}^{**}+R_{a}^{**} \bigr) +\gamma_{e} I_{e}^{**}+\gamma _{a} I_{a}^{**} . \end{aligned}$$
(16)
When we put (5) and (16) in (10), it implies that
$$\begin{aligned} N_{h}'(t) =&-f_{h} \bigl( V_{se}(t)+V_{ee}(t)+ V_{ie}(t)+V_{sa}(t)+V _{ea}(t)+V_{ia}(t)+V_{ra}(t) \bigr) \\ &{}-\gamma_{e} V_{ie}(t)-\gamma_{a} V _{ia}(t). \end{aligned}$$
It follows that
$$\begin{aligned} \mathcal{V}_{1}'(t) =&-f_{h} \bigl( V_{se}(t)+V_{ee}(t)+ V_{ie}(t)+V _{sa}(t)+V_{ea}(t)+V_{ia}(t)+V_{ra}(t) \bigr) ^{2} \\ &{}- \bigl( V_{se}(t)+V_{ee}(t)+ V_{ie}(t)+V_{sa}(t)+V_{ea}(t)+V_{ia}(t)+V _{ra}(t) \bigr)\\ &{}\times \bigl(\gamma_{e} V_{ie}(t)+ \gamma_{a} V_{ia}(t)\bigr). \end{aligned}$$
Therefore, \(\mathcal{V}_{1}'(t)\leq 0\).
Moreover,
$$\begin{aligned} E_{m}'(t)+I_{m}'(t) =&-k_{m} A^{*}-k_{m}\bigl(E_{m}(t)+I_{m}(t) \bigr)-f_{m}\bigl(E _{m}(t)+I_{m}(t)\bigr) \\ =& -k_{m} \bigl( V_{em}(t)+V_{im}(t) \bigr) +k_{m} S_{m}^{**}-f_{m}\bigl(E _{m}(t)+I_{m}(t)\bigr) \\ \leq & -k_{m} \bigl( V_{em}(t)+V_{im}(t) \bigr) +k_{m} S_{m}^{**}. \end{aligned}$$
It then follows that
$$\begin{aligned} \mathcal{V}_{2}'(t) \leq & \bigl( V_{em}(t)+V_{im}(t) \bigr) \times \bigl( -k_{m} \bigl( V_{em}(t)+V_{im}(t) \bigr) +k_{m} S_{m}^{**} \bigr) \\ \leq & -k_{m} \bigl( V_{em}(t)+V_{im}(t) \bigr) ^{2}+k_{m} S_{m}^{**} \bigl( V_{em}(t)+V_{im}(t) \bigr) \\ \leq & -k_{m} \bigl( V_{em}(t)+V_{im}(t) \bigr) ^{2} \\ \leq & 0, \end{aligned}$$
since \(V_{em}(t)\leq 0\) and \(V_{im}(t)\leq 0\) on \(\Delta^{**}\).

Hence, \(\mathcal{V}'(t)\leq 0\) and then LaSalle’s invariant principle [20] implies that the endemic equilibrium is globally asymptotically stable on \(\Delta^{**}\). □

Remark 3.2

The local stability of the equilibrium point \(DFE_{0}\) can be established using the mosquitoes growth rate r. We have previously obtained
$$ \mathcal{R}_{0}=\sqrt{K_{27}K_{72}+K_{47}K_{74}+K_{57}K_{75}}, $$
thus,
$$ \mathcal{R}_{0}=f_{\kappa }(r)= \sqrt{\kappa \biggl( 1- \frac{1}{r} \biggr) }, $$
where
$$\begin{aligned} \kappa =& \frac{\nu_{e}}{B^{*} C^{*}}\times c_{me}n_{a} \frac{S _{e}^{*}}{N_{h}^{*}} \times \frac{\nu_{m}}{H^{*}}\times \frac{1}{f _{m}^{2}}\times c_{em}n_{a} \frac{s_{P} K_{P}}{\chi_{P} N_{h}^{*}} \\ &{}+ \frac{\nu_{a}}{D^{*} E^{*}}\times c_{ma}n_{a} \frac{S_{a}^{*}}{N _{h}^{*}}\times \frac{\nu_{m}}{H^{*}}\times \frac{1}{f_{m}^{2}} \times c_{am}n_{a} \frac{s_{P} K_{P}}{\chi_{P} N_{h}^{*}} \\ &{}+ \frac{\alpha_{a} \nu_{a}}{D^{*} E^{*} G^{*}}\times c_{ma}n_{a} \frac{S _{a}^{*}}{N_{h}^{*}}\times \frac{\nu_{m}}{H^{*}}\times \frac{1}{f _{m}^{2}}\times \tilde{c}_{am}n_{a} \frac{s_{P} K_{P}}{\chi_{P} N _{h}^{*}} . \end{aligned}$$

Remark 3.3

Suppose that \(r>1\). Hence, the function \(f_{\kappa }\) is continuous and derivable. Moreover, some easy calculations give
$$ f_{\kappa }'(r)=\frac{\kappa }{r^{2} \sqrt{\kappa ( 1- \frac{1}{r} ) }} . $$
It is clear that \(f_{\kappa }'(r)\) is positive for \(r>1\); it follows that the larger the threshold r is, the larger the basic reproduction number \(\mathcal{R}_{0}\) becomes.

Lemma 3.2

Let consider the following threshold parameter: \(r_{0}=\frac{\kappa }{\kappa -1}\), \(\kappa \neq 1\).
  1. (i)

    \(\mathcal{R}_{0}<1\) is equivalent to \(1< r< r_{0}\).

     
  2. (ii)

    \(\mathcal{R}_{0}>1\) is equivalent to \(r>r_{0}\).

     

Proof

Indeed, we have
$$ \mbox{(i)} \quad \mathcal{R}_{0}< 1 \quad \Leftrightarrow\quad f_{\kappa }< 1 \quad \Leftrightarrow\quad \kappa \biggl( 1-\frac{1}{r} \biggr) < 1\quad \Leftrightarrow\quad r< \frac{\kappa }{\kappa -1}=r_{0}, \quad\kappa \neq 1, $$
and
$$ \mbox{(ii)} \quad \mathcal{R}_{0}>1\quad \Leftrightarrow\quad f_{\kappa }>1 \quad \Leftrightarrow \quad \kappa \biggl( 1-\frac{1}{r} \biggr) >1 \quad \Leftrightarrow\quad r> \frac{\kappa }{ \kappa -1}=r_{0}, \quad\kappa \neq 1. $$
 □

Theorem 3.5

  1. (i)

    If \(r\leq 1\) then the disease-free equilibrium \((0,0,0,0,S _{e}^{*},0,0,S_{a}^{*},0,0,0,0,0)\) of the system (9) is globally asymptotically stable.

     
  2. (ii)

    If \(1< r< r_{0}\) then the disease-free equilibrium point \((E^{*},L^{*},P^{*},A^{*},S_{e}^{*},0,0,S_{a}^{*},0,0,0,0,0)\) of the system (9) is globally asymptotically stable.

     
  3. (iii)

    If \(r>r_{0}\) then the endemic equilibrium \((E^{*},L^{*},P ^{*},A^{*},S_{e}^{**},E^{**},I_{e}^{**},S_{a}^{**},E_{a}^{**},I_{a} ^{**},R_{a}^{**},E_{m}^{**},I_{m}^{**})\) of the system (9) is globally asymptotically stable.

     

Proof

The key idea to prove this theorem is to use the theory of internally chain transitive sets [21, 36].

Let \(\Phi (t)\) be the solution of the system (9) on \(\Omega_{m}\times \Omega_{h}\), that is, \(\Phi (t) ( E(0),L(0),P(0),S_{e}(0), E_{e}(0),I_{e}(0),S_{a}(0),E _{a}(0),I_{a}(0),R_{a}(0),A(0),E_{m}(0),I_{m}(0) ) \). Then \(\Phi (t)\) is compact for each \(t>0\).

Let \(\omega =\omega ( E(0),L(0),P(0),A(0),S_{e}(0),E_{e}(0),I_{e}(0),S _{a}(0),E_{a}(0),I_{a}(0),R_{a}(0),E_{m}(0),I_{m}(0) ) \) be the omega limit set of \(\Phi (t) ( E(0),L(0),P(0),S_{e}(0),E_{e}(0),I_{e}(0),S_{a}(0),E _{a}(0),I_{a}(0),R_{a}(0),A(0), E_{m}(0),I_{m}(0) ) \) It then follows that ω is an internally chain transitive set for \(\Phi (t)\).
  1. (i)
    In the case where \(r\leq 1\), then
    $$\begin{aligned} E(t)\rightarrow 0,\qquad L(t)\rightarrow 0,\qquad P(t)\rightarrow 0 \quad \text{and}\quad A(t) \rightarrow 0 \quad \text{as } t\rightarrow +\infty . \end{aligned}$$
    Since \(A(t)\rightarrow 0\) as \(t\rightarrow +\infty \), \(E_{m}(t)\rightarrow 0\) and \(I_{m}(t)\rightarrow 0\) as \(t\rightarrow +\infty \). Thus, we have \(\omega ={(0,0,0,0,0,0)}\times \omega_{1}\) with \(\omega_{1} \subset \mathbb{R}^{7}\).
    Moreover, we have \(\Phi (t)|_{\omega } ( 0,0,0,0,0,0 ,S_{e}^{*},0,0,S _{a}^{*},0,0,0 ) = ( 0,0,0,0,0,0,\Phi_{1}(t)(S_{e}(0),E_{e}(0),I_{e}(0),S_{a}(0),E _{a}(0),I_{a}(0),R_{a}(0)) ) \) associated with the following system:
    $$\begin{aligned} \textstyle\begin{cases} S_{e}'(t)=p\Lambda_{h}-f_{h}S_{e}(t), \\ E_{e}'(t)=-(f_{h}+\nu_{e})E_{e}(t), \\ I_{e}'(t)=\nu_{e} E_{e}(t)-(f_{h}+\gamma_{e}+\alpha_{e})I_{e}(t), \\ S_{a}'(t) =(1-p)\Lambda_{h}+\beta_{a} R_{a}(t) - f_{h}S_{a}(t), \\ E_{a}'(t) = - (f_{h}+\nu_{a})E_{a}(t), \\ I_{a}'(t) = \nu_{a} E_{a}(t)-(f_{h}+\gamma_{a}+\alpha_{a})I_{a}(t), \\ R_{a}'(t) = \alpha_{e} I_{e}(t) +\alpha_{a} I_{a}(t)-(f_{h} +\beta _{a})R_{a}(t). \end{cases}\displaystyle \end{aligned}$$
    (17)
    From the second and the fifth equation of the system (17), we have \(E_{e}(t)\rightarrow 0\) and \(E_{a}(t)\rightarrow 0\) as \(t\rightarrow +\infty \). Using the limit system of the system (17), it then follows that \(I_{e}(t)\rightarrow 0\) and \(I_{a}(t)\rightarrow 0\). Hence, we deduce from the last equation that \(R_{a}(t)\rightarrow 0\). Finally, we obtain the following limit system:
    $$\begin{aligned} \begin{aligned} &S_{e}'(t)=p \Lambda_{h}-f_{h}S_{e}(t), \\ &S_{a}'(t) =(1-p)\Lambda_{h} - f_{h}S_{a}(t). \end{aligned} \end{aligned}$$
    (18)
    It is easy to see from the above system that \(S_{e}(t)\rightarrow 0\) and \(S_{a}(t)\rightarrow 0\) as \(t\rightarrow 0\).
     
  2. (ii)
    In the case where \(r>1\), then from Theorem 2.5, we have
    $$\begin{aligned} E(t)\rightarrow E^{*},\qquad L(t)\rightarrow L^{*},\qquad P(t)\rightarrow P^{*},\quad \text{and}\quad A(t)\rightarrow A^{*} \quad \text{as } t\rightarrow +\infty , \end{aligned}$$
    for any \(E(0)>0\), \(L(0)>0\), \(P(0)>0\), \(A(0)>0\). Hence, we have \(\omega ={(E^{*},L^{*},P^{*},A^{*})}\times \omega_{2}\) with \(\omega_{2}\subset \mathbb{R}^{9}\).

    It is easy to see that \(\Phi (t)|_{\omega } ( E^{*},L^{*},P^{*},A^{*} ,S_{e}^{*},0,0,S _{a}^{*},0,0,0,0,0 ) = ( E^{*},L^{*},P^{*},A^{*},\Phi_{2}(t)(S_{e}(0),E_{e}(0),I_{e}(0),S _{a}(0),E_{a}(0),I_{a}(0),R_{a}(0),E_{m}(0),I_{m}(0)) ) \) associated with the system (12).

    Since ω is an internally limit set for \(\Phi (t)\), it is easy to see that \(\omega_{2}\) is an internally chain transitive set for \(\Phi_{2}(t)\). Since \(r< r_{0}\), thanks to Theorem 3.2, the disease-free equilibrium \((S_{e}^{*},0,0,S_{a}^{*},0,0,0,0,0)\) is globally asymptotically stable for the limit system (12). It then follows from Theorem 3.2 and Remark 4.6 in [16] that \(\omega_{2}=\lbrace (S_{e}^{*},0,0,S_{a}^{*},0,0,0,0,0)\rbrace \) and \(\omega =\{(E^{*},L^{*},P^{*},A^{*}, S_{e}^{*},0,0,S_{a}^{*},0,0,0,0,0) \}\). Hence, if \(1< r< r_{0}\), then the disease-free equilibrium \((E^{*},L^{*},P^{*},A^{*}, S_{e}^{*},0,0,S_{a}^{*},0,0,0,0,0)\) is globally asymptotically stable through the system (9).

     
  3. (iii)
    In the case where \(r>r_{0}\), thanks to Theorem 2.5, we have
    $$\begin{aligned} E(t)\rightarrow E^{*},\qquad L(t)\rightarrow L^{*},\qquad P(t)\rightarrow P^{*} \quad \text{and}\quad A(t)\rightarrow A^{*} \quad \text{as } t\rightarrow +\infty , \end{aligned}$$
    for any \(E(0)>0\), \(L(0)>0\), \(P(0)>0\), \(A(0)>0\).

    Hence, we have \(\omega =\{(E^{*},L^{*},P^{*},A^{*})\}\times \omega _{3}\) with \(\omega_{3}\subset \mathbb{R}^{9}\) and \(\Phi (t)|_{\omega } ( E^{*},\ldots,A^{*}, S_{e}(0),\ldots,I_{m}(0) ) = ( E^{*},\ldots,A^{*},\Phi_{2}(t)(S_{e}(0),\ldots,I_{m}(0)) ) \) where \(\Phi_{2}(t)\) is the solution semiflow of the system (12). Thanks to Lemma 3.2, \(r>r_{0}\), implies that \(\mathcal{R}_{0}>1\) and then \(\omega_{2}\neq 0_{\mathbb{R}^{9}}\).

    Since \((S_{e}^{**},E_{e}^{**},I_{e}^{**},S_{a}^{**},E_{a}^{**},I_{a} ^{**},R_{a}^{**},E_{m}^{**},I_{m}^{**})\) is globally asymptotically stable for the system (17) in \(\mathbb{R}^{*9}\), \(\omega_{3} \cap W^{s}((S_{e}^{**},E_{e}^{**},I_{e}^{**},S_{a}^{**},E _{a}^{**},I_{a}^{**},R_{a}^{**},E_{m}^{**},I_{m}^{**}))\neq \emptyset \). Hence, the statement (iii) is valid.

     
 □

4 Numerical simulations

In this section we perform some numerical results in order to illustrate theoretical results which were previously established. Our numerical simulation will be performed using the MATLAB technical computing software with the fourth-order Runge–Kutta method [1, 23, 26, 30]. The values of the parameters are given in Table 7.
Table 7

Human and vector parameter’s values for the malaria model

Parameters

Value for extinction

Value for persistence

Reference

Dimension

p

0.25

0.8

estimated

dimensionless

\(c_{me}\)

0.021

0.03

[13]

dimensionless

\(c_{em}\)

0.11

0.45

[13]

dimensionless

\(c_{ma}\)

0.012

0.022

[13]

dimensionless

\(c_{am}\)

0.08

0.35

[13]

dimensionless

\(\tilde{c}_{am}\)

0.008

0.002

[13]

dimensionless

\(\nu _{e}\)

0.10

0.10

[13]

/days

\(\nu _{a}\)

0.06

0.09

[13]

/days

\(\nu _{m}\)

0.083

0.083

[13]

/days

\(\alpha _{e}\)

0.001

0.001

[13]

/days

\(\alpha _{a}\)

0.01

0.01

[13]

/days

\(\gamma _{e}\)

0.000018

0.000018

[13]

/days

\(\gamma _{a}\)

0.00003

0.00003

[13]

/days

\(\beta _{a}\)

0.0055

0.0027

[13]

/days

\(\Lambda _{h}\)

50

85

estimated

humans/week

\(f_{h}\)

0.00063

0.00063

[13]

/human/days

\(f_{m}\)

0.1

0.1

[13]

/mosquito/days

\(n_{a}\)

0.25

0.5

[13]

human/days

\(K_{E}\)

10,000

10,000

estimated

space

\(K_{L}\)

5000

5000

estimated

space

\(K_{P}\)

4000

4000

estimated

space

b

2

10.7

[1]

/days

\(s_{E}\)

0.6

0.4

[1]

dimensionless

\(d_{E}\)

0.3

0.36

[1]

dimensionless

\(S_{L}\)

0.4

0.5

[1]

dimensionless

\(d_{L}\)

0.3

0.34

[1]

dimensionless

\(s_{P}\)

0.25

0.3

[1]

dimensionless

\(d_{P}\)

0.15

0.17

[1]

dimensionless

\(f_{m}\)

0.6

0.15

[1]

dimensionless

4.1 Dynamical model for vector population growth

Firstly, using the following initial conditions: \(E(0)=35\), \(L(0)=25\), \(P(0)=30\), \(A(0)=45\) and the mosquito’s parameter values for extinction given in Table 7, we obtain Fig. 3. These values lead to the condition \(r\leq 1\); that is, the mosquitoes’ population disappears. This result confirms Theorem 2.2.
Figure 3
Figure 3

Evolution of eggs, larvae, pupae and adults for \(b=2\), \(S_{E}=0.6\), \(d_{E}=0.3\), \(s_{L}=0.4\), \(d_{L}=0.3\), \(s_{P}=0.25\), \(d_{P}=0.15\) and \(f_{m}=0.6\). We get \(r=0.7937\), which is less than unity

Secondly, using the following initial conditions: \(E(0)=50\), \(L(0)=40\), \(P(0)=8\), \(A(0)=6\) and the numerical values of parameters in Table 7 which lead to the threshold r greater than unity, we get Fig. 4. We observe that when r is greater than unity, the mosquitoes’ population persists. This result supports Theorem 2.3.
Figure 4
Figure 4

Evolution of eggs, larvae, pupae and adults for \(b=10.7\), \(S_{E}=0.4\), \(d_{E}=0.36\), \(s_{L}=0.5\), \(d_{L}=0.34\), \(s_{P}=0.3\), \(d_{P}=0.17\) and \(f_{m}=0.15\). We get \(r=14.2644\), which is greater than unity

These two observations show that the threshold parameter r may provide conditions in order to control the proliferation of the mosquito population.

4.2 Global model of malaria transmission

The initial conditions used here are \(E(0)=50\), \(L(0)=40\), \(P(0)=30\), \(A(0)=80\), \(S_{e}(0)=50\), \(S_{a}(0)=200\), \(I_{e}(0)=500\), \(S_{a}(0)=25\), \(E_{a}(0)=150\), \(I_{a}(0)=350\), \(R_{a}(0)=400\), \(S_{m}(0)=50\), \(E_{m}(0)=15\) and \(I_{m}(0)=100\). Using the above initial conditions and the numerical values of human parameters in Table 7, which lead to \(\mathcal{R}_{0}\) less than unity, we get Fig. 5. This figure shows the extinction of the different infected classes of global disease dynamics model. These observations confirm Theorem 3.1. With the same above initial conditions for vectors, taking \(S_{e}(0)=50\), \(S_{a}(0)=200\), \(I_{e}(0)=500\), \(S_{a}(0)=25\), \(E_{a}(0)=150\), \(I_{a}(0)=350\), \(R_{a}(0)=400\), \(S_{m}(0)=50\), \(E_{m}(0)=15\), \(I_{m}(0)=100\) for humans and the above numerical values of human parameters in Table 7, which lead to \(\mathcal{R}_{0}\) greater than unity, we get Fig. 6. This figure shows the persistence of the different infected classes of global disease dynamics model. These observations confirm Theorem 3.2.
Figure 5
Figure 5

Evolution of the different clinics classes of the human population for \(p=0.25\), \(\Lambda_{h}=10\), \(c_{me}=0.021\), \(c_{ma}=0.012\), \(c_{em}=0.11\), \(c_{am}=0.08\), \(\tilde{c}_{me}=0.008\), \(\nu_{e}=0.10\), \(\nu_{a}=0.06\), \(\nu_{m}=0.083\), \(\alpha_{a}=0.01\), \(\alpha_{e}=0.001\), \(\gamma_{e}=0.000018\), \(\gamma_{a}=0.00003\), \(\beta_{a}=0.0055\), \(f_{h}=0.00063\), \(f_{m}=0.1\), \(n_{a}=0.25\), \(b=10.7\), \(S_{E}=0.4\), \(d_{E}=0.36\), \(s_{L}=0.5\), \(d_{L}=0.34\), \(s_{P}=0.3\), \(d_{P}=0.17\) and \(f_{m}=0.15\). These values lead to \(r=14.26\) and \(\mathcal{R}_{0}=0.81\)

Figure 6
Figure 6

Evolution of the different clinics classes of the human population for \(p=0.8\), \(\Lambda_{h}=50\), \(c_{me}=0.03\), \(c_{ma}=0.022\), \(c_{em}=0.45\), \(c_{am}=0.35\), \(\tilde{c}_{me}=0.002\), \(\nu_{e}=0.10\), \(\nu_{a}=0.09\), \(\nu_{m}=0.083\), \(\alpha_{a}=0.01\), \(\alpha_{e}=0.001\), \(\gamma_{e}=0.000018\), \(\gamma_{a}=0.00003\), \(\beta_{a}=0.0027\), \(f_{h}=0.00063\), \(f_{m}=0.1\), \(n_{a}=0.5\), \(b=10.7\), \(S_{E}=0.4\), \(d_{E}=0.36\), \(s_{L}=0.5\), \(d_{L}=0.34\), \(s_{P}=0.3\), \(d_{P}=0.17\) and \(f_{m}=0.15\). These values lead to \(r=14.26\) and \(\mathcal{R}_{0}=1.26\)

5 Discussions and conclusion

In this paper, we have proposed mathematical models to describe the vector population growth global dynamics and the malaria virus transmission to human population.

In the first part, we have proposed the vector population dynamics model including auto-regulation phenomena of eggs, larvae and pupae. We used the Verhulst–Pearl logistic functions in order to gain insight into its qualitative features. For this model, we found that the mosquito growth rate r is the threshold condition for the existence of the endemic state. Besides, for r greater than unity, we proved using the Lyapunov function that the endemic equilibrium is globally asymptotically stable. The study of this model shows that the effect of immature stages is very important on the mosquitoes’ population proliferation [1, 23, 30].

Moreover, we have proposed a model to describe the malaria virus transmission to human population including some biological complexities as human host susceptibility. We divide the human hosts into two major types: the first type is called non-immune and comprised all humans who have not acquired the immunity against malaria; the second type is called semi-immune and represents all the people who have at least once acquired immunity during his life. For a better understanding of the mosquitoes population proliferation effect on the malaria disease spread, the two models were associated and the global study done. For this global model, the common basic reproduction number was determined using the next generation matrix idea [17, 18, 32]. As a further new insight, there was established a very interesting relationship between the common basic reproduction number and the regulatory threshold parameters of mosquito population, r, and its implications for malaria management analyzed. We found another threshold parameter, called \(r_{0}\), which is expressed using the two model parameters. Using the threshold \(r_{0}\), we established the global transmission model stability.

Finally, numerical simulations are carried out to support all the above theoretical results and provide conditions in order to control the proliferation of mosquito population and its implication on malaria spread control. It shows clearly that malaria management is concerned firstly lowering the mosquito threshold parameters to a value less than unity. This condition leads to the mosquito population’s disappearance. This is an ideal case in the fight against malaria but it can have some environment mistakes. In the other case, keeping the threshold r between 1 and \(r_{0}\) leads to the basic reproduction number less than unity, so the disease disappears.

In this study, we consider a homogeneously mixed population; however, each individual may have a heterogeneous number of contacts in the population. Meanwhile, the contact network structure is ignored. For future work, it would be fair to include the network structure as in [33, 34], which will make the model more realistic.

Declarations

Acknowledgements

The authors would like to thank the referees for a careful reading and several constructive comments and making some useful corrections that have improved the presentation of the paper.

Availability of data and materials

Not applicable.

Funding

This work was supported by Department of Mathematics, Université Nazi Boni, Burkina Faso.

Authors’ contributions

All authors worked together to produce the results and read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics and Informatics, UNB, Bobo Dioulasso, Burkina Faso

References

  1. Abdelrazec, A., Gumel, A.B.: Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics. J. Math. Biol. 74, 1351–1395 (2017) MathSciNetView ArticleMATHGoogle Scholar
  2. Aron, J.L., May, R.M.: The population dynamics of malaria. In: Anderson, R.M. (ed.) The Population Dynamics of Infectious Disease: Theory and Applications, pp. 139–179. Chapman & Hall, London (1982) View ArticleGoogle Scholar
  3. Asamoah, J.K., Oduro, F.T., Bonyah, E., Seidu, B.: Modelling of rabies transmission dynamics using optimal control analysis. J. Appl. Math. 2017, Article ID 2451237 (2017) MathSciNetView ArticleGoogle Scholar
  4. Athitan, S., Ghosh, M.: Stability analysis and optimal control of a malaria model with larvivorous fish as biological control agent. Appl. Math. Inf. Sci. 9(4), 1893–1913 (2015) MathSciNetGoogle Scholar
  5. Beck-Johnson, L.M., Nelson, W.A., Paaijmans, K.P., Read, A.F., Thomas, M.B., Bjørnstad, O.N.: The effect of temperature on Anopheles mosquito population dynamics and the potential for malaria transmission. PLoS ONE 8(11), Article ID e79276 (2013) View ArticleGoogle Scholar
  6. Birkhoff, G., Rota, G.C.: Ordinary Differential Equations. Ginn, Boston (1982) MATHGoogle Scholar
  7. Chitnis, N., Cushing, J.M., Hyman, J.M.: Bifurcation analysis of a mathematical model for malaria transmission. SIAM J. Appl. Math. 67, 24–25 (2006) MathSciNetView ArticleMATHGoogle Scholar
  8. Chitnis, N., Hyman, J.M., Cushing, J.M.: Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 70(5), 1272–1296 (2008) MathSciNetView ArticleMATHGoogle Scholar
  9. Chitnis, N.R.: Using mathematical models in controlling the spread of malaria. PhD Thesis (Applied Mathematic), University of Arizona (2005) Google Scholar
  10. Chiyaka, C., Tchuenche, J.M., Garira, W., Dube, S.: A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria. Appl. Math. Comput. 195(3), 641–662 (2008) MathSciNetMATHGoogle Scholar
  11. Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990) MathSciNetView ArticleMATHGoogle Scholar
  12. Dietz, K., Molineaux, L., Thomas, A.: A malaria model tested in the African savannah. Bull. World Health Organ. 50, 347–357 (1974) Google Scholar
  13. Ducrot, A., Sirima, S., Somé, B., Zongo, P.: A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host. J. Biol. Dyn. 3(6), 574–598 (2009) MathSciNetView ArticleMATHGoogle Scholar
  14. Forouzannia, F., Gumel, A.B.: Mathematical analysis of an age-structured model for malaria transmission dynamics. Math. Biosci. 247, 80–94 (2014) MathSciNetView ArticleMATHGoogle Scholar
  15. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977) View ArticleMATHGoogle Scholar
  16. Hirsch, M.W., Smith, H.L., Zhao, X.-Q.: Chain transitivity, attractivity and strong repellors for semidynamic system. Journ. Dyn. Differ. Eq. 13, 107–131 (2001) View ArticleMATHGoogle Scholar
  17. Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations. Oxford University Press, New York (2004) MATHGoogle Scholar
  18. Kolmanovskii, V., Shaikhet, E.: Some peculiarities of the general method of Lyapunov functionals construction. Appl. Math. Lett. 15, 355–360 (2002) MathSciNetView ArticleMATHGoogle Scholar
  19. Lakshmikantham, V., Leela, S., Kaul, S.: Comparison principle for impulsive differential equations with variable times and stability theory. Nonlinear Anal. 22(4), 499–503 (1994) MathSciNetView ArticleMATHGoogle Scholar
  20. LaSalle, J.P.: The Stability of Dynamical Systems. SIAM, Philadelphia (1976) View ArticleMATHGoogle Scholar
  21. Lou, Y., Zhao, X.-Q.: Modelling malaria control by introduction of larvivorous fish. Bull. Math. Biol. 73, 2384–2407 (2011) MathSciNetView ArticleMATHGoogle Scholar
  22. Macdonald, G.: The Epidemiology and Control of Malaria p. 3, 31, 48, 96. Oxford University Press, London (1957) Google Scholar
  23. Moulay, D., Aziz-Alaoui, M.A., Cadivel, M.: The chikungunya disease: modeling, vector and transmission global dynamics. Math. Biosci. 229(1), 50–63 (2011) MathSciNetView ArticleMATHGoogle Scholar
  24. Ngwa, G.A., Shu, W.S.: A mathematical model for endemic malaria with variable human and mosquito populations. Math. Comput. Model. 32, 747–763 (2000) MathSciNetView ArticleMATHGoogle Scholar
  25. Olaniyi, S., Obabiyi, O.S.: Mathematical model for malaria transmission dynamics in human and mosquito populations with nonlinear forces of infection. Int. J. Pure Appl. Math. 88(1), 125–156 (2013) View ArticleMATHGoogle Scholar
  26. Ouedraogo, W., Sangaré, B., Traoré, S.: Some mathematical problems arising in biological models: a predator-prey model fish-plankton. J. Appl. Math. Bioinform. 5(4), 1–27 (2015) MATHGoogle Scholar
  27. Ranson, H., N’Guessan, R., Lines, J., Moiroux, N., Nkuni, Z., Corbel, V.: Pyrethroid resistance in African anopheline mosquitoes: what are the implications for malaria control? Trends Parasitol. 27, 91–98 (2011) View ArticleGoogle Scholar
  28. Ross, R.: The Prevention of Malaria p. 3, 31, 48. Murray, London (1911) Google Scholar
  29. The world health report (2015) http://www.who.int/malaria/worl-malariareport-2016/report/en/
  30. Traoré, B., Sangaré, B., Traoré, S.: Mathematical model of malaria transmission with structured vector population and seasonality. J. Appl. Math. 2017, Article ID ID6754097 (2017) MathSciNetView ArticleGoogle Scholar
  31. Traoré, B., Sangré, B., Traoré, S.: A mathematical model of malaria transmission in a periodic environment. J. Biol. Dyn. 12(1), 400–432 (2018) MathSciNetView ArticleGoogle Scholar
  32. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002) MathSciNetView ArticleMATHGoogle Scholar
  33. Wang, Y., Cao, J., Alsaedi, A., Ahmad, B.: Edge-based SEIR dynamics with or without infectious force in latent period on random networks. Commun. Nonlinear Sci. Numer. Simul. 45, 35–54 (2017) MathSciNetView ArticleGoogle Scholar
  34. Wang, Y., Jin, Z., Yang, Z., Zhang, Z.-K., Zhou, T., Sun, G.-Q.: Global analysis of an SIS model with an infective vector on complex networks. Nonlinear Anal., Real World Appl. 13, 543–557 (2012) MathSciNetView ArticleMATHGoogle Scholar
  35. Yang, H.M.: Malaria transmission model for different levels of acquired immunity and temperature-dependent parameters (vector). Rev. Saude Publica 34, 223–231 (2000) View ArticleGoogle Scholar
  36. Zhao, X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003) View ArticleMATHGoogle Scholar

Copyright

© The Author(s) 2018

Advertisement