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A coinfection model of dengue and leptospirosis diseases
Advances in Difference Equations volume 2020, Article number: 664 (2020)
Abstract
In this paper an SIR deterministic mathematical model for coinfection of dengue and leptospirosis is proposed. We use a compartment model by using ordinary differential equations. The positivity of future solution of the model, the invariant region, and the stability of diseasefree equilibrium point as well as endemic equilibrium point are studied. To study the stability of the equilibria, a basic reproduction number is obtained by using the next generation matrix. The robustness of the model is also investigated. To identify the effect of each parameter on the expansion or control of the diseases, sensitivity analysis is performed. The effects of treating dengue infected only, leptospirosis infected only, and coinfected individuals have been identified by using the numerical simulation. Therefore, increasing the rate of recovery and decreasing the contact rate of dengue, leptospirosis, and their coinfection have a great influence in fighting dengue, leptospirosis, and their coinfection in the community.
Introduction
Dengue is the world’s fastest spreading mosquito transmitted infectious disease. An estimated 50 million dengue infections occur annually, and nearly 2.5 billion people live in regions with potential risk of dengue transmission [1–3]. Roughly half of the world’s population probably lives in regions ecologically suitable for dengue transmission [3]. Dengue is transmitted to humans by mosquitoes of Aedes species, which thrive around the globe in tropical and subtropical urban centers [1, 2]. The virus is transmitted by the infected female mosquito called Aedes aegypti [3]. There are four serotypes, DENV1–DENV4 [2, 3]. On the other hand, a zoonotic bacterial disease leptospirosis is one of the most significant neglected tropical bacterial diseases and one of the leading zoonotic causes of morbidity in the world [1, 4]. The most important reservoirs are small mammals, with large herbivores as additional important sources of infection. Pathogenic species of Leptospira were isolated from hundreds of mammalian species including bats and pinnipeds [5]. Leptospirosis is a global occurrence particularly in tropical and subtropical countries [5, 6]. It mainly affects vulnerable populations, with an estimated global annual incidence of 1.03 million people and 58.900 deaths [4, 6]. It is transmitted by contact with infected animal products and with water, wet surfaces contaminated with animal urine [6]. Leptospirosis and dengue coinfections (LDCI) have been reported in many countries [7–10]. Coinfections occur due to simultaneous transmission of both these infections during the rainy season, and both of these infections are characterized by fever with myalgia.
Mathematical modeling has a great role in describing the dynamics of infectious diseases in a community. Several scholars have developed different models for dengue, leptospirosis, and their coinfection with other diseases to study their dynamics. For example, many mathematical models have been developed to understand dengue [11–16] and leptospirosis [17–20]. Also some coinfection models for dengue and chikungunya [21, 22] and for dengue–Zika [23, 24] have been proposed and analyzed. However, to the best of our knowledge, no one has investigated coinfection of dengue and leptospirosis. Therefore, in this paper we are interested in filling this gap.
The paper is organized as follows: Sect. 2 is devoted to the description and formulation of the model. In Sect. 3, the analysis of dengue only model, leptospirosis only model, and the coinfection model is performed. In Sect. 4 numerical simulations to give a better interpretation of the analytical results are reported. Our discussion and conclusion are presented in Sect. 5.
Model description and formulation
In this paper, we consider a deterministic mathematical model for coinfection of dengue and leptospirosis in a human population. The total population is divided into seven subclasses, which are susceptible population (S), dengue infectious \((I_{d})\), leptospirosis infectious \((I_{l})\), dengue and leptospirosis coinfectious \((I_{dl})\), dengue recovered \((R_{d})\), leptospirosis recovered \((R_{l})\), and dengue leptospirosis coinfectious recovered \((R_{dl})\). Each vector population of both diseases is also divided into two susceptible vectors of dengue \((S_{vd})\), infected vector of dengue \((I_{vd})\), susceptible vector of leptospirosis \((S_{vl})\), and infected vector of leptospirosis \((I_{vl})\).
Susceptible are recruited with the rate of Π through birth or immigration, and their number increases from individuals that come from subclasses of dengue recovered, leptospirosis recovered, and coinfectious recovered by losing their temporary immunity at a rate of \(\sigma _{1}\), \(\sigma _{2}\), and \(\sigma _{3}\), respectively.
The entire susceptible population join \(I_{d}\) compartment, when individuals can get dengue with contact rate of \(\beta _{1}\) from a dengue only infected vectors. In a similar way, individuals can get leptospirosis by the contact rate of \(\alpha _{1}\) from a leptospirosis only infected or coinfected person with force of infection of leptospirosis \(\alpha _{1}(I_{l}+I_{dl})\) or by the contact rate \(\alpha _{2}\) from the infected vector of leptospirosis \((I_{vl})\) and join \(I_{l}\) compartment.
Dengue only infected individuals also get an additional leptospirosis infection by the contact rate \(\alpha _{1}\) from a leptospirosis only infected and coinfected person with force of infection of leptospirosis \(\alpha _{1}(I_{l}+I_{dl})\) or by the contact rate \(\alpha _{2}\) from the infected vector of leptospirosis \((I_{vl})\) and join coinfected compartment \((I_{dl})\). On the other hand, leptospirosis only infected individuals get an additional dengue infection by the contact rate \(\beta _{1}\) from the infected vector of dengue \((I_{vd})\) and join coinfected compartment \((I_{dl})\).
Dengue only infected individuals recover with the rate of \(\gamma _{1}\) and join dengue only recovered compartment \((R_{d})\). In a similar way, leptospirosis only infected individuals also recover with the rate of \(\gamma _{2}\) and join leptospirosis only recovered compartment \((R_{l})\). The coinfected compartment also recovers with ε rate, but those individuals either recover only from dengue disease and join dengue only recovered compartment with probability of εη, or recover only from leptospirosis only disease and join leptospirosis only recovered compartment with probability of \(\varepsilon k(1 \eta )\), or recover from both diseases and join coinfected recover compartment with probability of \(g=\varepsilon (1 k)(1\eta )\). In all compartments μ is a natural death rate of human population. Moreover, \(\delta _{1}\) is a dengue only caused death rate, and \(\delta _{2}\) is a leptospirosis caused death rate. Ω and Λ are the recruitment rates for vector population of dengue and leptospirosis, respectively. The leptospirosis and dengue vector populations have the natural death rate \(\mu _{vl}\) and \(\mu _{vd}\), respectively. The description of parameters of the model are found in Table 1.
With regards to the above considerations, we have the compartmental flow diagram shown in Fig. 1. From the flow chart, the model will be governed by the following system of differential equations:
with the initial condition \(S(0)=S_{0} \geq 0 \), \(I_{d}(0)=I_{d,0} \geq 0 \), \(I_{l}(0)=I_{l,0} \geq 0 \), \(I_{dl}(0)=I_{dl,0} \geq 0\), \(R_{d}(0)=R_{d,0} \geq 0\), \(R_{l}(0)=R_{l,0} \geq 0 \), \(R_{dl}(0)=R_{dl,0} \geq 0 \), \(S_{vd}(0)=S_{vd,0} \geq 0\), \(I_{vd}(0)=I_{vd,0} \geq 0\), \(S_{vl}(0)=S_{vl,0} \geq 0 \), \(I_{vl}(0)=I_{vl,0} \geq 0\).
Qualitative analysis
In this section, for simplification of the work, we split the full model into submodels, which are dengue and leptospirosis only models. The qualitative behavior of the submodel is studied first and the full model follows.
Dengue fever only model
To get this model from equation (1), we set \(I_{l}=I_{dl}=R_{l}=R_{dl}=0\), \(\sigma _{2}=\sigma _{3}=0\), \(I_{l}+I_{dl}=0\), and then we get
Invariant region
To get an invariant region, which shows as the solution is bounded, the total human population of the model is \(N_{h}=S+I_{d}+R_{d}\). Differentiating \(N_{h}\) both sides and substituting respective expressions of \(\frac{dS}{dt}\), \(\frac{dI_{d}}{dt}\), and \(\frac{dR_{d}}{dt}\) from equation (2), we get
If we are not considering death from dengue fever, then equation (3) becomes
Solving equation (4), we get
Similarly, for the vector population \(N_{vd}=S_{vl}+I_{vl}\), we have
Therefore, all the solution set of (2) is bounded in \(\mathcal{D}=\mathcal{D}_{hd}\times \mathcal{D}_{vd}\).
Positivity of solutions
Theorem 3.1
If the initial value \(S_{0} \geq 0 \), \(I_{d0} \geq 0 \), \(I_{dl0} \geq 0\), \(R_{d0} \geq 0 \), \(S_{vd0} \geq 0\), \(I_{vd0} \geq 0\), then the solutions of the dengue only infected model are nonnegative.
Proof
From the first equation of the system in equation (2)
which can be taken as
after evaluating equation (6), we obtain
Similarly, we obtain
Therefore, all the solution sets are nonnegative for \(t \geq 0\). □
Diseasefree equilibrium (DFEP)
By equating equation (2) to zero and substituting \(I_{d}=R_{d}=I_{vd}= 0\), we obtain DFEP \(E_{0d}=(\frac{\Pi }{\mu }, 0, 0 ,\frac{\Omega }{\mu _{vd}},0)\).
Basic reproduction number
The basic reproduction number is an average number of secondary cases of infection when a single infectious individual is introduced into the total susceptible population. To obtain (\(\mathcal{R}_{0d}\)), the next generation matrix method that was formulated by [25] is used, and it is given by
Local stability of DFEP
Theorem 3.2
DFEP is locally asymptotically stable if \(\mathcal{R}_{0d}<1\) and unstable if \(\mathcal{R}_{0d}>1\).
Proof
The Jacobian matrix of the dengue only model at DFEP is given by
The eigenvalues of the matrix in (8), −μ, \((\sigma _{1}+ \mu )\) and \(\mu _{vd}\) are clearly negative. The other two eigenvalues can be obtained from the quadratic equation given by
where
Here, it follows from Routh–Hurwitz criteria \(\psi _{1}>0\) and \(\psi _{2}>0\). Thus, all the eigenvalues of the matrix in equation (9) have negative real parts if \(\mathcal{R}_{0d} < 1\). Hence DFEP is locally asymptotically stable. □
Existence of endemic equilibrium point (EEP)
The endemic equilibrium point is denoted by \(E^{*}=(S^{*},I^{*}_{d},R^{*}_{d},S^{*}_{vd},I^{*}_{vd} )\) and it occurs when the disease persists in the community.
Lemma 3.3
For \(\mathcal{R}_{0d} > 1\) a unique EEP \(E^{*}\) exists and no EEP otherwise.
Proof
To obtain it, we equate all the model equations in equation (2) to zero. Then we obtain
where
□
Existence of backward bifurcation of dengue only model
We investigated the nature of the bifurcation by using the method introduced in [26], which is based on center manifold theory [26]. In order to apply theory, the following rearrangement and modification of variables are done on the submodel in equation (2). We let \(S = z_{1}\), \(I_{d} = z_{2}\), \(R_{d} = z_{3}\), \(S_{vd} = z_{4}\), and \(I_{vd}= z_{5}\). In addition, using vector notation \(z = (z_{1}, z_{2}, z_{3}, z_{4}, z_{5} )^{T}\), formulated as \(\frac{dz}{dt} = F(z)\), with \(F = ( f_{1}, f_{2}, f_{3}, f_{4}, f_{5})^{T}\) given in the following, we chose \(\beta _{1}\) as a bifurcation parameter and solve \(\mathcal{R}_{0d} = 1\), which leads to
where
The Jacobian matrix evaluated at DFEP is given by \(( z_{1}=\frac{\Pi }{\mu }, z_{2}=0; z_{3}=0 , z_{4}= \frac{\Omega }{\mu _{vd}}, z_{5}=0)\):
The right eigenvector \(u = (u_{1}, u_{2}, u_{3}, u_{4}, u_{5})^{T}\) associated with the simple zero eigenvalue can be obtained as follows:
Similarly, we obtain the left eigenvector \(v = (v_{1}, v_{2}, v_{3}, v_{4}, v_{5})\) associated with the simple zero eigenvalue and given by
Since the first, third, and fourth components of v are zero, we do not need the derivatives of \(f_{1}\), \(f_{3}\), and \(f_{4}\). From the derivatives of \(f_{2}\) and \(f_{5}\), the only ones that are nonzero are the following:
and all the other partial derivatives are zero. The direction of the bifurcation at \(\mathcal{R}_{0d} = 1\) is determined by the signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given respectively by
and
By the fact that coefficient b is positive, it shows that the sign of coefficient a determines the local dynamics around DFEP for \(\beta _{1}\). It is clearly seen that system (11) undergoes backward bifurcation, and therefore we propose the following theorem.
Theorem 3.4
The model in system (1) exhibits backward bifurcation at \(\mathcal{R}_{0d}=1\) and the diseasefree equilibrium may not be globally stable.
Leptospirosis only model
To get this model from equation (1), we set \(I_{d}=I_{dl}=R_{d}=R_{dl}=0\), \(\sigma _{1}=\sigma _{3}=0\), and then we get
Invariant region
To get an invariant region, the total population of the model is \(N_{hl}=S+I_{l}+R_{l}\). Then the derivative of \(N_{hl}\) with respect to time gives
If we are not considering death from leptospirosis, then equation (15) becomes
After solving equation (16), we get
Similarly, for the vector population \(N_{vl}=S_{vl}+I_{vl}\), we have
Therefore, all the solution set of the model in equation (14) is bounded in \(\mathcal{D}_{2}=\mathcal{D}_{l}\times \mathcal{D}_{vl}\).
Positivity of solutions
Theorem 3.5
If the initial value \(S_{0} \geq 0\), \(I_{l0} \geq 0 \), \(R_{l0} \geq 0 \), \(S_{vl0} \geq 0\), \(I_{vl0} \geq 0\) and the solutions of the leptospirosis only infected model are nonnegative.
Proof
To show solutions of the model are positive, first we take \(\frac{dS}{dt} \) from equation (14):
which can be rewritten as
Solving equation (18), we get
Similarly, we obtain
Therefore, all the solution sets are positive for \(t \geq 0\). □
Diseasefree equilibrium (DFEP)
Equating equation (14) to zero and substituting \(I_{l}= 0\), \(R_{l}= 0\), \(I_{vl}= 0\), we obtain the DFEP \(E_{0l}=(\frac{\Pi }{\mu }, 0, 0 ,\frac{\Lambda }{\mu _{vd}},0)\).
Basic reproduction number
The basic reproduction number (\(\mathcal{R}_{0l}\)) is obtained using the next generation matrix method that was formulated by [25], and it is given by
where \(\mathcal{R}_{0hl}= \frac{\alpha _{1}\Pi }{\mu (\mu +\delta _{2}+\gamma _{2})}\) and \(\mathcal{R}_{0vl}= \frac{\alpha _{2}\rho _{2}\Pi \Lambda }{\mu \mu ^{2}_{vl} (\mu +\delta _{2}+\gamma _{2})}\).
Local stability of DFEP
Theorem 3.6
DFEP is locally asymptotically stable if \(\mathcal{R}_{0l}<1\) and unstable if \(\mathcal{R}_{0l}>1\).
Proof
The Jacobian matrix at DFEP is given by
The eigenvalues −μ, \((\sigma _{2}+\mu )\) and \(\mu _{vl}\) are clearly negative. The other two eigenvalues can be obtained from the quadratic equation given by
where
We applied Routh–Hurwitz criteria, and by the principle, the quadratic equation has a strictly negative real root iff \(\psi _{1}>0\), \(\psi _{2}>0\), and \(\psi _{1}\psi _{2}>0\).
Hence DFEP is locally asymptotically stable if \(\Re _{0l}<1 \). □
Global stability of DFEP
Theorem 3.7
The equilibrium point \(E_{0} = (X^{*}, 0)\) of system (14) is globally asymptotically stable if \(\mathcal{R}_{l0} < 1\) and conditions (i) and (ii) from the theorem in [26] are satisfied.
Proof
To investigate the global stability of DFEP, we use the technique implemented by CastilloChavez and Song [26]. From equation (14), we can get \(F(X, Z)\) and \(G(X, Z)\), where \(X=(S,R_{l}, S_{vl})\in \mathcal{R}^{3}\) denotes uninfected populations and \(Z=(I_{l}, I_{vl})\in \mathcal{R}^{2}\) denotes infected populations:
and
Now consider the reduced system \(\frac{dX}{dt} = F(X,0)\):
\(X^{*} = (\frac{\Pi }{\mu },0 , \frac{\Lambda }{\mu _{vl}})\) is a globally asymptotically stable equilibrium point for the reduced system \(\frac{dX}{dt} = F(X,0)\). This can be verified. From the solution of the first equation in equation (21) we obtain \(S(t) = \frac{\Pi }{\mu }+(S(0)\frac{\Pi }{\mu })e^{\mu t} \), which approaches \(\frac{\Pi }{\mu }\) as \(t \longrightarrow \infty \); and from the second equation of equation (21) we get \(S_{vl}=\frac{\Lambda }{\mu _{vl}}+(S_{vl}(0) \frac{\Lambda }{\mu _{vl}})e^{\mu _{vl}t}\), which approaches \(\frac{\Omega }{\mu _{vd}}\) and \(\frac{\Lambda }{\mu _{vl}}\) as \(t \longrightarrow \infty \). We note that this asymptomatic dynamics is independent of the initial conditions in Ω, therefore the convergence of the solutions of the reduced system (21) is global in Ω. Now we compute
Then \(G(X,Z)\) can be written as
and we want to show \(\hat{G}(X;Z)\geq 0\), which is obtained as follows:
Here, \(\frac{ \Pi }{\mu }\geq S\) and \(\frac{\Lambda }{ \mu _{vl}}\geq S_{vl}\). Hence it is clear that equation \(\hat{G}(X,Z)\geq 0\) for all \((X,Z) \in \Omega \). Therefore, this proves that DFEP is globally asymptotically stable when \(\Re _{0l} < 1\). Hence the endemic equilibrium point does not coexist with the diseasefree equilibrium point when \(\mathcal{R}_{0l}<1\). This implies that the model exhibits forward bifurcation at \(\mathcal{R}_{0l}=1\). From this we can say that the disease can be eliminated from the population irrespective of the initial infectious population. □
Endemic equilibrium (EEP)
The endemic equilibrium is denoted by \(E^{*}_{p}=(S^{*},I^{*}_{l},R^{*}_{l},S^{*}_{vl},I^{*}_{vl} )\), and it occurs when the disease persists in the community.
Lemma 3.8
A unique endemic equilibrium exists if one of the following holds true:

(i)
if \(C < 0 \Longleftrightarrow \mathcal{R}_{0l} > 1\);

(ii)
if \(B<0\) and \(C=0\) or \(B^{2} 4AC=0\).
Proof
To obtain it, we equate all the model equations (14) to zero. Then we obtain
When we substitute \(S^{*}\) in the first equation of the model in (14), we get
where
It is important to note that the coefficient A is positive with B and C having different signs. There is unique EE whenever condition \((i)\) or \((ii)\) holds true. Also the other condition we did not mention is that there are two endemic equilibria if \(C>0\), \(B<0\), and \(B^{2}4AC>0\). However, this does not happen because we proved that DEFP of leptospirosis only model is globally asymptotically stable, which means that there is no way that the endemic equilibrium point coexists with DFEP when \(\mathcal{R}_{0l}<1\). □
Dengue–leptospirosis coinfection model
The model in equation (1) is dengue and leptospirosis coinfection equation with \(S(0)=S_{0} \), \(I_{d}(0)=I_{d0} \), \(I_{l}(0)=I_{l0} \), \(I_{dl}(0)=I_{dl0}\), \(R_{d}(0)=R_{d0} \), \(R_{l}(0)=R_{l0}\), \(R_{dl}(0)=R_{dl0} \), \(S_{vd}(0)=S_{vd0}\), \(I_{vd}(0)=I_{vd0}\), \(S_{vl}(0)=S_{vl0} \), \(I_{vl}(0)=I_{vl0} \) being nonnegative initial values.
Invariant region
In this section, we obtain a region in which the solution of (1) is bounded. For this model the total human population is \(N=S+I_{d}+I_{l} +I_{dl}+R_{d}+R_{l} +R_{dl}\). Then, after differentiating N with respect to time and substituting each equation from equation (1) to the respective place, we get
If we do not consider death from leptospirosis, dengue, and the coinfection, then equation (25) becomes
After solving equation (26) and evaluating it as time tends to infinity, we get
Similarly, for vector population of dengue, if there is no discharge of virus from the infectious, then
Also, for vector population of leptospirosis, if there is no discharge of bacteria from the infectious, then
Therefore, the feasible solution set for the dengue–leptospirosis coinfection model is given by
Therefore, all the solution set of equation (1) is bounded in \(\mathcal{D}=\mathcal{D}_{h}\times {\mathcal{D}_{vl}\times \mathcal{D}_{vl}}\).
Positivity of solutions
Theorem 3.9
If \(S_{0} > 0\), \(I_{d0} > 0 \), \(I_{l0} > 0 \), \(I_{dl0} > 0\), \(R_{d0} > 0 \), \(R_{l0} > 0\), \(R_{dl0} > 0\), \(S_{vd0} > 0\), \(I_{vd0} > 0\), \(S_{vl0} > 0\), \(I_{vl0} > 0\), then the future time solutions of the dengue–leptospirosis coinfection population are positive.
Proof
We let \(\tau =sup\{ t>0: S_{0}(t_{1}) >0,I_{d0}(t_{1}) > 0 ,I_{l0}(t_{1}) > 0 ,I_{dl0}(t_{1}) >0,R_{d0}(t_{1}) >0 , R_{l0}(t_{1}) > 0, R_{dl0}(t_{1}) > 0,S_{vd0}(t_{1}) > 0, I_{vd0}(t_{1}) > 0,S_{vl0}(t_{1}) > 0 , I_{vl0}(t_{1}) > 0\text{ for all }t_{1} \in [0,t] \}\). Since \(S_{0} \geq 0\), \(I_{d0} \geq 0 \), \(I_{l0} \geq 0 \), \(I_{dl0} \geq 0\), \(R_{d0} \geq 0 \), \(R_{l0} \geq 0\), \(R_{dl0} \geq 0\), \(S_{vd0} \geq 0\), \(I_{vd0} \geq 0\), \(S_{vl0} \geq 0\), and \(I_{vl0} \geq 0\), hence \(\tau >0\). If \(\tau <\infty \), then \(S_{0}(t) \), \(I_{d0}(t)\), \(I_{l0}(t) \), \(I_{dl0}(t) \), \(R_{d0}(t)\), \(R_{l0}(t)\), \(R_{dl0}(t)\), \(S_{vd0}(t)\), \(I_{vd0}(t)\), \(S_{vl0}(t)\), \(I_{vl0}(t)\) are zero at \(t_{1}\). To show that solution of the model is positive, first we take \(\frac{dS}{dt}\) from Eq. (1).
Using the variation of constants formula, the solution of equation (30) at τ is given by
Moreover, since all the variables are positive in \([0,\tau ]\), then \(S(\tau )>0\).
It can be shown in a similar way that \(I_{d}(\tau ) > 0 \), \(I_{l}(\tau ) > 0 \), \(I_{dl}(\tau ) > 0\), \(R_{d}(\tau ) > 0 \), \(R_{l}(\tau ) > 0\), \(R_{dl}(\tau ) > 0\), \(S_{vd}(\tau ) > 0\), \(I_{vd}(\tau ) > 0\), \(S_{vl}(\tau ) > 0 \), and \(I_{vl}(\tau ) > 0\), which is a contradiction. Hence \(\tau =\infty \). □
Diseasefree equilibrium (DFEP)
Equating equation (1) to zero and substituting \(I_{d}= 0\), \(I_{l}= 0\), \(I_{dl}= 0\), \(R_{d}= 0\), \(R_{l}= 0\), \(R_{dl}= 0\), \(I_{vd}= 0\), \(I_{vl}= 0\), we obtain the DFEP \(E_{0}=(\frac{\Pi }{\mu }, 0,0,0,0,0,0 ,\frac{\Omega }{\mu _{vd}},0, \frac{\Lambda }{\mu _{vl}},0)\).
Basic reproduction number
To obtain (\(\mathcal{R}_{0}\)), we use the next generation matrix method [25]. Let us consider the infective compartments of the model
Then, by the principle of nextgeneration matrix, we obtain \(\mathcal{F}\) and \(\mathcal{V}\), and evaluating the Jacobians at DFEP, we get
where
Then the basic reproduction number from the product matrix \(\mathcal{FV}^{1}\) is the maximum of the eigenvalues listed as follows:
Therefore, the basic reproduction number of the coinfection is given by
Local stability of DFEP
Theorem 3.10
DFEP is locally asymptotically stable if \(\mathcal{R}_{0}<1\) and unstable if \(\mathcal{R}_{0}>1\).
Proof
The Jacobian matrix of the model at DFEP is obtained as follows:
From equation (34), we can get the following characteristic polynomial:
Therefore, DFEP is locally asymptotically stable iff \(\mathcal{R}_{0}=\max \{\mathcal{R}_{0d}, \mathcal{R}_{0l}\} <1 \). □
Global stability of DFE
Theorem 3.11
The equilibrium point \(E_{0} = (X^{*}, 0)\) of system (1) is globally asymptotically stable if \(\mathcal{R}_{0} \leq 1\) and conditions (i) and (ii) in the theorem [26] are satisfied.
Proof
We use the technique implemented by CastilloChavez and Song [26]. From equation (1), we can get \(F(X, Z)\) and \(G(X, Z)\), where \(X=(S,R_{d},R_{l},R_{dl},S_{vd}, S_{vl})\in \mathcal{R}^{6}\) denotes uninfected populations and \(Z=(I_{d}, I_{l},I_{dl}, I_{vd}, I_{vl})\in \mathcal{R}^{5}\) denotes infected populations.
and
Now consider the reduced system \(\frac{dX}{dt} = F(X,0)\):
\(X^{*} = (\frac{\Pi }{\mu },\frac{\Omega }{\mu _{vd}} , \frac{\Lambda }{\mu _{vl}})\) is a globally asymptotically stable equilibrium point for the reduced system \(\frac{dX}{dt} = F(X,0)\). This can be verified. From the solution of the first equation in equation (35) we obtain \(S(t) = \frac{\Pi }{\mu }+(S(0)\frac{\Pi }{\mu })e^{\mu t} \), which approaches \(\frac{\Pi }{\mu }\) as \(t \longrightarrow \infty \); and from the second and third equation of equation (35) we get \(S_{vd}=\frac{\Omega }{\mu _{vd}}+(S_{vd}(0)\frac{\Omega }{\mu _{vd}})e^{ \mu _{vd}t}\) and \(S_{vl}=\frac{\Lambda }{\mu _{vl}}+(S_{vl}(0) \frac{\Lambda }{\mu _{vl}})e^{\mu _{vl}t}\), which approaches \(\frac{\Omega }{\mu _{vd}}\) and \(\frac{\Lambda }{\mu _{vl}}\) as \(t \longrightarrow \infty \). We note that this asymptomatic dynamics is independent of the initial conditions in Ω, therefore the convergence of the solutions of the reduced system (35) is global in Ω. Now we compute
Then \(G(X,Z)\) can be written as
and we want to show \(\hat{G}(X;Z)\geq 0\), which is obtained as follows:
In equation (36), \(\hat{G}_{3}(X,Z)<0\), which leads to \(\hat{G}(X,Z)< 0\) for all \((X,Z) \in \Omega \). Therefore, this proves that DFEP may not be globally asymptotically stable when \(\mathcal{R}_{0} < 1\). Hence the endemic equilibrium point coexists with DFEP when \(\mathcal{R}_{0}<1\). This implies that the model exhibits backward bifurcation at \(\mathcal{R}_{0}=1\). □
Sensitivity analysis
Sensitivity is performed to identify the most dominant parameters for the spreading out as well as control of infection in the community. To go through sensitivity analysis, we follow the technique described in [27]. The sensitivity index of \(\mathcal{R}_{0}\) with respect to a parameter, say y, is given by
Since \(\mathcal{R}_{0}=\max \{\mathcal{R}_{0d}, \mathcal{R}_{0l}\}\), we obtain the sensitivity analysis of \(\mathcal{R}_{0d}\) and \(\mathcal{R}_{0l}\) separately in the following way:
The above computation of sensitivity analysis is summarized in Table 2.
From Table 2, we understand that the parameters with positive sensitivity indices, particularly \(\beta _{1}\), \(\rho _{1}\), c, \(\alpha _{1}\), \(\alpha _{2}\), and \(\rho _{2}\), have great potential in expanding dengue, leptospirosis, and their coinfection in the community. However, the parameters with negative indices, particularly \(\delta _{1}\), \(\gamma _{1}\), \(\delta _{2}\), \(\gamma _{2}\), μ, \(\mu _{vd}\), and \(\mu _{vl}\), have a great contribution in controlling the expansion of dengue, leptospirosis, and their coinfection in the community if their values are increased. From this policy makers and stakeholders are expected to act accordingly in combating dengue,leptospirosis, and their coinfection in the community.
Numerical simulations
Analytical results cannot be complete without numerical result verification. In this section, the full dengue–leptospirosis coinfection model numerical simulation is performed using Maple 18. The simulation is used for checking the effect of some parameters in the expansion as well as the control of dengue only, leptospirosis only, and coinfection of dengue and leptospirosis. For simulation purpose the parameter values in Table 3 are used.
Effect of recovery rate \((\gamma _{1})\) on dengue infectious population
In Fig. 2, we experimented with the effect of \(\gamma _{1} \) in reducing dengueonly infectious population by maintaining the contact rate \((\beta _{1})\) constant. The figure reflects that when the values of \(\gamma _{1}\) increase, the number of dengue only infectious population is going down. From this we should concentrate on improving recovery rates either by treating infected populations or by raising individuals’ immunity levels to dengue disease in the population. It should be viewed by policy makers as a mitigation strategy.
Effect of recovery rate \((\gamma _{2})\) on leptospirosis infectious population
In Fig. 3, it is shown that \(\gamma _{2}\) plays a significant role in bringing down the leptospirosis infection. When the value of \(\gamma _{2}\) increased from 0.017 to 0.087, the amount of infectious population due to leptospirosis decreased, where the contact rate is kept constant, which is \(\alpha _{1}=\alpha _{2} \). It can also be used by policy makers as a tool for mitigation.
Effect of dengue contact rate \((\beta _{1})\) on coinfectious population
As it is shown in Fig. 4, the contact rate of leptospirosis (\(\alpha _{1}=\alpha _{2}\)) and the recovery rate of coinfectious population (ε) are kept constant. The figure reflects that as the value of contact rate of dengue increased, the coinfectious population increased, which means increased expansion of coinfection. From this we can see that decreasing of the dengue contact rate is significant in the controlling of coinfection transmission. Therefore, stakeholders are expected to work on decreasing the contact rate of susceptible humans and dengue vector by using either bednet or chemical or using an appropriate method of prevention mechanism to bring down the expansion of coinfection in the community.
Effect of leptospirosis contact rate (\(\alpha _{1} \) and \(\alpha _{2}\)) on coinfectious population
Similarly, we investigated the effect of leptospirosis contact rate (\(\alpha _{1}=\alpha _{2}\)) in the expansion of dengue–leptospirosis coinfection while keeping the recovery rate of coinfection (ε) constant. Figure 5 shows that coinfectious population decreases as the leptospirosis contact rate is decreasing, by keeping dengue contact rate (\(\beta _{1}\)) and (ε) constant. This implies that, in order to mitigate the coinfection, it is advisable to reduce the rate of contact with infectious humans, and that the host vector of leptospirosis is crucial.
Effect of recovery rate of dengue–leptospirosis \((\varepsilon )\) on coinfectious population
Here, we experimented on the effect of recovery rate of dengue and leptospirosis (ε) on the coinfectious population. As we explained in the model description, due to treatment or other mechanisms, coinfectious population either recover from dengue only or from leptospirosis only or from both diseases with their own probability and join their respective recovered compartment. Therefore, Fig. 6 shows that increasing the rate of recovery of the coinfectious population has a great advantage in reducing both diseases in the population.
Conclusions
The deterministic coinfection model for dengueleptospirosis disease was developed using ordinary differential equations, and the population is subdivided into eleven compartments. The qualitative analysis of the model was done by splitting the full model into two, which are dengue only and leptospirosis only models. The analysis of the model shows that there exists a region where the model is mathematically and epidemiologically well posed. Basic reproduction numbers, diseasefree equilibrium, endemic equilibria, stability analysis of equilibrium points, and sensitivity analysis of basic reproduction of dengue only, leptospirosis only, and the full model were analyzed in their respective order. Numerically, we experimented on the effect of basic parameters in the expansion or control of dengue only, leptospirosis only, and coinfectious diseases. From the result, we conclude that an increase in the rate of dengue recovery contributes greatly to reducing dengue infection in the community. Similarly, increasing the recovery rate for leptospirosis also contributes to the reduction of leptospirosis. The rate of recovery for coinfection also has an effect on reducing coinfectious population if its value has been increased. The other finding obtained in this section is that decreasing either dengue or leptospirosis contact rate has a great influence on controlling coinfection of dengue and leptospirosis in the population.
Availability of data and materials
The data supporting this deterministic model are from previously published articles.
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Acknowledgements
I would like to express my heartfelt appreciation to University of Gondar for financial support and the reviewers for their valuable comments, which improved the paper.
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Alemneh, H.T. A coinfection model of dengue and leptospirosis diseases. Adv Differ Equ 2020, 664 (2020). https://doi.org/10.1186/s13662020031266
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Keywords
 Confection
 Dengue fever
 Leptospirosis
 Stability analysis
 Numerical simulation