New mathematical model of vertical transmission and cure of vector-borne diseases and its numerical simulation
- Abdullah^{1},
- Aly Seadawy^{2, 3}Email author and
- Wang Jun^{1}
https://doi.org/10.1186/s13662-018-1516-z
© The Author(s) 2018
Received: 29 November 2017
Accepted: 6 February 2018
Published: 21 February 2018
Abstract
In this research article, a new mathematical model for the transmission dynamics of vector-borne diseases with vertical transmission and cure is developed. The non-negative solutions of the model are shown. To understand the dynamical behavior of the epidemic model, the theory of basic reproduction number is used. As this number increases, the disease invades the population and vice versa. The effect of vertical transmission and cure rate on the basic reproduction number is shown. The disease-free and endemic equilibria of the model are found and both their local and global stabilities are presented. Finally, numerical simulations are carried out graphically to show the dynamical behaviors. These results show that vertical transmission and cure have a valuable effect on the transmission dynamics of the disease.
Keywords
1 Introduction
Vector-borne diseases are infectious diseases transmitted to humans and animals by blood-feeding arthropods. Some common vector-borne diseases are West Nile virus, dengue fever, Rift Valley fever, malaria, and viral encephalitis caused by pathogens such as bacteria, viruses, and parasites. The arthropods are blood sucking insects and arachnids such as ticks, mosquitoes, biting flies, and lice called vectors [1]. The vectors receive pathogens from an infected host and transmit them to a human host, as humans are the major host, or animals. However, direct transmissions, such as transplantation related transmission, transfusion related transmission, and needle-stick-related transmission, are also possible [2]. In case of some diseases such as AIDS and Hepatitis B, it is possible for the offspring of infected parents to be born infected. This type of transmission is called vertical transmission. Now it is found that vector-borne diseases can also be transmitted vertically [3, 4]. Also new research shows that virus is transmitted from female mosquitos to their eggs at a high rate [5], which causes vertical transmission of the disease.
Vector-borne diseases are prevalent in hot areas, such as tropics and subtropics, and are relatively rare in temperate zones. Vector-borne infectious diseases remain amongst the most important cause of global health illness and are major killers, particularly of children. The World Health Organization reports the numbers of deaths in different regions of the world annually. Nearly 700 million people get mosquito-borne illnesses that cause about one million deaths each year. Worldwide, malaria is the leading cause of premature mortality, particularly in children under the age of five. Nearly half of the world’s population is at risk of malaria, and every year 198 million cases (uncertainty range: 124–283 million) and 584,000 deaths (range: 367,000–755,000) occur according to the World Malaria Report 2014 [6]. According to WHO, an estimated of 3.3 billion people in 97 countries are at risk of malaria. Currently, dengue threatens up to 40% of the world’s population, and there may be 50–100 million infections annually [7]. More than 2.5 billion people over 40% of the world’s population are now at risk of dengue.
From the above discussion it is clear that it is necessary to control such epidemic diseases. Control measures for vector-borne diseases are important because most are zoonoses. For the control measure, it is necessary to understand the dynamical features of diseases and treat the infected hosts. Therefore, deciphering the mechanisms and modeling of such diseases are of great interest. Our paper involves such an epidemic model for the transmission dynamics of vector-borne diseases that incorporates both horizontal and vertical transmission in the vector–host population.
Up to date, many mathematical models have been investigated to understand the mechanism of real world phenomena. Researchers investigate different methods to solve these models both analytically and numerically (e.g., see [8–21]). Several models of infectious diseases have been developed in the literature [22–27]. The model first proposed by Ross [28] and subsequently modified by Macdonald [29] has influenced both the modeling and the application of control strategies to a vector-borne disease. The model presented in [30] studied the analysis of a simple vector–host epidemic model with horizontal transmission. We extend their model by including vertical transmission in both vector and host populations, and treatment class in the host population with different interaction rates.
The structure of this paper is as follows: Section 1 represents the introductory remarks with a brief history. Section 2 is about the derivation of SITR epidemic model and shows the non-negative solutions of the proposed model. In Section 3, we find the disease-free and endemic equilibria and prove their local stability. In Section 4, we use mathematical analysis to establish global stability results for the proposed model. We use Lyapunov function theory to show global stability of both disease-free and endemic equilibria. Parameter estimation and numerical results are discussed in Section 5. Finally, we give conclusion.
2 Model framework
2.1 Properties of solutions
The proposed model (1) is a system of nonlinear ordinary differential equations with the initial conditions (2). To be epidemiologically and mathematically meaningful, it is important to prove that all the solutions with the given initial conditions will remain non-negative and bounded for all finite time. The model shall be analyzed in a biologically meaningful feasible region governed by a positive invariant set.
Theorem 2.1
There exists a unique and bounded solution of the system of equations (1), in a positively invariant set, that remains for all finite time \(t\geq0\).
Proof
The right-hand side of each equation is continuous in the convex domain \(E=(t,S(t),I(t), T(t),R(t),V(t),W(t))\) of \((6+1)\)-dimensional space \(R_{+}^{6+1}\) with continuous partial derivatives. So problem (1) has a unique solution in \(R_{+}^{6}\) which exists for a given finite time \(t\in[0,\infty)\) and initial conditions (2).
The above theorem shows that model (1) is well posed epidemiologically and mathematically in a positively invariant set Φ. We shall study the dynamics of this basic model in Φ, so, all the solutions of system (1) start and remain in Φ for all \(t\geq0\). All the parameters and state variables for the model should be non-negative for all time because they represent the number of the population sizes of humans and vectors.
3 Equilibrium points
3.1 Disease-free equilibrium
Theorem 3.1
The disease-free equilibrium point \(E_{1}\) is locally asymptotically stable if \(R_{0}<1\), otherwise unstable.
Proof
3.2 Endemic equilibrium
Theorem 3.2
The endemic equilibrium point \(E_{2}\) is locally asymptotically stable if \(R_{0} > 1\), otherwise unstable.
Proof
4 Global stability analysis
In this section, we study the global analysis of the disease-free and endemic equilibria using the direct Lyapunov method which requires the construction of a function with specific properties. In order to do this, we derive the following results.
Theorem 4.1
When \(R_{0}<1\), then the disease-free equilibrium \(E_{1}\) of system (1) is globally asymptotically stable on Φ.
Proof
Theorem 4.2
For \(R_{0}>1\), the endemic equilibrium \(E_{2}\) is globally asymptotically stable.
Proof
5 Numerical simulation and graphs
6 Conclusion
The spread of different infectious diseases causes very high mortality rates in a population. Vector-borne diseases are infectious diseases transmitted to humans and animals through vectors. These diseases propagate from the infected to the susceptible population in different ways. This paper formulated an epidemic model for the transmission dynamics of vector-borne diseases with both vertical and horizontal transmissions with treatment strategy. The equilibrium points and the basic reproduction of the model are found. The basic reproduction number, which is a threshold quantity, has an important role in the epidemiology of the disease. As this number increases the disease invades the population, and as it decreases the disease simply dies out. Figure 2 shows that \(R_{0}\) decreases as treatment strategies increase and increases as vertical transmission increases. Figure 3 shows the threshold behavior of \(R_{0}\) and the critical value \(R_{0}=1\). As \(R_{0}\) increases, the infected population increases with time. For \(R_{0}<1\), the number of infected population decreases; for \(R_{0}=1\), the infected population remains constant; and for \(R_{0}>1\), the number of infected population increases. It is also shown that when \(R_{0}<1\) the disease-free equilibrium is locally and globally asymptotically stable; and for \(R_{0}>1\), the positive endemic equilibrium is locally and globally asymptotically stable.
Numerical simulations are carried out graphically to show the dynamical behavior of the diseases. Figure 5 shows the effect of cure rate on the transmission dynamics of the disease. As treatment strategy increases, the susceptible population and the recovered human population increase while the infected population decreases. Figure 6 shows the effect of vertical transmission. As vertical transmission increases, the susceptible population decreases and the infected population increases. Finally, Figures 7 and 8 show the phase portraits of the susceptible populations versus the infected populations which move towards the stable points.
Declarations
Acknowledgements
This work was supported by NNSFC (Grants 11571140, 11671077), Fellowship of Outstanding Young Scholars of Jiangsu Province (BK20160063), the Six Big Talent Peaks Project in Jiangsu Province (XYDXX-015), and NSF of Jiangsu Province (BK20150478).
Authors’ contributions
All authors 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
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