Bifurcation analysis in models for vector-borne diseases with logistic growth.

Guihua Li,Zhen Jin

doi:10.1155/2014/195864

Abstract

We establish and study vector-borne models with logistic and exponential growth of vector and host populations, respectively. We discuss and analyses the existence and stability of equilibria. The model has backward bifurcation and may have no, one, or two positive equilibria when the basic reproduction number R 0 is less than one and one, two, or three endemic equilibria when R 0 is greater than one under different conditions. Furthermore, we prove that the disease-free equilibrium is stable if R 0 is less than 1, it is unstable otherwise. At last, by numerical simulation, we find rich dynamical behaviors in the model. By taking the natural death rate of host population as a bifurcation parameter, we find that the system may undergo a backward bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, and cusp bifurcation with the saturation parameter varying. The natural death rate of host population is a crucial parameter. If the natural death rate is higher, then the host population and the disease will die out. If it is smaller, then the host and vector population will coexist. If it is middle, the period solution will occur. Thus, with the parameter varying, the disease will spread, occur periodically, and finally become extinct.

Highlights

The dynamical modeling and studies of vector-borne disease have been in many literatures
Dengue fever, and West Nile virus, the three typical vector-borne diseases, the dynamics of most of the related models can be characterized by the reproduction number, but some compartmental models for these three mosquito-borne diseases undergo a backward bifurcation, an important feature of mosquitoborne diseases
After R0 is more than a threshold, the two equilibria will disappear, and all of trajectories tend to equilibrium E(Sh, Ih, Rh, SV, IV) = (0, 0, 0, NV0, 0), which means that the host population will be extinct If we fix R0 and let dh change, we find that Hopf bifurcation maybe occurs from the blue curve of Figure 4 for system (13)

Summary

Introduction

The dynamical modeling and studies of vector-borne disease have been in many literatures. For West Nile virus (WNV), there are many literatures, [13,14,15,16,17,18], and so forth. Dengue fever, and West Nile virus, the three typical vector-borne diseases, the dynamics of most of the related models can be characterized by the reproduction number, but some compartmental models for these three mosquito-borne diseases undergo a backward bifurcation, an important feature of mosquitoborne diseases. For compartmental models for malaria, dengue fever, and WNV, different forms of birth functions have been used; we summarize and find that the logistic birth function which describes the self-limiting growth of a host population was rarely considered. We will build the following model and analyse the dynamic behaviors: dSh dt

Disease Malaria

Stability and Bifurcation Analysis

Findings

Discussion