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Abstract: Host migration among discrete geographical regions is demonstrated as an important factor that brings about the diffusion and outbreak of many vector–host diseases. In the paper, we develop a mathematical model to explore the effect of host migration between two patches on the spread of a vector–host disease. Analytical results show that the reproduction number provides a threshold condition that determines the uniform persistence and extinction of the disease. If both the patches are identical, it is shown that an endemic equilibrium is locally stable. It is also shown that a unique endemic equilibrium, which exists when the disease cannot induce the death of the host, is globally asymptotically stable. Finally, two examples are given to illustrate the effect of host migration on the spread of the vector–host disease. [Copyright &y& Elsevier]

In this paper, we propose a mathematical model to describe the transmission dynamics of infectious diseases with targeted antiviral prophylaxis strategy. Our model incorporates seasonal driving force since seasonal force has a great effect on the spread of infectious diseases. Based on the local stability of disease free equilibrium we derive the control reproduction number $\mathcal R_0$. Sufficient conditions for the global stability of the disease free equilibrium are obtained. Using the persistence theory for discrete dynamical system, we prove that the infectious disease will remain endemic if \mathcal R_0>1$. Simulation results are also provided to study the effect of targeted antiviral prophylaxis on transmission dynamics of infectious disease and investigate the influence of seasonality on the efficiency of targeted antiviral prophylaxis strategy. [ABSTRACT FROM AUTHOR]

Due to the increasing risk of drug resistance and side effects with large-scale antiviral use, it has been suggested to provide antiviral drugs only to susceptibles who have had contacts with infectives. This antiviral distribution strategy is referred to as 'targeted antiviral prophylaxis'. The question of how effective this strategy is in infection control is of great public heath interest. In this paper, we formulate an ordinary differential equation model to describe the transmission dynamics of infectious disease with targeted antiviral prophylaxis, and provide the analysis of dynamical behaviours of the model. The control reproduction number Rc is derived and shown to govern the disease dynamics, and the stability analysis is carried out. The local bifurcation theory is applied to explore the variety of dynamics of the model. Our theoretical results show that the system undergoes two Hopf bifurcations due to the existence of multiple endemic equilibria and the switch of their stability. Numerical results demonstrate that the system may have more complex dynamical behaviours including multiple periodic solutions and a homoclinic orbit. The results of this study suggest that the possibility of complex disease dynamics can be driven by the use of targeted antiviral prophylaxis, and the critical level of prophylaxis which achieves Rc=1 is not enough to control the prevalence of a disease. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a nine-dimensional sexually transmitted disease model proposed by Castillo-Chavez et al. [C. Castillo-Chavez, Wenzhang Huang and Jia Li, The effects of female’s susceptibility on coexistence of multiple pathogen strains of sexually transmitted diseases, J. Math. Biol. 35 (1996) 503–522] is studied. The model involves two competing strains 1 and 2 in a two-sex heterosexually active population that includes a single group of males and two different groups of females. The first and second reproduction numbers , are defined for strain respectively. By applying the theory of type- monotone dynamical systems, a complete classification for the dynamics of this model is presented in terms of the first and second reproduction numbers . The classification not only is different to the complete classification given by Castillo-Chavez et al. [C. Castillo-Chavez, Wenzhang Huang and Jia Li, Competitive exclusion and coexistence of multiple strains in an SIS STD model, SIAM J. Appl. Math. 59 (1999) 1790–1811], but also can be easily explained in biology. Our results also show that the dynamics of the model is completely determined by the ubiety of two functions. [Copyright &y& Elsevier]

Abstract: In the paper, we propose a model that tracks the dynamics of many diseases spread by vectors, such as malaria, dengue, or West Nile virus (all spread by mosquitoes). Our model incorporates demographic structure with variable population size which is described by nonlinear birth rate and linear death rate. The stability of the system is analyzed for the existence of the disease-free and endemic equilibria points. We find the basic reproduction number in terms of measurable epidemiological and demographic parameters is the threshold condition that determines the dynamics of disease infection: if the disease fades out, and for the disease remains endemic. The threshold condition provides important guidelines for accessing control of the vector diseases, and implies that it is an efficient way to halt the spread of vector epidemic by reducing the carrying capacity of the environment for the vector and the host. Moreover, sufficient conditions are also obtained for the global stability of the unique endemic equilibrium . [Copyright &y& Elsevier]