Xamxinur Abdurahman

Resonance of the epidemic threshold in a periodic environment

Resonance between some natural period of an endemic disease and a seasonal periodic contact rate has been the subject of intensive study. This paper does not focus on resonance for endemic diseases but on resonance for emerging diseases. Periodicity can have an important impact on the initial growth rate and therefore on the epidemic threshold. Resonance occurs when the Euler–Lotka equation has a complex root with an imaginary part (i.e., a natural frequency) close to the angular frequency of the contact rate and a real part not too far from the Malthusian parameter. This is a kind of continuous-time analogue of work by Tuljapurkar on discrete-time population models, which in turn was motivated by the work by Coale on continuous-time demographic models with a periodic birth. We illustrate this resonance phenomenon on several simple epidemic models with contacts varying periodically on a weekly basis, and explain some surprising differences, e.g., between a periodic SEIR model with an exponentially distributed latency and the same model but with a fixed latency.

Persistence and Extinction for General Nonautonomous n-Species Lotka–Volterra Cooperative Systems with Delays

The general non-autonomous n-species LotkaVolterra cooperative systems with varying-time delays are studied. Sufficient conditions for the uniform strong persistence, uniform weak average persistence, uniform strong average persistence, and extinction of populations are obtained by applying the method of Liapunov functionals. Particularly, when the intrinsic growth rate of a species is non-positive, the species can be persistent or extinct under the same given assumptions.

Global Stability of HIV-1 Infection Model with Two Time Delays

We investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in vivo. The model has two time delays describing time needed for infection of cell and CTLs generation. Our model admits three possible equilibria: infection-free equilibrium, CTL-absent infection equilibrium, and CTL-present infection equilibrium. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied.

On the persistence of a nonautonomous n-species Lotka-Volterra cooperative system

In this paper a special nonautonomous n-species Lotka-Volterra cooperative system is studied. Sufficient conditions for the uniform strong persistence, uniform weak average persistence and uniform strong average persistence of population are obtained.

Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington–DeAngelis incidence and three delays

ABSTRACT In this paper, the dynamical behaviours for a five-dimensional virus infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and Beddington–DeAngelis incidence are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization method, the threshold conditions on the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and interior, respectively, are established. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are presented to justify the analytical results.

Global stability of a diffusive and delayed virus infection model with general incidence function and adaptive immune response

In this paper, the dynamical behaviors for a five-dimensional virus infection model with diffusion and two delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and a general incidence function are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization methods, the threshold conditions on the global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and antibody and CTL responses, respectively, are established if the space is assumed as homogeneous. When the space is inhomogeneous, the effects of diffusion, intracellular delay and production delay are obtained by the numerical simulations.

Global Dynamics of a Discretized Heroin Epidemic Model with Time Delay

We derive a discretized heroin epidemic model with delay by applying a nonstandard finite difference scheme. We obtain positivity of the solution and existence of the unique endemic equilibrium. We show that heroin-using free equilibrium is globally asymptotically stable when the basic reproduction number , and the heroin-using is permanent when the basic reproduction number .