Abstract
In this paper, we establish a discrete-time analog for coupled within-host and between-host systems for an environmentally driven infectious disease with fast and slow two time scales by using the non-standard finite difference scheme. The system is divided into a fast time system and a slow time system by using the idea of limit equations. For the fast system, the positivity and boundedness of the solutions, the basic reproduction number and the existence for infection-free and unique virus infectious equilibria are obtained, and the threshold conditions on the local stability of equilibria are established. In the slow system, except for the positivity and boundedness of the solutions, the existence for disease-free, unique endemic and two endemic equilibria are obtained, and the sufficient conditions on the local stability for disease-free and unique endemic equilibria are established. To return to the coupling system, the local stability for the virus- and disease-free equilibrium, and virus infectious but disease-free equilibrium is established. The numerical examples show that an endemic equilibrium is locally asymptotically stable and the other one is unstable when there are two endemic equilibria.
Highlights
As is well known, viruses have caused abundant types of epidemic and occur almost everywhere on Earth, infecting humans, animals, plants, and so on
Many authors have established and investigated the various kinds of viral infection dynamical systems which are described by differential equations and difference equations
We propose a discrete-time analog for above continuous-time system (2) by using discretization method of Micken’s non-standard finite difference scheme, the model is given as follows:
Summary
Viruses have caused abundant types of epidemic and occur almost everywhere on Earth, infecting humans, animals, plants, and so on. 3. In this paper, for fast system (6) we will investigate the dynamical behaviors, including the positivity, boundedness, basic reproduction number, the existence of equilibria and the local stability of equilibria by using the discretization method. For slow system (7), we will investigate the dynamical properties, including the positivity, boundedness, the existence of disease-free equilibrium, only a unique endemic equilibrium, and two endemic equilibria, and the local asymptotic stability for the disease-free and endemic equilibria. We will investigate the dynamical behaviors for the coupled systems (3)–(4) basing on the research results obtained for the fast and slow subsystems. Lemma 5 Let E > 0, fast system (6) always has a unique infected equilibrium. When E = 0 we see that equilibrium B∗ is local asymptotically stable
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