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A reaction-diffusion model for nested within-host and between-host dynamics in an environmentally-driven infectious disease
Ning Wang,Long Zhang,Zhidong Teng
Keywords:Reaction-diffusion equation; nested model; basic reproduction number; stability; backward bifurcation.
Abstract:
      A reaction-diffusion model for nested within-host and between-host dynamics in an environmentally-driven infectious disease is proposed. The model is composed of the within-host virus infectious fast time model of ordinary differential equations and the between-host disease transmission slow time model of reaction-diffusion equations. The isolated fast model has been investigated in previous literature, and the main results are summarized. For the isolated slow model, the well-posedness of solutions, and the basic reproduction number Rb are obtained. When Rb ≤ 1, the model only has the disease-free equilibrium which is globally asymptotically stable, and when Rb> 1 the model has a unique endemic equilibrium which is globally asymptotically stable. For the nested slow model, the positivity and boundedness of solutions, the basic reproduction number Rc and the existence of equilibrium are firstly obtained. Particularly, the nested slow model can exist two positive equilibrium when Rc < 1 and a unique endemic equilibrium when Rc > 1. When Rc < 1 the disease-free equilibrium is locally asymptotically stable, and when Rc > 1 and an additional condition is satisfied the unique endemic equilibrium is locally asymptotically stable. When there are two positive equilibria, then a positive equilibria is locally asymptotically stable under an additional condition and the other one is unstable, which implies that the nested slow model occurs the backward bifurcation at Rc = 1. Lastly, numerical examples are given to verify the main conclusions. The research shows that the nested slow model has more complex dynamical behavior than the corresponding isolated slow model.