Abstract
We consider a class of non-cooperative reaction-diffusion system, which includes different types of incidence rates for virus dynamical models with nonlocal infections. Threshold dynamics are expressed by basic reproduction number \begin{document}$ R_0 $\end{document} in the following sense, if \begin{document}$ R_0 , the infection-free steady state is globally attractive, implying infection becomes extinct; while if \begin{document}$ R_0 > 1 $\end{document} , virus will persist. To study the invasion speed of virus, the existence of travelling wave solutions is studied by employing Schauder's fixed point theorem. The method of constructing super-solutions and sub-solutions is very technical. The mathematical difficulty is the problem constructing a bounded cone to apply the Schauder's fixed point theorem. As compared to previous mathematical studies for diffusive virus dynamical models, the novelty here is that we successfully establish the general existence result of travelling wave solutions for a class of virus dynamical models with complex nonlinear transmissions and nonlocal infections.
Highlights
Mathematical models have been shown to be an effective and valuable approach to understand virus infection dynamics in the within-host environment
We further consider threshold dynamics and travelling wave solutions for the following virus dynamical model with more complex, nonlinear transmissions and nonlocal infections, which is constituted by four equations
We study the existence of travelling wave solutions of model (10), which has the form U (x, t) = φ(x+ct), V (x, t) = φ(x+ct), M (x, t) = ψ(x+ct), ω(x, t) = γ(x+ct), where φ, φ, ψ, γ ∈ cl(R+2 ) to R. (C2)(R, R4) and c > 0 is the wave speed
Summary
Mathematical models have been shown to be an effective and valuable approach to understand virus infection dynamics in the within-host environment. [20] and [36] have developed the general theory on the existence of travelling wave solutions for monotonic (or cooperative) systems This method can not be applied to the non-monotonic model (1). [17] developed a geometric method to study the existence of travelling wave solutions for a class of non-monotonic systems consisting of two equations with general functions. The results obtained in [58] cannot be directly applied to study the existence of travelling wave solutions for a class of non-cooperative reaction-diffusion systems with a discrete delay and spatial non-locality. We further consider threshold dynamics and travelling wave solutions for the following virus dynamical model with more complex, nonlinear transmissions and nonlocal infections, which is constituted by four equations.
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