Abstract
We consider a spatially-heterogeneous generalization of a well-established model for the dynamics of the Human Immunodeficiency Virus-type 1 (HIV) within a susceptible host. The model consists of a nonlinear system of three coupled reaction-diffusion equations with parameters that may vary spatially. Upon formulating the model, we prove that it preserves the positivity of initial data and construct global-in-time solutions that are both bounded and smooth. Finally, additional results concerning the local and global asymptotic behavior of these solutions are also provided.
Highlights
Over the past few decades, considerable effort has been devoted to modeling the in-host dynamics of viral infection, and in particular, Human Immunodeficiency Virus-type 1 (HIV) infection within humans
Over a time period that can vary from weeks to months, an eventual balance of viral replication and clearance of the virus by the immune system occurs, leading to a state known as chronic infection
In order to investigate the impact of spatial dynamics in a simple mathematical model of HIV infection we extend the standard lumped or three-component model of in-host viral dynamics [1, 13, 19, 20, 26] to include spatially random diffusion and a spatially-dependent T-cell supply rate
Summary
Over the past few decades, considerable effort has been devoted to modeling the in-host dynamics of viral infection, and in particular, HIV infection within humans. Though it is assumed throughout that λ(x) is smooth, one may relax this assumption and arrive at similar conclusions regarding global-in-time solutions and their dynamical properties Another reasonable assumption to include within the model would be to take DI = DT , as infection of susceptible cells should not influence the rate of diffusion. I(t) ∞ + V (t) ∞ ≤ C0e−at for every t ≥ 0 Combining this result with Theorem 2.7 yields a sufficient condition under which the viral clearance state is a globally asymptotically stable equilibrium point of the system with exponential decay. Assume that u(t, x) satisfies the differential inequality (∂t − D∆)u ≤ g(t, x), x ∈ Ω, t > 0 u(x, 0) = u0(x), x ∈ Ω With these results in place, we may prove the global existence and positivity theorem.
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