A reaction-diffusion model for viral infection and immune response
Résumé
In this work we extend the ODE model for virus infection and immune response proposed by P. Getto, ("Modelling and analysis of dynamics of viral infection of cells and of interferon resistance", J. Math. Anal. Appl., No. 344, 2008, pp. 821-850) to account for the spatial effects of the processes, such as diffusion transport of virions, biomolecules and cells. This leads to two different nonlinear PDE models, a first one where the cells and the biomolecules diffuse (which we call the reaction-diffusion model) and a second one where only the biomolecules can diffuse (the hybrid model). We show that both the reaction-diffusion and the hybrid models are well-posed problems, i.e., they have global unique solutions which are non-negative, bounded, and depend continuously on the initial data. Moreover, we prove that there exists a ''continuous'' link between these two models, i.e., if the diffusion coefficient of the cells tends to zero then the solution of the reaction-diffusion model converges to the solution of the hybrid model. We also prove that the solutions are uniformly bounded and integrable for all times. We characterize the asymptotic behavior of the solutions of the hybrid system and present several relations concerning the survivability of viruses and cells. Finally, we show that the solutions of the hybrid model converge to the steady state solutions, which implies that the latter are globally stable. We finish with several numerical simulations performed in Matlab.
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