Mathematical Analysis and Pattern Formation for a Partial Immune System Modeling the Spread of an Epidemic Disease

Abstract

Our motivation is a mathematical model describing the spatial propagation of an epidemic disease through a population. In this model, the pathogen diversity is structured into two clusters and then the population is divided into eight classes which permits to distinguish between the infected/uninfected population with respect to clusters. In this paper, we prove the weak and the global existence results of the solutions for the considered reaction-diffusion system with Neumann boundary. Next, mathematical Turing formulation and numerical simulations are introduced to show the pattern formation for such systems.

Solvability of reaction–diffusion model with variable exponents

“…After the pioneering work of Kermack and McKendrick [1] in 1927, epidemiological models have been receiving much attention from the mathematicians and biologists. During the last few decades, considerable amount of work has been performed on epidemiology models [2][3][4][5][6] and their related predator-prey models (see [7][8][9] and the references therein). The cancer cell invasion disease and chemotaxis are also interesting mathematical biology models (e.g., see [10][11][12][13][14]).…”

“…In particular, Aliziane and Langlais [2] established the existence, uniqueness and uniform bounds of solutions using approximation method for Laplacian diffusion operator. Bendahmane and Saad studied the existence of solutions and numerical pattern formation for epidemic disease reaction-diffusion model in [3]. Bendahmane et al established the existence results for cross diffusion type reaction-diffusion epidemic model in [4] and also the existence of solutions of degenerate bidomain model of cardiac tissue reaction-diffusion system was proved by Bendahmane and Karlsen in [31].…”

Stationary Patterns of a Cross-Diffusion Epidemic Model

“…More recently, many studies have shown that a spatial epidemic model is an appropriate tool for investigating the fundamental mechanism of complex spatiotemporal epidemic dynamics. In these studies, reaction-diffusion equations have been intensively used to describe spatiotemporal dynamics [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Spatial epidemiology with diffusion has become a principal scientific discipline aiming at understanding the causes and consequences of spatial heterogeneity in disease transmission [20].…”

Stability analysis in a diffusional immunosuppressive infection model with delayed antiviral immune response

“…In , Turing patterns were induced by the anomalous diffusion in both Brusselator chemical system and Boissonade chemical system. Pattern formation were influenced by cross‐diffusion in population dynamics and epidemic dynamics . Additionally, in systems with Lévy flights, investigated both the emergence of spiral waves and chemical turbulence from the nonlinear dynamics of oscillating reaction–diffusion patterns.…”