Spatiotemporal Dynamics in a Diffusive Bacterial and Viral Diseases Propagation Model with Chemotaxis
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Spatiotemporal Dynamics in a Diffusive Bacterial and Viral Diseases Propagation Model with Chemotaxis Xiaosong Tang1 · Peichang Ouyang1 Received: 22 April 2020 / Accepted: 9 September 2020 © Springer Nature Switzerland AG 2020
Abstract In this article, we study the effect of chemotaxis on the dynamics of a diffusive bacterial and viral diseases propagation model. From three aspects: χ > 0, χ = 0 and χ < 0, we investigate the existence of Turing bifurcations and stability of positive equilibrium under Neumann boundary conditions. We find that Turing bifurcations can induced by chemotaxis, which does not occur in the original model. Moreover, for the model with diffusion and chemotaxis, we need explore the new expression of the normal form on Turing bifurcation. By the newly obtained normal form, we can determine the properties of Turing bifurcation. Finally, we perform some numerical simulations to verify the theoretical analysis and obtain stable steady state solutions, spots pattern and spots-strip pattern, which also expand the main results in this article. Keywords Bacterial and viral diseases propagation model · Chemotaxis · Turing bifurcation · Steady state solutions · Pattern formation Mathematics Subject Classification 35B32 · 35B35 · 35B36
1 Introduction Nowadays, more and more people in the world are dying of various diseases such as AIDS, avian influenza, cholera, Ebola, Zika virus and so on. To explore the mechanisms of these diseases, scientists have proposed many mathematical models describing the transmission of disease such as infectious disease model (SI, SIR, SEI, SIS) and withinhost virus model (HBV, HCV, HIV, Ebola, Zika), see [1–7]. For example, in order to reveal the spatial transmission of epidemics through the infective human population’s
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Peichang Ouyang [email protected] School of Mathematics and Physics, Jinggangshan University, Ji’an 343009, Jiangxi, China 0123456789().: V,-vol
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living environment, Capasso and Maddalena [8] have given out the reaction-diffusion model as follows: ∂u(x,t) = d1 u x x − a11 u(x, t) + a12 v(x, t), x ∈ , t > 0, ∂t (1.1) ∂v(x,t) = d2 vx x + g(u(x, t)) − a22 v(x, t), x ∈ , t > 0, ∂t where the densities of bacteria is represented by u(x, t) and the densities of infective population is represented by v(x, t) at a location x in a habitat ⊂ R and time t ≥ 0. d1 and d2 represent the diffusion speed of bacteria and infective population, respectively. The mortality of bacteria denoted by a11 > 0. a22 > 0 is the natural declining rate of infective individuals. The contribution of infected population to the bacteria density denoted by a12 v. g(u) represents the “force of infection” on human population due to the concentration of bacteria. They have discussed the convergence to equilibrium states of model (1.1) by using mathematics analysis method. Since then, considering the latent period of bacterial, Thieme and Zhao [9] inserted distributed time delay into model (1.1) and have proved the asymptotic spreading speeds for
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