In this article, we suggest a novel nonlinear cross-diffusion to understand the distribution of susceptible and infected individuals in their habitat through an SIR epidemic model with saturated treatment under zero-flux boundary conditions. First, we examine the boundedness of solutions to the proposed spatio-temporal model using the theory of parabolic partial differential equations. We ensure that at least one non-constant positive steady state is admissible in the reduced SI model with the suggested cross-diffusion. We draw the conclusion that susceptible and infected populations may coexist in the presence of cross-diffusion. Furthermore, we have shown that the system exhibits Turing instability. Through the existence of codimension-2 Turing-Hopf bifurcation we determine the Turing space in the spatial domain. Due to various seasonal effects of diseases, we introduced two novel perturbations. A detailed numerical simulation is performed with these perturbations to illustrate various types of scenarios of pattern formation like holes, strips, and holes-strips mixtures inside the Turing space. Non-Turing and Turing-bogdanov-Takens patterns have also been observed for suitable parameter choices. The corresponding spatial series and surfaces are also plotted to clarify the results obtained for various patterns. This work provides a significant insight into how epidemic models can be influenced by cross-diffusion.
MSC 2020 No.: 35K57, 35B36, 92D25