High order discretization methods for spatial dependent SIR models Bálint Takács * Yiannis Hadjimichael † July 13, 2020 Abstract In this paper, an SIR model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of integro-differential equations. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different choices of spatial and temporal discretizations are employed, and step-size restric- tions for population conservation, positivity and monotonicity preservation of the discrete model are investigated. We provide sufficient conditions under which high order numerical schemes preserve the discrete properties of the model. Computational experiments verify the convergence and accuracy of the numerical methods. 1 Introduction During the millenia of the history of mankind, many epidemics have ravaged the population. Since the plague of Athens in 430 BC described by historian Thucydides (one of the earliest description of such epidemics), researchers tried to model and describe the outbreak of illnesses. More recently, the outbreak of COVID-19 pandemic revealed the importance of epidemic re- search and the development of models to describe the public health, social, and economic impact of major virus diseases. Nowadays many of the models used in science are derived from the original ideas of Kermack and McKendrick [26] in 1927, who constructed a compartment model to study the process of epidemic propagation. In their model the population is split into three classes: S being the group of healthy individuals, who are susceptible to infection; I is the compartment of the ill species, who can infect other individuals; and R being the class of recovered or immune individuals. * Applied Analysis and Computational Mathematics, MTA-ELTE Numerical Analysis and Large Net- works Research Group, Eötvös Loránd University, Pázmány P. s. 1/C, Budapest 1117, Hungary, (email: takacsbm@caesar.elte.hu). † MTA-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Pázmány P. s. 1/C, Budapest 1117, Hungary, (email: hadjimy@cs.elte.hu). 1 arXiv:1909.01330v2 [math.NA] 9 Jul 2020