An asymptotic solution for the SIS epidemic model, taking into account migration and diffusion

The purpose of this work is to propose and investigate a simple and effective model of an epidemic in an animal population that takes into account migration along the plane of both diseased and healthy individuals. Within the framework of this model, the spatial migration of a population is described by introducing both diffusion and advective terms into its equations. Methods. In this paper, a method of many scales was used to find an asymptotic solution to the system of equations of the epidemic. Solutions of auxiliary linear equations of the parabolic type arising during this procedure were found using the Poisson integral. The simplification of the initial system of equations of the model is based on the assumption that the sum of densities of healthy and sick individuals on a single-connected region of large diameter on the plane is constant at the initial moment of time. Results. It is shown that in this case, designed for a slowly changing initial density of sick individuals concentrated inside this area at a considerable distance from its boundaries, the asymptotic solution of the model describes the effect of merging several spatially spaced small outbreaks of the disease into one large outbreak during migration of the entire population as a whole. In particular, for such an initial density obtained by the functional transformation of a Gaussian, a circular plateau is formed over long periods with an effective radius that grows linearly over time. Conclusion. The constructed asymptotic solution of the epidemic model proposed in this paper is simple in form and describes the transfer of the disease on a locally flat area of the earth’s surface without the use of numerical methods. This solution is convenient when describing the migration of a sick population under the influence of flooding, forest fire, man-made disaster with contamination of the area, etc.  

moving coordinate system, small parameter, slow variables, the logistic curve, the method of variation of the constant, the Cauchy problem, the Kardar-Parisi-Zhang equation, the uniformity of the asymptotic expansion, formation of structures