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In this paper, we are concerned with two SIS epidemic reaction-diffusionmodels with mass action infection mechanism of the form $SI$, and study thespatial profile of population distribution as the movement rate of the infectedindividuals is restricted to be small. For the model with a constant totalpopulation number, our results show that the susceptible population alwaysconverges to a positive constant which is indeed the minimum of the associatedrisk function, and the infected population either concentrates at the isolatedhighest-risk points or aggregates only on the highest-risk intervals once thehighest-risk locations contain at least one interval. In sharp contrast, forthe model with a varying total population number which is caused by therecruitment of the susceptible individuals and death of the infectedindividuals, our results reveal that the susceptible population converges to apositive function which is non-constant unless the associated risk function isconstant, and the infected population may concentrate only at some isolatedhighest-risk points, or aggregate at least in a neighborhood of thehighest-risk locations or occupy the whole habitat, depending on the behaviorof the associated risk function and even its smoothness at the highest-risklocations. Numerical simulations are performed to support and complement ourtheoretical findings.

Novel Spatial Profiles of Population Distribution of Two Diffusive SIS  Epidemic Models with Mass Action Infection Mechanism and Small Movement Rate  for the Infected Individuals