Reduction of discrete‐time infectious disease models

Abstract

In this work, we propose the construction of discrete‐time systems with two time scales in which infectious diseases dynamics are involved. We deal with two general situations. In the first, we consider that individuals affected by the disease move between generalized sites on a faster time scale than the dynamics of the disease itself. The second situation includes the dynamics of the disease acting faster together with another slower general process. Once the models have been built, conditions are established so that the analysis of the asymptotic behavior of their solutions can be carried out through reduced models. This is done using known reduction results for discrete‐time systems with two time scales. These results are applied in the analysis of two new models. The first of them illustrates the first proposed situation, being the local dynamics of the SIS‐type disease. Conditions are found for the eradication or global endemicity of the disease. In the second model, a case of co‐infection with a primary disease and an opportunistic disease is treated, the latter acting faster than the former. Conditions for eradication and endemicity of co‐infection are proposed.