Analysis of solutions to mathematical models of epidemic processes with common structural properties

We present the equations of a family of mathematical models describing the process of the spread of infectious diseases among the population of one or more regions. The variables of the models are the numbers of various groups of individuals involved in the spread of the epidemic (groups of susceptible, infected, diseased individuals, etc.). The change rates in the number of groups of individuals are defined using abstract functions that take into account the current state and the history of the spread of the epidemic process. For analyzing the solutions of the models, we use the results of the theory of monotone operators and the properties of $M$-matrices. Sufficient conditions for the existence of bounded solutions of a family of models and the limit of these solutions at infinity are obtained. The results of the study of the solutions of the models of the spread of HIV-infection and tuberculosis are formulated.