The construction of mathematical models of infection spread is an important element in the study of complex disease dynamics. In the paper, we investigating the classical mathematical SIR (Susceptible-Infected-Recovered) model of Kermack-McKendrick, in which a population is divided into three distinct groups: susceptible individuals who can contract the infection (S), infected individuals who carry the disease (I), and recovered individuals who are no longer infectious (R). The simplest SIR models rely on fundamental assumptions, such as that everyone has the same chance of contracting the infection from an infected person due to the population being perfectly and evenly mixed, and that all infected individuals are equally contagious until they recover or die.

The conditions for local Lipschitz continuity of the right-hand sides of the considered mathematical models are investigated. Based on this, it is established that the classical Kermack--McKendrick SIR model, which describes the dynamics of the disease at the macro level, has a unique solution. The limit properties of the SIR model solutions are also investigated, and a relationship is derived that allows estimating the maximum possible number of infected individuals.

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