Mathematical model of the parasite – host system with distributed immunity retention time

The COVID-19 pandemic has caused increased interest in mathematical models of the epidemic process, since only statistical analysis of morbidity does not allow medium-term forecasting in a rapidly changing situation.
Among the specific features of COVID-19 that need to be taken into account in mathematical models are the heterogeneity of the pathogen, repeated changes in the dominant variant of SARS-CoV-2, and the relative short duration of post-infectious immunity.
In this regard, solutions to a system of differential equations for a SIR class model with a heterogeneous duration of post-infectious immunity were analytically studied, and numerical calculations were carried out for the dynamics of the system with an average duration of post-infectious immunity of the order of a year.
For a SIR class model with a heterogeneous duration of post-infectious immunity, it was proven that any solution can be continued indefinitely in time in a positive direction without leaving the domain of definition of the system.
For the contact number $R_0\leq1$, all solutions tend to a single trivial stationary solution with a zero share of infected people, and for $R_0>1$, in addition to the trivial solution, there is also a non-trivial stationary solution with non-zero shares of infected and susceptible people. The existence and uniqueness of a non-trivial stationary solution for $R_0>1$ was proven, and it was also proven that it is a global attractor.
Also, for several variants of heterogeneity, the eigenvalues of the rate of exponential convergence of small deviations from a nontrivial stationary solution were calculated.
It was found that for contact number values corresponding to COVID-19, the phase trajectory has the form of a twisting spiral with a period length of the order of a year.
This corresponds to the real dynamics of the incidence of COVID-19, in which, after several months of increasing incidence, a period of falling begins. At the same time, a second wave of incidence of a smaller amplitude, as predicted by the model, was not observed, since during 2020–2023, approximately every six months, a new variant of SARS-CoV-2 appeared, which was more infectious than the previous one, as a result of which the new variant replaced the previous one and became dominant.