A mathematical model of an epidemic with an arbitrary law of recovery

Abstract

The paper proposes a mathematical model of the epidemic process, taking into account the dependence of the rates of cure and loss of immunity on time. Today, mathematical models of epidemics based on the basic Kermack–McKendrick model have become widespread. The most famous models are Susceptible-Infected-Recovered (SIR) and Susceptible-Exposed-Infected-Recovered (SEIR). These models are based on dividing the population into separate groups that are in different epidemic conditions. The description of the models is based on differential equations similar to the equations of birth and death in the process of radioactive transformations of elements in a radioactive chain. However, this approach does not take into account the dependence of the probabilities of the transition of the population from group to group on time spent in the treatment process and in the process of loss of acquired immunity. The known models do analyze the nature of the course of the epidemic for long periods of time, when the process can enter a stationary state. The paper proposes a mathematical model based on dividing the population into separate groups. The first group consists of healthy people susceptible to infection due to contact with members of the second group, which includes the infected population. Members of the third group are being treated, the fourth group includes members of the society who have recovered with antibodies and are vaccinated. The fifth group consists of deceased members of society. In contrast to the SIR and SEIR models, the proposed approach takes into account that immunity is lost over time, and people who survived again move to the group susceptible to infection. The dependences of the probabilities of transition from group to group on time spent both during the treatment process and in the loss of acquired immunity have been taken into account. Thus, the proposed mathematical model is based on five integro-differential equations, two of which are partial differential equations. A new mathematical model has been formulated that makes it possible to take into account the dependence of the cure rate and the probability of transition from the vaccinated state to the initial state on time spent in the corresponding state. It is shown that the proposed model is autocatalytic. With increasing time, a state of bistability is observed, when, under the same boundary conditions, two stable states coexist. Switching between states is determined by the epidemic spread control parameter found in the work. One of the stable states is stationary and leads to the end of the epidemic, the second one leads to the population’s extinction. It has been shown that, for the stationary regime, the form of the distribution function in terms of treatment time and exposure time in the vaccinated state does not affect the final result in any way. The conditions for suppressing the epidemic for managing the process of its development are formulated. Numerical experiments were carried out to simulate the epidemic spreading process, taking into account the constancy of all transition probabilities. Integration of the original system of equations was carried out using the Radaus algorithm for stiff differential equations. The results of numerical simulations have confirmed that the experimental data agree with the theory of the control parameter. The results of the work can be used to organize the management of the epidemic spreading process in order to suppress it as soon as possible by changing the value of the control parameter.

Keywords

• epidemic
• mathematical model
• distribution density over recovery time
• bistable state
• control parameter
• suppression conditions