The Study of Extreme Conditions for Destructive Processes Using the SIR Model

Abstract

The article focuses on a study resulted in a mathematical tool to predict the most unfavorable scenario that describes how destructive processes are being developed under given initial conditions. The investigation has been worked out on the example of the way the infection in a closed population group is spreading. The main parameters of the extreme progression of the epidemic process have been studied that are the period of time during which the greatest number of subjects can be infected, the corresponding values of the infection rate and recovery rate, as well as the end time of the active phase of the epidemic. For this purpose, we used the SIR model by William Kermack and Anderson McKendrick, in which the borderline values of the function of the infection transmission were taken into account. The values mentioned above characterize both 100% and 0% probability for each subject to become diseased while contacting with an infected person. Conducting the research, we offered the minimax criterion that determines how the time, when the peak number of infected people is reached, depends on the recovery rate. The criterion expressed in graphical forms is the locus of time points showing the maximum number of infected subjects at a given ratio of healthy and infected persons when the monitoring process of the epidemic situation starts. The practical importance of the minimax criterion lies in probability to predict the least destructive consequences of the spread of the infection under the most unfavorable initial conditions. The results obtained are not limited only to the field of epidemiology, and the mathematical apparatus can be used to study many other processes that occur in extreme conditions and cause a negative reaction from the milieu they impact on.

Keywords

• sir model
• kermack-mckendrick theory
• mathematical epidemiology
• destructive processes
• active phase of the epidemic