ON CALCULATING THE MEAN INFECTED TIME USING A DISCRETE MARKOV EPIDEMIOLOGICAL MODEL WITHOUT TREATMENT

Abstract

Modeling of the spread of viruses is a relevant research field. There are a lot of «continuous» epidemic models based on the use of systems of differential equations. The disadvantage of such models lies in their error in describing the initial stage of virus propagation and in the fact that they ignore the specific features of inter-individual connections. «Discrete» models, in which the time and the number of infected and susceptible nodes are discrete values, provide a more accurate picture of the epidemic process. In this work, we study a discrete Markov model in the case when there is no treatment. This is an important case, since it can be viewed as either an approximation to the initial phase of an epidemic or as a model for epidemics of viruses that are difficult to treat. The first section provides a detailed description of the properties of the Markov model used in this study. In the second section, using Markov approach, we define the mean infected time, i.e. the number of time steps taken to infect all individuals in the population. However, calculating the mean infected time in populations with a large number of individuals (or in networks with a large number of nodes) is computationally difficult problem, so in the third section we propose the corresponding approximate formula for this parameter. This approximation is designed for conditions of low network connectivity and а low probability of virus spread. In the fourth section, to validate our approximate formula, we compare its results against both exact calculations (using the fundamental matrix M) and data from simulation modeling. For the simulations, we developed a custom C++ console application. Our analysis demonstrates that all three methods yield consistent results under the specified conditions, confirming the practical utility of the simpler approximate formula