A comprehensive probabilistic analysis of approximate SIR‐type epidemiological models via full randomized discrete‐time Markov chain formulation with applications

Abstract

This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR‐type epidemiological models based on discrete‐time Markov chain formulation. The randomization is performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed.

Novel Method to Analytically Obtain the Asymptotic Stable Equilibria States of Extended SIR-type Epidemiological Models

“…The stochastic representation of pandemic dynamics allows more flexibility and credibility than treating model parameters as deterministic values due to the fact that data often involves uncertainty [15]. This approach allows simulating pandemic dynamics that based on an extended SIR model using distributed systems models which allows adding upon the epidemic dynamics additional social [3], non-pharmaceutical and pharmaceutical intervention (NPI / PI) policies [27], and economical policies [7].…”

“…Eqs (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) describe the epidemic's dynamics. In Eq (1),…”

“…One method demonstrated to be useful to model dynamics appearing in epidemics is Markov chain [14,15]. In the context of epidemics, the Markov chain method represents the dynamics, that is, the rapid spread of the disease in the population using a transmission matrix [14].…”

“…Since the pandemic spread is subject to multiple complex factors whose nature is uncertain [15], and that these factors change based on the current state of the disease spread, Markov chain is a natural approach to use to model such dynamics [16][17][18]. It was shown that Markov chain approximates the deterministic SIR model very well [19,20].…”

Comparison of Pandemic Intervention Policies in Several Building Types Using Heterogeneous Population Model

“…Mathematical models of epidemiological dynamics usually represent the transformations of individuals in the population between several epidemiological states [52,11,26,5]. These models can be roughly divided into two main groups: diseases with short-term and long-term immunity memory when more often than not an immunity memory is considered long-term if it is longer than the average duration of two infection waves in the population.…”