Abstract
We present a new analytical method to find the asymptotic stable equilibria states based on the Markov chain technique. We reveal this method on the Susceptible-Infectious-Recovered (SIR)-type epidemiological model that we developed for viral diseases with long-term immunity memory. This is a large-scale model containing 15 nonlinear ordinary differential equations (ODEs), and classical methods have failed to analytically obtain its equilibria. The proposed method is used to conduct a comprehensive analysis by a stochastic representation of the dynamics of the model, followed by finding all asymptotic stable equilibrium states of the model for any values of parameters and initial conditions thanks to the symmetry of the population size over time.
Highlights
Introduction and Related WorkA large group of epidemiological models are extensions of the Susceptible-InfectedRecovered (SIR) model [1]
One can model the pandemic dynamics using a stochastic process due to the unstable nature of the parameters of the pandemic used in the model, such as the infection rates ( β cs, β ca, β as, β aa, β es, β ea ) and recovery rates, which differ over time
We present this method for an extended SIR model, for the three age groups SIIRD
Summary
A large group of epidemiological models are extensions of the Susceptible-InfectedRecovered (SIR) model [1] These models were used for both prediction of the pandemic spread and to find optimal intervention policies for multiple types of diseases, such as polio [2], COVID-19 [3], ebola [4], and influenza [5]. These models use a wide range of analyses and extensions for the SIR model to properly represent the epidemiological and biological properties unique to each disease. We discuss the main advantages and limitations of the proposed method
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