Maximum entropy approach to reliability analysis based epidemic disease model

Abstract

This study analyzes the dynamic model of epidemic disease system based on the maximum entropy approach to reliability. The epidemic disease system is assumed to be composed of several classes, namely susceptible class, infective class, resistance class, exposed class and so on. The dynamics of the populations in each class are depicted by a number of differential equations, where the populations transmitting between the classes are the averages of transmitting populations in microscopic stochastic transmission process. In microscopic transmission process, several basic variables are assumed to be random, such as incubation period, hospitalization duration and so on. To infer the probability distributions of these variables is a main task of this study. Inspired by the recent study of reliability, the maximum entropy based approach is applied to determine the parameters in the dynamic model. The basic idea of this study is presented below. In this study, degrade (hazard) function, which is a fundamental quantity in the disciplines of risk and reliability analysis, is associated with death rate, incidence rate and healing rate. Specifically, death rate and healing rate of the infective people are associated with hazard rate and repair rate of the repaired system in reliability theory, respectively; the incidence rate is associated to the hazard rate during incubation period. By means of the maximum entropy principle, the moments of the period from onset to death, the period from onset to recovery and the incubation period are fused to infer the most probable death rate, incidence rate and healing rate. Applying the maximum-entropy based statistical inference, the information of macroscopic transmission process is fused, such as average values, fluctuations and median values. To the best of our knowledge, the traditional fitting approaches to determine parameters usually rely on the information of macroscopic phenomenon. It is different from these fitting approaches that the maximum-entropy based approach applied in this study relies on the information of microscopic process. Thus this approach is adapted to practice scenario where the limitations of information access and number of samples both exist. And the parameter determination is independent on the choice of the macroscopic epidemic disease dynamic model. After applying the maximum entropy principle based inference and the SEIR (susceptible-infective-exposed-removed) model, several discussions associated with coronavirus disease 2019 (COVID-19) are made. With the help of recent information of microscopic transmission process, the parameters in SEIR model are determined directly by the maximum-entropy principle. Then with numerical calculation, dynamics of the populations in infective class, resistance class and exposed class are obtained. Besides, some typical phenomena are revealed by the analytical and numerical results. For example, the calculation shows that the peak of the infectious ratio in transmission process is unique; the infectious ratio of steady state tends to zero when the immunity duration is much larger than the healing period. Additionally, limited immunity duration model is also considered. The relationship between the infectious ratio of steady state and the proportion of immunization duration to healing period is presented.