Stochastic Approach to Epidemic Spreading
- PDF / 498,927 Bytes
- 12 Pages / 595.224 x 790.955 pts Page_size
- 68 Downloads / 244 Views
STATISTICAL
Stochastic Approach to Epidemic Spreading ˆ Tania Tome´ 1
´ J. de Oliveira1 · Mario
Received: 7 July 2020 / Accepted: 7 September 2020 / Published online: 10 October 2020 © Sociedade Brasileira de F´ısica 2020
Abstract We analyze four models of epidemic spreading using a stochastic approach in which the primary stochastic variables are the numbers of individuals in each class. The stochastic approach is described by a master equation and the transition rates for each process such as infection or recovery are set up by using the law of mass action. We perform numerical simulations as well as numerical integration of the evolution equations for the average number of each class of individuals. The onset of the epidemic spreading is obtained by a linear analysis of the disease free state, from which follows the initial exponential increase of the infected and the frequency of new cases. The order parameter and the variance in the number of individuals are also obtained characterizing the onset of epidemic spreading as a critical phase transition. Keywords Stochastic epidemic models · Epidemic spreading models · SIR model · SIS model
1 Introduction The theoretical study of the epidemic spreading [1–5] started with the employment of ordinary differential equations of the first order in time, which became known as the deterministic approach [1]. The individuals of a population are classified in accordance with their condition in relation to the infectious disease and these equations give the evolution equations on the number of individuals belonging in each class. The deterministic approach, however, does not describe, in an explicit manner, the random fluctuations occurring in a real epidemic spreading. This observation may have given way to the need of a stochastic approach to the epidemic spreading as that developed by Bartlett [6, 7] and by Bailey [8, 9]. A stochastic version of the deterministic model proposed by Kermack and McKendrick [10] was developed by Bartlett in 1949 [7]. The model, called susceptible-infectiveremoved, describes the spread of an infectious disease in a community of individuals who acquire permanent immunization. There are three classes of individuals: the susceptible, the infective, and the recovered. The approach advanced
Tˆania Tom´e
[email protected] 1
Institute of Physics, University of S˜ao Paulo, Rua do Mat˜ao, 1371, S˜ao Paulo, SP, 05508-090, Brazil
by Bartlett treated the numbers of individuals in each class as stochastic variables from which he developed a time evolution equation for the generating function corresponding to the probability distribution of these variables. The evolution equation for the probability distribution, or master equation, of the model analyzed by Bartlett was obtained by Bailey [9]. The stochastic approach they employed was based on the use of a continuous time Markov process in a discrete space in which the variables increase or decrease by one unit. In 1955, Whittle [11] presented a stochastic version of the Kermack and McKendrick theorem
Data Loading...
