Modeling of epidemic spreading with white Gaussian noise

Abstract

Motivated by the need to include the different characteristics of individuals and the damping effect in predictions of epidemic spreading, we build a model with variant coefficients and white Gaussian noise based on the traditional SIR model. The analytic and simulation results predicted by the model are presented and discussed. The simulations show that using the variant coefficients results in a higher percentage of susceptible individuals and a lower percentage of removed individuals. When the noise is included in the model, the percentage of infected individuals has a wider peak and more fluctuations than that predicted using the traditional SIR model. The history of Homo sapiens is always closely interrelated with the study of diseases. Whether it be the Black Death in Europe in the fourteenth century or the superbug that recently spread across the globe like wildfire [1,2], scientists have always looked at diseases with great interest [3][4][5][6][7][8][9][10]. In the well-known theoretical SIS model, the population is divided into two disjoint classes, susceptible individuals and infected individuals, for which the percentages at time t are denoted by s(t) and i(t), respectively [11][12][13][14]. However, this model is not suitable for describing diseases like flu or malaria. The subsequent has a third class with percentages denoted by r(t). They are the recovered individuals who are immune to the infection. This is a simple model commonly used for many infectious diseases like measles, mumps and rubella. In these two models, the spread of an infectious disease in a population depends mainly on the character of the disease. The most suitable model can be chosen according to the particular case. However, these models do not work well for some special cases. For example, in the spreading of SARS, although an individual cannot develop lasting immunity, there is only a very tiny chance of being infected again. By mean field theory, the course of an epidemic spreading is determined by the contact rates among susceptible, infected and removed individuals, which are assumed to be proportional to the number of encounters among susceptible, infected and removed individuals. Each individual is treated in the same way, even at different times. A more realistic and interesting model of infectious diseases should take into account a change in the environment and the variety of characteristics of individuals [21]. Along with the spreading process, the preventive effect should become stronger because people could find useful methods such as taking pills and avoiding close contacts with infected individuals to prevent the disease from spreading [22,23]. Moreover, how easy or difficult it is for an individual to be infected by others depends on their own characteristic, such as age, nutritional status, sex and so on. Therefore, each individual should be distinguished by different transmission coefficients and recovery coefficients. Furthermore, the part of the model that measures the coefficient of infection during the w...

The impact of individual differences on the spreading of epidemic

“…Obviously, if we can prove the existence of S , then the invasive influence of the disease can be measured by equation (13).…”

“…If a disease breaks out, then we get 0 1 R That is to say, R can be measured by equations (12)- (13). Using Matlab software, Figure 2 shows that if 0 1 R , it is impossible to exist a 0 (0,1) …”

“…Moreover, due to different individual characteristics, the classic constant transmission, hospitalization, recovery and disease induced death rates are considered to be infected by white Gaussian noise [13,14,15]. For simplicity, we assume ( Based on the previous creative idea of Liu et.al [8], we know that the variant transmission rate ( ) t , hospitalization rate ( ) t , recovery rate ( ) t and disease induced death rates 0 ( ) d t and 1 ( ) d t are exponentially influenced by the number of both infectious and hospitalized individuals.…”

“…However, this is not the unique method to convert a deterministic model to a stochastic one as some scholars have included the parameter perturbation into the traditional deterministic models. Moreover, the traditional SIR model have been combined with different variant coefficients and additive Gaussian noise which can interpret the damping effect of individual differences with a purpose to study the SARS outbreak in Beijing in 2003 [13,14]. A year later, they proposed a multiple-compartment model in consideration of white Gaussian noise by involving not only local spreading but also longdistance infection which was caused by the convenient public transportation system [15].…”

The Evolutionary Dynamics of Stochastic Epidemic Model with Nonlinear Incidence Rate

“…In fact, the white noise intensity may be less than the critical intensity σ * in the real word. In Gu et al (2011), authors formulated stochastic SIR epidemic model with the transmission rate β disturbed by white Gaussian noise in the similar form to (2) (see, P. 3685 in Gu et al 2011); the real data of SARS in Beijing in 2003 are nicely fitted by their model with the white noise intensity σ = 2.31 × 10 −5 < σ * , which has been converted into the time unit (week −1 ) in this paper. This realistic noise intensity does not exceed the critical intensity σ * for both cases above in (38) and (39).…”