Discrete Dynamics in Nature and Society

Volume 2018, Article ID 3187807, 7 pages

https://doi.org/10.1155/2018/3187807

## Parameter Estimation on a Stochastic SIR Model with Media Coverage

^{1}Department of Basic Science, Army Military Transportation University, Tianjin 300161, China^{2}School of Science, Tianjin Polytechnic University, Tianjin 300161, China^{3}School of Computer Science and Software Engineering, Tianjin Polytechnic University, Tianjin 300387, China

Correspondence should be addressed to Yongzhen Pei; moc.361@iepnehzgnoy

Received 11 March 2018; Accepted 20 May 2018; Published 12 June 2018

Academic Editor: Guang Zhang

Copyright © 2018 Changguo Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Media coverage reduces the transmission rate from infective to susceptible individuals and is reflected by suitable nonlinear functions in mathematical modeling of the disease. We here focus on estimating the parameters in the transmission rate based on a stochastic SIR epidemic model with media coverage. In order to reduce the computational load, the Newton-Raphson algorithm and Markov Chain Monte Carlo (MCMC) technique are incorporated with maximum likelihood estimation. Simulations validate our estimation results and the necessity of a model with media coverage when modeling the contagious diseases.

#### 1. Introduction

The spread of a contagious disease can trigger responses of people so as to minimize the effect of the disease onto them, and that prevents themselves from contracting the disease [1, 2]. Many scholars have explored the influence of media awareness from mathematical models. Cui et al. [3] modeled the transmission rate involving the media effect by the function , in which the parameters and represent the media impact (hence we call these parameters media parameters) and the transmission rate before media alert, respectively, and denotes the number of infected individuals at time . Liu et al. [4] proposed another transmission rate to capture the impact of media on disease spread. Here the media parameter refers to the reduced maximum value of the transmission rate when the number of infected individuals approaches infinite and reflects the reactive velocity of media coverage to the epidemic disease. Literatures concerning with media coverage mainly focus on the dynamic behaviors of epidemic models with media coverage by way of surveys, qualitative approaches, or numerical simulations [2–7].

Comparatively the problem of parameter estimation concerning transmission rate involving media effect has not been fully discussed. One main challenge in these inferential tasks is attributed to the burden of computational load of minimizing/maximizing corresponding object functions, including likelihood functions or the squared differences [8–11]. In this paper, we consider a stochastic Susceptible-Infected-Removed (SIR) model [12] with media coverage. The advantage of the model [12] is that an explicit likelihood functions can be formulated, which enables us to estimate parameters of the model. Newton-Raphson algorithms and Bayesian inference techniques are adopted to alleviate the computational load.

The paper is organized as follows. The stochastic SIR model with media coverage effect is introduced in Section 2. Then the likelihood function of the model is derived, from which unknown parameters can be estimated via Newton-Raphson algorithm or Markov Chain Monte Carlo (MCMC) technique in Section 3. Finally, some simulations are included to help illustrate the necessity to take media coverage into account when modeling the transmission dynamics of infectious disease in Section 4.

#### 2. Model Formulation

Katriel [12] proposed a stochastic discrete time model with infection age. Suppose that the population of size is partitioned into three classes, the susceptible, infectious, and recovered, with the numbers denoted by , , and at day , respectively. Furthermore, newly infected number is introduced on day . Suppose the infected individuals can remain “infective” for days. And the period from the moment an individual became infected to the present is named “age-of-infection” of the individuals which is denoted by . Hence the number of the infective with infection age on day is .

Faced with growing number of the infective, people will reduce their chances of contacting with others for fear of being infected. For this reason, we adopt the nonlinear contact rate . Here parameter is constant and is the intensity of infection occurring on day , and parameter reflects the extent to which media coverage affects society. If , the contact rate is constant which has been used by many classical models. As increases the alertness of the public to the disease, the public will be more aware of the diseases which reduces the contact rate as such.

In order to deduce the likelihood, we divide into discrete “days” and each day is divided into small intervals of length . In view of the impacts of media, a susceptible encounters an individual with infection age in each small interval with probability and gets infected with probability Here we assume, when a susceptible contacts with an infective individual whose infection age is , the susceptible becomes infective with probability . Therefore the possibility that this susceptible escapes the infection in each day is as As a result, the probability that at least one susceptible becomes infected during day t is given by Given , the number of newly infected individuals at day is binomially distributed with parameters and . That is, Since the number of infected individuals each day is equal to the reduction of susceptible of that day, we readily have

Based on (5) and (6) not only can we construct the likelihood so as to estimate the parameters, they are also crucial for simulations afterward. In order to realize this process, we can produce a binomial random number by applying (5) and (6) iteratively for when starting values and are known.

When the population is large, a deterministic model is obtained from the stochastic one in the sense of “thermodynamic limit”. Let and be the proportions of the individuals which turn into infective and susceptible by day , respectively. Given information about the infection before day , the conditional expected size of infected individuals on day is Hence we derive the infected fraction at day by the formula Then a deterministic model is yielded as In order to check the relationship between the stochastic model and its deterministic counterpart, we simulate the stochastic models with different values of population sizes. Together with solutions of model (9) and (10), the result (Figure 1) shows the model (9) is indeed the limit of its stochastic model as increases.