Open Access ISSN: 2155-6180 Journal of Biometrics & Biostatistics Research Article Volume 12:1, 2021 Bayesian Sir Modeling in the Case of Undocumented Infectious Individuals of COVID 19 Abstract Covid 19 that began in the end 0f 2019 has spread rapidly across continents in a short of time. This event makes most countries do not report and record the disease properly. There is considerable number of undocumented infectious individuals that should be considered when we want to estimate the Susceptible, Infectious, and Removal (SIR) parameters and to predict the future cases. The covid 19 data obtained from City of Surabaya were analyzed using Bayesian SIR modeling by BaySIR package in RStudio. The estimated and predicted medians of SIR compartment tend to decrease over time. The estimated and predicted medians of effective reproduction number tend to decrease over time. The results depend on the assumptions of SIR model to use such as the dynamic of covid 19, local government policies, and individual behavior in the community. Keywords: Covid 19 Undocumented individuals • Bay SIR • Infectious Disease Kuntoro Kuntoro* Department of Biostatistics and Population Study, Airlangga University School of Public Health, Indonesia *Address for Correspondence: Kuntoro Kuntoro, Department of Biostatistics and Population Study, Airlangga University School of Public Health, Indonesia, Tel: 62318700289; E-mail: kuntoro@fkm.unair.ac.id Copyright: © 2021 Kuntoro K. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Received 16 September 2020; Accepted 29 January 2021; Published 05 February 2021 Introduction Covid 19 is an infectious disease phenomenon in the 21th century. It began in the end of 2019 in Wuhan province, China [1]. It has spread out across continents in a short of time. It affects individuals in the developing countries as well as the developed countries. It shakes socio-economic-political-psychological order in most countries. The occurrence of covid 19 that is so fast makes most countries lack of medical infrastructures in the hospitals as well as health infrastructures in the public health facilities. Lack of ventilators in the hospitals enhances death among severe cases. Lack of public health personnel who promote better health behaviors and perform mass screening in the community enhances asymptomatic cases. These cases make high undocumented cases. They are not reported officially so that they cannot be controlled properly. Over years epidemiological experts implement Susceptible, Infectious, and Recovery (SIR) Compartment Model in understanding the dynamics of infectious diseases. Yang et al. [2] used SIR model as basis for estimating effective reproduction number of pandemic for pandemic influenza H1N1. This model is categorized as deterministic model in closed epidemic. It holds assumption of closed population in which there is no emigration as well as immigration, birth and death do not change [3]. In fact, the development of infectious diseases is dynamic and changes across time [4]. Change of health behaviors of individuals in the community as well as change of health policies affects the course of infectious diseases. Hence, deterministic model of SIR cannot accommodate these conditions. Besides an infectious disease is a random process, it needs probabilistic model to understand its progression. Theoretical Frame-Work Classical SIR Model Kermack and MacKendrik [5] began their works concerning the difficulty of finding a causal factor that accounted for epidemic of a disease in the population. When an infected individual was introduced into a population, more or less susceptible to the disease in question. Disease spreading occurred when an infectious individual contacted unaffected one. Finally, an infected individual was removed from the number of infected individuals by recovery or by death. These stories inspired them to develop concepts of Susceptible, Infectious, and Removal compartments as part of SIR model. According to Zhou and Ji [4]; Bjørnstad et al. [3], a population is divided into three compartments based on SIR model. First of all, susceptible individuals(S) are those who do not have the disease but they may be infected. Second, infectious individuals (I) are those who have the disease and they may infect the susceptible individuals. Third, recovered/removed individuals are those who had the disease, however, they are removed from the possibility of being reinfected or spreading the disease. They are removed due to several reasons such as recovery with immunity against reinfection, quarantine and isolation from the rest of the population, and death. Moreover, the SIR model can be expressed as differential system equations as follows.   ⁄ = −⁄      ⁄ = ⁄    −   (1)   ⁄ =   Transmission rate of the disease is denoted by β, while removal rate is denoted by α. When an infectious individual is doing effective contact with individuals per unit time, then I infectious individuals result to a rate of new infections (βS=N) · I . This is the explanation of the first part of equation (1). Moreover, the infectious individuals leave the infectious compartment at a rate of αI. This is the explanation of the third part of equation (1). Hence, the explanation of second part of equation follows immediately the explanations of first and second parts. According to Zhou and Ji [4], to make easy to compute S,I, and Rina given time t, we use a discrete-time approximation of equation (1) as follows.   =  −1 −  −1  −1 ⁄   = (1 − ) −1 +  −1  −1 ⁄ (2)   =  −1 +  −1 Stochastic sir model When the spreading of a disease is not deterministic, then Classical SIR Model is not recommended. The spreading of a disease is a random process. Disease transmission among individuals is more random than deterministic, it is called