This project aims to study the parameters of the Deterministic SIR(Susceptible ? Infected ? Recovered) model of COVID-19 in a Bayesian MCMC framework. Several deterministic mathematical models are being developed everyday to forecast the spread of COVID-19 correctly. Here, I have tried to model and study the parameters of the SIR Infectious disease model using the Bayesian Framework and Markov-Chain Monte-Carlo (MCMC) techniques. I have used Bayesian Inference to predict the Basic Reproductive Rate R_t in real time using and following this, demonstrated the process of how the parameters of the SIR Model can be estimated using Bayesian Statistics and Markov-Chain Monte-Carlo Methods.
The first COVID-19 case was detected in November 2019 and since then, the world is in a turmoil. Scientists around the globe are continuously trying to come up with multiple deterministic as well as stochastic models to understand and predict the spread of this virus. Bayesian Analysis is a method of Statistical Inference that allows us to combine prior information about a population parameter with evidence from information contained in a sample to guide statistical inference process.
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains and in this paper, I have taken help of the Metropolis–Hastings algorithm. Taking help of the Bayesian Framework and the MCMC Techniques, I have tried to model the parameters of the SIR model for Infectious Diseases, which would assist us to correctly understand the spread of COVID-19, based on real time data.
II. THE BAYESIAN FRAMEWORK
The origin of compartmental models, such as the SIR model, originated with the works of Kermack and McKendrik in 1927. The SIR Model is one of the simplest compartmental models and consists of 3 compartments:
S: The number of Susceptible Individuals. When a susceptible and an infectious individual come into infectious} contact, the susceptible individual contracts the disease and transitions into the infectious compartment
I: The number of Infected Individuals. The individuals are infected and are capable of spreading the infection to a susceptible group
R: The number of deceased or recovered individuals. These individuals have either been infected and completely recovered and entered the recovered compartment, or have died
The dynamics of an epidemic are often much faster than the dynamics of birth and death, therefore, birth and death are omitted in simple compartmental models, like the SIR Model. The SIR system without the vital dynamics can be expressed by the following set of Ordinary Differential Equations:
VI. BAYESIAN INVERSION APPROACH FOR MODEL PARAMETER ESTIMATION
We propose Bayesian inversion methods, in which probabilities are used as a general concept to represent the uncertainty in the model parameters in order to solve the backward/inverse problem of COVID-19, i.e., the problem of accurate estimation of the epidemiological model parameters. Bayesian inference in the context of the statistical inversion theory is based on Bayes’ Theorem and quantifies the uncertainty involved in the model parameters by defining a probability distribution over the possible values of the parameters and uses sample data to update this distribution. Bayesian analysis, in contrast to traditional inverse methods, is a robust inversion technique for determining parameters, yields the (a posteriori) probability distribution, and has the advantage of updating the prior knowledge about the unknown quantity using the measurement/observation data, giving confidence intervals for the unknowns instead of providing a single estimate.
In this paper, I have tried to demonstrate how the deterministic modelling of the COVID-19 is performed based on real time data. I have used Python 3.8 with the package PyMC3 to carry out the simulations performed during the course of the project. I chose the MCMC algorithm to implement the statistical estimation and prediction because of the consideration on the prediction uncertainly. The spread dynamics of the COVID-19 virus is significantly complex and potential inaccuracy and incompleteness have also crept in in the process of collecting and storing the data. The data has been collected from the open-source project https://covidtracking.com/ . The aim of this paper was to demonstrate how the parameters of the SIR Model are distributed with respect to data obtained from a region. Similar densities and values are used when statisticians are modelling the spread of an infectious disease using the SIR Model. This paper also includes a brief Bayesian Analysis on the Basic Reproductive Rate R_t for the COVID-19 virus.
However, a thing to note is that high dimension of the parameter space might affect the convergence of Markov Chain to the posterior density. But we can always construct more efficient algorithms. Although this type of methods somehow takes us away from our original purpose which was to improve upon an existing algorithm, they still make sense within this survey in that they allow for almost automated implementations.
Based on the availability of computational power, several other Monte Carlo Techniques like the Hamiltonian Monte Carlo (HMC), scalable MCMC methods, where algorithms manage to handle large scale targets by breaking the problem into manageable or scalable pieces(divide and conquer & sub-sampling), parallelization schemes etc. can be applied to obtain better results.
 Bayesian Inference for Dynamical Systems - Weston C. Roda, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
 A Mathematical Model of Epidemics—A Tutorial for Students - Yutaka Okabe and Akira Shudo, Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397
 Bayesian Inference for Stochastic Epidemic Models using Markov chain Monte Carlo Methods - Nicholas Demiris, The University of Nottingham
 The Covid-19 outbreak in Spain. A simple dynamics model, some lessons, and a theoretical framework for control response - Antonio Guirao, Department of Physics, Universidad de Murcia, Ed. CIOyN, Campus de Espinardo, 30100, Murcia, Spain
 Uncertainty quantification in epidemiological models for the COVID-19 pandemic - Leila Taghizadeh, Ahmad Karimi, Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040, Vienna, Austria, Clemens Heitzinger, School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA
 A simple model for COVID-19 - Julien Arino, Department of Mathematics & Data Science NEXUS, University of Manitoba, Canada, Canadian COVID-19 Mathematical Modelling Task Force, Canada, Stephanie Portet, Department of Mathematics \\& Data Science NEXUS, University of Manitoba, Canada