International Journal of Innovative Research in Engineering & Management (IJIREM) ISSN: 2350-0557, Volume-6, Issue-4, July 2019 DOI: 10.21276/ijirem.2019.6.4.3 Copyright © 2019. Innovative Research Publications. All Rights Reserved 38 A SIR Model for Measles Disease Case for Albania Elfrida DISHMEMA Department of Mathematics and Informatics, Agriculture University of Tirana, Tirana, Albania e.dishmema@gmail.com Lulezim HANELLI Department of Mathematical Engineering, Polytechnic University of Tirana, Tirana, Albania lh9355@yahoo.co.uk ABSTRACT Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. In this paper we have used SIR model to study the outbreak of measles epidemic in Albania during the year 2018. Two basic parameters of SIR model are determined using least square principle and other numerical techniques. The model can be used to predict in advance the dynamics of the disease and this can help finding the best possible strategies to control its spread. Keywords Measles Mpidemic, SIR Model, Susceptible Population, Reproduction Function, Differential Equation, Least Square Principle. 1. INTRODUCTION Infectious diseases like influenza, measles, tuberculosis, to name a few, have been a great concern of humankind since the very beginning of the history [3]. Nowadays, they are still the major causes of mortality in many countries [4], [5]. Mathematical models provide helpful tools in studying a large range of such diseases. One of them, known as Susceptible-Infected-Removed (SIR) model, was firstly developed in [4]. It is used in epidemiology to compute the number of susceptible, infected, and recovered people in a closed population at any time. The whole population is divided into three classes, S: the number of susceptible, I: the number of infective and R: the number of recovered during an epidemic. The dynamic of the SIR model is greatly affected by the modeling way of the disease transmission mechanism from infective to susceptible individuals. Many different epidemic models are related to different kinds of these transmission mechanisms. In this paper we study the SIR model related to measles disease and some specific and reasonable assumptions related to this last. The basic and analytic discussion of the model, terminology and biological meaning of parameters and variables used, are given in section 2. A brief description of measles disease history in Albania and especially the epidemic of year 2018, hereafter EpAL18, is given in section 3. Facts, data and helpful conclusions of this section are used in section 4. In this section we collate measles SIR model with EpAL18 and implement it in Matlab using a range of numerical and computer techniques. The results are presented and analyzed graphically. Relevant conclusions are drawn in section 5. 2. MEASLES SIR MATHEMATICAL MODEL When a disease is modeled, it is needed to understand the way this disease is spread, to be able to predict the progression of the disease and to understand how it’s spread may be controlled. The following model is used to study population dynamics of susceptible, infected and recovered individuals from measles disease and to describe the temporal evolution in the number of individuals in need of medical care during this illness. The compartments used for the SIR model consist as usually of three classes: Infective ( ): denotes the number of individuals at time t (days) who have been infected with the disease and can spread the disease to those in the susceptible category. Overall, measles usually resolves after statistically 21 days, but it is supposed, like in [2], that an ill individual spreads its illness only in 3-4 days before the characteristic measles rash and 3-4 days after it. We have selected the onset of rash as reference time in our model for the important fact that individuals are able to report this event exactly to their best. Each other moment would be probably reported wrong. So symbolically, an individual is considered infective only in the time interval (t(rash)-3.5, t(rash)+3.5). Susceptible ( ): denotes the number of individuals at moment t, not yet infected with the disease, i.e., not able to spread the illness to other individuals. Recovered ( ): denotes the individuals who have been infected from the disease, then recovered, and not able to be infected again or to transmit the infection to others. Measles SIR model is presented schematically in Figure 1. Figure 1: Kermack-McKendrick Schematic Model. Individuals move in direction from class S to class I and R. S I R