An Introduction to Mathematical Modeling of Infectious diseases: The S-I-R Model

The S-I-R Model

Mathematical models have become important tools in analysing the transmission and control of infectious diseases. Governments across the world are depending on mathematical models and predictions to help in the decision making as seen in this current global Covid-19 pandemic.

The earliest mathematical model was formulated by Kermack and Mckendrick in the early part of the twentieth century and their work was published in three articles between 1927 and 1933 [1]. This model called as the S-I-R Model is a model wherein the population is divided into compartments S, I, and R and where there is a flow of the individuals between the different compartments.

S: denotes the population of individuals who are susceptible to the infection, I: denotes the number of infectives (i.e) the individuals who are infectious and R: denotes the number of people who have recovered or have died. This compartment also called as the removed class will also consist of individuals who are immune to the disease. The total population is given by S+I+R = N. All these variables are functions of time t.

The basic mechanism driving a contagious phenomenon is the interaction between susceptibles and infectives. Therefore, the way this interaction is described is very important. Most of the epidemiological models are modeled by the law of mass action [2]. The rate at which effective contacts occur is taken to be proportional to the number of susceptibles and the number of infectives. This model assumes that the population is homogeneously mixing.

As implied by the variable, function of t, the model is dynamic and has been applied to various infectious diseases which have occured in the twentieth and the twenty first century. This model has been applied to the current Covid-19 Pandemic as well and there are a number of articles published on the same.

During an epidemic, the number of susceptible individuals decreases as more of them are infected and thus enter the infectious and recovered compartments. The disease cannot break out again until the number of susceptibles has built back up.

For e.g. In the current Covid-19 scenario, it is seen that social distancing and complete lockdown has resulted in the reduction in the number of susceptible individuals and thus has helped in the control of spread of the infection in the community.

Each individual of the population typically move from susceptible to infectious to recovered classes: S ->I ->R .

This classic S-I-R model is described by a system of ordinary differential equations given by

We see that

Though this system of equations is non-linear, an analytical solution exists for the same [3].

From the susceptible ‘S’ class to the infective ‘I’ class , the transition rate is given by βI/N where β denotes the contact rate of the infection. The infectives from the infective ‘I’ class move to the removed/recovered ‘R’ class with a recovery or a mortality rate γ. Since the dynamics of an epidemic, for example the Covid-19, are faster when compared with the dynamics of birth and death, hence, birth and natural death rates are not included in the simple S-I-R models.

From the above model, the basic reproduction number is calculated, given by

The basic reproduction number is the number of new infectives produced from a single infected individual in a population where all subjects are susceptible.

If Ro > 1, the infection spreads in the population and if Ro < 1, the infection is under control. The reproduction number is a very important parameter for controlling the transmission of a disease in a population.

For e.g. The basic reproduction number of the Covid-19 Pandemic is estimated to be 3.28 which exceeds WHO estimates of 1.4 — 2.5. Hence, it is very much necessary to bring down this value so that the disease can be brought under control [5].

References

1. Kermack, W. O.; McKendrick, A. G. (1927). “A Contribution to the Mathematical Theory of Epidemics”. Proceedings of the Royal Society A. 115 (772): 700–721.
2. Hethcote H (2000). “The Mathematics of Infectious Diseases”. SIAM Review. 42 (4): 599–653.
3. Harko, Tiberiu; Lobo, Francisco S. N.; Mak, M. K. (2014). “Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates”. Applied Mathematics and Computation. 236: 184–194.
4. Bailey, Norman T. J. (1975). The mathematical theory of infectious diseases and its applications (2nd ed.). London: Griffin. ISBN 0–85264–231–8.
5. Liu, Ying et al(2020). “ The Reproductive Number of Covid-19 is higher compared to SARS Coronavirus”. Journal of Travel Medicine.27(2):1–4 DOI: 10.1093/jtm/taaa021