** ****MATHEMATICAL MODELING OF INFECTIOUS DISEASES. Case Solution**

### Introduction

The words “diseases”, “infection”, and “outbreak”can be heard almost everywhere ever since the Covid19 pandemic has hit the world. Medical experts and researchers around the globe, are working to limit the spread of the virus and to find a cure for this disease. The way this or other diseases’ spread is an ongoing concern, and it is apparent that there is still much more that we do not realize about the dynamics of the infectious diseases. Our understanding of infectious diseases and how they spread can impact each of our lives.

The team of experts and researchers who are currently working on understanding the dynamics of Covid19 or any other infectious diseases, not only consist of doctors, or medical experts but also consist of mathematicians and engineers. Researchers from different disciplines have come together to use the mathematical modeling and computer software to tackle the infectious diseases.Mathematicians are the unsung heroes who are also helping us overcome a global pandemic. Mathematical modeling or mathematics, in general,helps the scientists in understanding the dynamics of any infectious disease, modelers collect the data of current outbreak and previous outbreaks to predict who is most vulnerable, who may get infected, how to limit the disease and how fast will this disease spread, etc. etc. When a mathematical model is built, mathematical analysis is joint with the computer simulations, which-aids the researchers in investigating the behavior of the model.

Mathematics is very useful in identifying the patterns of how a virus spreads and in finding the underlying structures to help prevent and design strategies to govern the outbreaks. The Centre for disease control and prevention (CDC) reported that they used mathematical modeling techniques to respond to the 2019’s pandemic.(Prevention, 2020). CDC reported using mathematical modeling during the decision phase, planning phase, resource provision phase, and in the implementation of the social distancing, etc. during Covid 19.

### How a Mathematical Model Helps in Understanding Infectious Diseases Transmission

Let us consider an example of a mathematical model, which is designed to examine the spread of a disease in a population of masses. This particular model is designed to read the patterns of human contacts, the period from coming in contact with the virus to becoming infectious, the length of sickness from the infection and immunity, etc. Once all of these factors are expressed in a model; the Scientist can make the following predictions:

- Total number of individuals who are expected to be infected during an outbreak.
- Total epidemic duration.
- When will the peak period starts.
- Expected number of deaths etc.

### Types of Epidemic Modeling

The two types of epidemic modeling, are:

- Stochastic modeling, and
- Deterministic modeling.

A stochastic model is used for estimating the probability distributions of outcomes. This model depends on random variations (unknown change in hereditary traits). Deterministic mathematical models are used when dealing with huge populations. In this model,the population is categorized into different subgroups and each subgroup represents a different stage of the disease outbreak.

### The SIR Model

The SIR model (Kermack & McKendrick, 1927)was developed in the year 1927. In this model,a fixed population is considered which is divided into three different compartments i.e.susceptible ( ), infected ( ), and recovered ( ). Image below shows the SIR model graph (Source Internet).

**S (0) =997, I (0) =3, R (0) =0 rates for infection ** **=0.4 and for recovery ** **=0.04**

Susceptible (S) represents the individuals that are not yet infected with the disease at the given time (t). Infected (I) represents the individuals who are infected with the infectious disease and tend to spread it to the population of people categorized under the susceptible compartment. Recovered (R) represents the individuals who,after being infected, recovered from the disease. Recovered not only contains individuals who were able to successfully recover from the infection but also individuals who died from it as well as the individuals who cannot get infected again or transmit the disease………………….

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