# Introduction

News of the most recent outbreak of yellow fever passing 300 deaths in Angola has prompted fears of the World Health Organisation (WHO) regarding its epidemic progression. Vaccination in some regions of the outbreak are very hard to accomplish, as the process of vaccinating is both very expensive, and hard to enforce on a population to satisfy the necessary threshold (to be explained later). Tracking and predicting progression of disease has always been a difficult aim to accomplish. However, there are several mathematical models that may assist with this problem. These models will be explained and adapted to accommodate a range of underlying assumptions about the population and the method of transforming the disease, which will enable the research question to be answered. This is certainly area that is worthy of exploration, both personally and on a greater scale. I have been an avid follower of how diseases have captivated the attention of global media and prevention efforts, namely the Ebola virus and the Zika virus. I have specifically chosen the Yellow Fever virus as it is a currently developing situation, and has manageable statistics for an exploration of this magnitude.

# Theory of the SIR model, basic reproductive number, and herd immunity

## The ’basis’ or basic SIR model

The SIR model is a frequently used model to attempt to predict the rate at which a disease will spread through a population. SIR is an acronym for the three classes an individual may lie within: Susceptible, Infected, and Removed. An individual in the population may only exist within these three classes. There are also other assumptions in which the basic SIR model: There are a number of assumptions that are important to note in the basic SIR model. These are the following:

1. All individuals in the susceptible class have never been in contact with the disease and are all therefore susceptible to it

2. The population is constant; there are no natural births and natural deaths

3. In order to move from one section, an individual has to enter another. This would mean that the population is assumed to be constant. (This is acceptable as the removed section includes deceased individuals).

4. When an individual is removed, he/she is assumed to be immune for the remainder of his/her life, and is therefore unable to rejoin susceptible.

5. Every individual that is infected has equal probability of encountering an individual that is susceptible per unit time.

These assumptions will obviously limit the practical usages of the model, as the model’s outcomes would not represent results in a real world environment. However, the model may be adapted in order to incorporate dimensions from a real world scenario, such as that of Yellow Fever in Angola. In order to adapt the model to fit the situation, understand of the underlying, or basic model is required. This will be explained below. As stated above, individuals may be in one of three categories. The equations below show the number of people in each of the three categories as a function of time. \begin{aligned} S &= S(t) && \text{absolute number of people who are susceptible} \\ I &= I(t) && \text{absolute number of people who are infected} \\ R &= R(t) && \text{absolute number of people who are removed}\end{aligned}

The notation $$N$$ will be used for the total population value. It should be noted that as these are the only categories the population may be divided into, $$S+I+R=N$$ must be case for any value of $$t$$.
In order to determine the rate at which individuals transition between susceptible and infected and , a rate constant must be involved to state the percentage of the population that makes each of these transmissions. If values for $$t$$ are taken at discrete integer intervals, we can deduce the equation for the rate of transition, i.e. difference in the number of people in the susceptible category for time $$t$$ and $$t+1$$, or the number of people that have become infected in day $$t$$. This can be expressed in the equation below: $S_{t+1} - S_t = \frac{\mathrm{d} S}{\mathrm{d} t} = -\beta S_t I_t$ where $$\beta$$ is the aforementioned rate constant. $$Beta$$ is given a negative value, as the number of people who are classed as susceptible is decreasing, as they are moving into the infected classification. A similar method may be applied to calculated the rate at which individuals move from the infected to the removed category. The rate of people becoming infected would logically be the same as the rate of decrease in susceptible individuals, as individuals can only from susceptible to infected.
However, people who are already infected may have either recovered or died at this time, meaning they are moved into removed classification. This is explained mathematically below: \begin{aligned} I_{t+1} - I_t &= \frac{\mathrm{d} I}{\mathrm{d} t} = \beta S_t I_t && \text{This equation is incorrect as it does not take into account recovered individuals} \\ I_{t+1} - I_t &= \frac{\mathrm{d} I}{\mathrm{d} t} = \beta S_t I_t -\gamma S_t I_t && \text{ This equation includes individuals changing from infected to removed} \\\end{aligned}