The graph shows that initially, the number of people infected increases steeply, however, over a longer period of time, the numbers eventually decrease. This happens simultaneously as the number of people recovered increases because as those infected decreases, they are being transferred into the recovered category. The main reason is due to the increased awareness of the disease leasing to further medical support being given in order to help combat the transmission of the disease. Furthermore, an increased awareness results in more people being aware of methods of protection. The steep increase in the beginning of the first few days is most likely to be due to the great uncertainty that lied with YF allowing a greater rate of transmission. The peak of the graph illustrates the maximum number of people ever to be infected and after this point, there is a transition whereby the numbers decrease.
The number of people susceptible to the disease remains constant for the first 10 days, and then it steeply decreases to create a negative sigmoidal curve. This means that it is shaped like the letter S, but in reverse. It must be noted that the number of people never reaches 0, and only tends towards it allowing the epidemic to reoccur in the future. The only way for the number of susceptible to reach 0 is through the vaccination as this acts as a vehicle to remove the disease from the population. The number of people susceptible remains constant at the beginning, which is similar to the small increase in the number of people infected. However, as there are more people infected, there is a steep decline in the number of people susceptible to the disease. This is because being the number of people infected comes from the number of people susceptible and they are connected. Therefore, as the number of people infected begins to decline, the number of people susceptible begins to level off. This is due to the fact that everyone infected is eventually becoming recovered, thus reducing the numbers of those who are infected. Therefore, there is very little change in the number of people susceptible to the disease towards the end of the two month period.
The number of people recovered from the disease, slowly increase at the beginning with the slow rate of infection. However, as the number of people infected increases dramatically, this leads to a consequent steep increase with the number of people recovered, until eventually levelling off simultaneously to the number of people susceptible. The line illustrating the number of people recovering increases concurrently as the number of people susceptible decreases. This is because susceptibility and recovery are inversely proportional to one another. However, the number of people recovered from the disease, never reaches the total population, and only tends towards it. Furthermore, this graph illustrates cumulative distributions through the positive sigmoidal curve on the graph.
The graph also shows that the total population remains constant throughout the two month period via a linear correlation. This is because, as established earlier, and in order to detect a change in something we need to differentiate it. In this case, the graph suggests no change, so the differentiation must be equivalent to 0.
Therefore, and by substituting the differential equations 1, 2 and 3, we get the following:
Thus, there is no change in the population and it will remain constant in a given period of time.
The graph is useful because it allows me to see the interaction between the different variables and it is interesting to relate it to differentiation to determine the changes over time.
Evaluation
I have diveded my evaluation into two parts which consist of the advantages and disadvantages of the model used and therefore what could have been done to make it better.
The advantages to the model include:
1. It is very quick to model the data having found the values for the respective parameters and transition probabilities to allow immediate assessment of the condition that is present. This results in instant evaluation of the situation as well as valid prediction of the spread of disease in the future.
2. The model is widely used and also widely understood by the medical community making it easier to explain the effects of the epidemic
3. This model is clear and easy to understand in order to distinguish between the number of people susceptible, infected and recovered
4. The mechanism to create the data is flexible, allowing it to be easily altered if certain values are incorrect or have changed
5. It is computationally cheap and there are other software available with very small time intervals allowing it to be more accurate which I could not complete in excel.
The disadvantages to the model include:
1. The calculation of the beta values and gamma values are often inaccurate because small deviation from the ‘correct value’ can result in great changes in the overall model. Data itself can be fairly unreliable especially, in countries where death counts are difficult to manage.
2. In order for the model to be calculated correctly, you need the right form of data including the number of people infected, recovered and susceptible. This data can be very particular and calculating the number of people susceptible as the number of people left over from taking away the number of people infected and recovered from the total population, is not always the most reliable method.
3. This model is only effective for small environments with heterogeneous population density distribution
Conclusion
By using an SIR model, I was able to see the importance of modelling data, especially in the field of medicine. This is because, in order to cope with the rapid changes in the medical sector, many governments must find methods to sustain and maximise the efficiency of the available health care systems. One of these methods includes mathematical modelling which is becoming increasingly important in helping identify the future of certain diseases. The application of mathematical models on diseases can be extended to include the effects of vaccination and impacts of herd immunity on an outbreak as well. This can help to determine different factors which can help reduce the mortality rate.
From doing my exploration, I gained further insight in the ways in which modelling can be used to predict the apparent spread of diseases in order to inform health care superficial of the necessary precautions that must be in place. Nonetheless, similar to most models, the SIR is also subject to limitations as often a model is a simplified representation of the real situation and often this can lead to over simplification, creating conflicts between simplicity and complexity. Ultimately, the aim of modelling is to clarify certain concepts, but models often attempt to mimic a real life situation through introducing many variables and can lead to further confusion.
However, the results obtained from modelling data can lead to differing perspectives and interpretations. This is due to the unequal distribution of data across the world whereby in countries such as Angola, there is very little access to the statistics which makes it difficult to make constructive predictions concerning the outbreak. However, in countries such as UK, the data is more widely available making developing countries and their governments dependent on them. This caused an exaggerated media coverage leading to the development of irrational fears which promoted the prevalence of more resilient and contagious diseases such as tuberculosis. Due to the inflation of the situation, much research has been conducted in order to create a potential vaccine against it.
Through completing this exploration, I am able to see the impact of mathematical modelling and the influences it has in helping scientists to analyse epidemics and help prevent further disruption. The SIR model which I used showed the general trend of the epidemic, however due to its limitations which eventually outweighed the advantages, the model did not precisely correspond to the real life data, although they mostly illustrated similar correlation.
Therefore, through my exploration, I have gained further insight into the uses of mathematical modelling in order to determine the spread of diseases as well as evaluating its flaws. Having chosen YF as the disease of concentration, as it is very relevant to the current situation in Africa, it has enabled a realistic understanding of its rate of transmission. Moreover, this task had allowed me to combine my interests in maths alongside a disease with which I have great interest in, in order to simulate an analytical study and gain further understanding of the ways in which health care professions rely on mathematical studies to help them make important decisions in improving the healthcare of the population.