Introduction
Every country is susceptible to potential outbreaks of large scale diseases. These infectious diseases cause a multiplicity of problems, specifically extreme numbers of death with those afflicted. Diseases like SARS, HIV/AIDS, and Ebola have claimed the lives of many, but what if we could predict the spread of new infectious diseases? That is the purpose of this paper. Mathematics will be used to map how diseases could spread, as well as determining the susceptibility of infection in certain regions. The basic reproduction number of diseases will be calculated, followed by how said disease will spread throughout specific areas. By doing so, a plan of action can be established when the next pandemic arises. This plan encompasses providing greater amounts of vaccinations to those regions more susceptible, as well as ensuring an effective first response and containment.
In my lifetime, I have seen the outbreak of various diseases all over the world. Most notably, Ebola. I thought the outbreak of Ebola had been minimized, but upon watching the news I saw that it is still prevalent in the Democratic Republic of the Congo (WHO). I then wondered if there was a method to model the spread of the disease. Doing some research, I learned of the SIR model. A model of how diseases spread, along with a region’s susceptibility has the potential to save lives and reduce the hospitalization of many. This type of advancement in determining the spread of infectious diseases can provide intel that has the possibility of eradicating disease epidemics altogether.
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The mathematics that will be involved in this study are various topics in calculus, such as differential equations. Specifically, the SIR model will be used. Statistics, algebra, and functions will also be used in modeling the spread of diseases.
Investigation
The SIR model, as created by Kermack and McKendrick (MathWorld), calculates an estimated number of people that could be infected by a contagious disease in a specific area’s population. This model also has the capability of determining the recovery time necessary for patients within that selected area. The name SIR is an acronym for the variables used in this model: S refers to the number of susceptible people, I refers to the number of people infected, and R refers to the number of people who have recovered from the illness (MathWorld).
Variables
- The SIR model represents disease epidemics over a period of time thus making time the independent variable t, measured in days in this investigation (Smith and Moore).
- The SIR model divides the total population, N, into three classes which act as the dependent variables (Smith and Moore):
o represents the number of susceptible individuals to the disease
o represents the number of infected individuals
o represents the number of recovered individuals
The rate of change of the variables must be assumed in this investigation (Johnson).
- A constant population size must be assumed. The population size must be large and hold a stable number of people which excludes births, deaths, and immigration. No one can be added to the susceptible group, and the only way to leave this grouping is by becoming infected.
- Homogenous mixing must be assumed. Everyone has equal probability of encountering another and therefore face the same probability of exposure of disease by those already infected.
- A fixed fraction of recovery per day must also be assumed. For example, if the average infection duration is two days, it can be assumed that on average ½ of the infected population recovers each day.
Since the population is fixed and there are no outside influences that inflict death, the total of the number of people susceptible, infected, and recovered is equal to the total population. Thus, the equation below is derived:
There are two parameters relating to this model, and where both . is defined as the rate of contact between infected and susceptible persons and is defined as the rate of the duration of recovery. Both variables are impacted by the duration of the disease for those recovered, or D, and the mortality rate for those who die per day, or M (Modelling Infectious Diseases).
- is found using the equation, , where the mortality rate is divided by the number of susceptible individuals. It represents the rate at which a disease travels from a susceptible person to that of an infected person due to contact. , where 0 implies no infection rate and 1 implies a total infection rate. For example, if the mortality rate of a population is 50% and the number of susceptible individuals is 10, then the rate of infection or is
- is found using the equation, , where 1 is divided by the duration of the disease. An infected person can only recover once during the duration of an illness, thus verifying the equation. For example, if the duration of the disease is 5 days, then the recovery rate is .
The SIR model uses a system of three differential equations (Smith and Moore):
- where represents the rate of change of those susceptible to a disease over time. Susceptibility to the disease is determined by the number of susceptible and infected individuals, and the rate of infection. Once infected by the disease, the individual leaves the susceptible group therefore showing the proportional decrease between susceptibility () and infection (I).
- where represents the rate of change of those infected by a disease. Infection is determined by the number of people susceptible and infected, as well as the rate of recovery from a disease. shows the same inverse relationship as . The greater the population of I shows a greater decrease in the population of S.
- where represents the rate of change of those recovered from a disease over time. The recovery rate is dependent on the population of I because in order to recover, one must become infected first. As v
Using these equations, it can be determined that This shows that there is no net change in the total population meaning that the total population is constant.
Dengue Fever in the Philippines
Dengue is an easily transmittable disease originating in mosquitoes. There are an estimated 390 million dengue infections recorded each year globally, with most cases occurring in the tropical destinations of the world (WHO). Dengue fever is transmitted by the bite of an infected Aedes mosquito. Those infected will experience high fever, migraines, pain behind the eyes, joint and muscle pain, fatigue, nausea, vomiting, rash, and mild bleeding (WedMD). The first documented dengue fever epidemic was in the 1950s in the Philippines and Thailand (WHO). My extensive reading about and learning of the processes required to complete the SIR model will be applied to the Dengue Fever epidemic in order to gain an understanding of this model while utilizing an example.
The population of the Philippines in the 2014 will be considered in relation to the Dengue Fever epidemic (WHO). According to data from the WHO, the total population of the Philippines in 2014 is N=100,102,249, the infected population is I=121,580, deaths due to Dengue is 465, and mortality rate is 0.44. To find R, you must include both those who have died and those who have recovered seeing as both groups have achieved permanent immunity.
As seen in the graph above, there is a large drop in the susceptible group beginning at 17 days. The susceptible group began as a significant portion of the total population of the Philippines in 2014 and falls below half the total population after a duration of 32 days as reaches 50,000,000 people. After 52 days, levels off and remains at 20,000,000 susceptible individuals. Conversely, begins an increase after 17 days. levels off in tandem with after 52 days and remains at 80,000,000 recovered individuals. This shows an inverse relationship between the susceptible group and the recovered group. As the number of recovered individuals increase, the number of susceptible individuals decrease. Once exposed to and recovered from the disease, the individual has built up an immunity therefore not rendering them as susceptible. For , an increase in the infected number of individuals begins at day 17. A positive slope can be seen from day 1 to day 34, where the number of infected individuals reaches its maximum at about 18,000,000 people. After day 34, decreases to approach 0 between days 34 and 61. The decreasing trend of line works simultaneously with increase in line By creating this graph, I was able to see the relationship between the different variables, and successfully use differentiation to model the spread of Dengue fever on the Philippine population over the span of 60 days.
Wanting to further analyze the data, I individually graphed the susceptible, infected, and recovered groups. I then used polynomial regression to find a line of best fit for the data to then formulate an equation.
The shapes of the susceptible, infected, and recovered group graphs had a strange shape to me. I researched and I learned that the graphs take the shape of a sigmoid or logistic curve (Weisstein). Logistic functions use the equation (Balkew). Learning this, I realized my equations do not resemble the parent function. The limitations in Excel’s trendlines led me to come up with incorrect equations. Graphing the lines individually ended up providing me little to no information.
Conclusion
The SIR model is useful in the medicinal field. Not only does it estimate the spread of diseases within a certain population, but also dictates the recovery rate of said population. This can be used to determine necessity of vaccinations and the devising of plans to handle disease epidemics in specific areas. Through this exploration I realized the necessity of modeling data to provide advantages in the medical field. The information I gathered should be shared with the governments of countries who are prone to disease outbreak so they can take precautionary measures by way of vaccinations and health care. The model is widely used by scientists, but it holds many limitations. The model works under unrealistic assumptions like each individual will have an equal number of encounters with others in the population. In terms of tracking real-life epidemics, that is not the case. There is not an equal probability of everyone in the population encountering one another, thus there is greater room for error within the model’s estimates.
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