IntroductionDiseases are scary, but they’ve been around longer than most things alive today.
Sometimes these diseases can be minor like the common cold with small symptoms that are at times an annoyance, other times they can be very dangerous like the infamous names of Ebola, AIDS or Zika which often fill us with dread. What has often intrigued scientists has been the spread of these diseases, some people getting infected and other escaping the burden of disease. In the past century, mathematics has helped us predict the spread of diseases and reveal infection parameters. The mathematics behind these things are all data-driven with case reports from doctors and organizations being the most valuable sources of data. Which allows us to know the number of cases in a specific amount of time in a specific area.
This data also allows for the investigation of social effects, like changes in birth rate. This dynamic of the spread of diseases requires a variety of mathematical tools like model creation and differential equations. Exploration When I started high school, science has always been my favorite subject. I liked every single part my freshman biology class from cells to ecology, no matter what it was, I found everything interesting. The months following freshman year I began watching more science-related shows in my free time.
One show that truly sparked my interest in pathology was Helix. It was a Netflix show about epidemiologists going to investigate an outbreak at a laboratory in Antarctica. After finishing the show my new obsession was viruses and how they function and spread. Then a couple years later I’m in class and my teacher tells me to start thinking about interests or topics that we’d like to write about for our Internal Assessment and so I start panicking asking myself “what do I like”, “I don’t know!”.
So I made a list, and at first, it wasn’t a real list because the topics were things I wasn’t really passionate about. Then it comes to me, I could do something on pathology.I then asked myself if there as a way I could examine the way a disease spreads then that would be excellent. I searched what I could I do with pathology and I was stumped, but a few hours later I found an SIR model. An SIR model is an epidemiological model that can determine the theoretical number of people infected with a contagious disease in a closed system.Once I encountered this I knew that the basis of my exploration would be this SIR Model and how to investigate a spread of a disease. It was 1854 and a dangerous new disease called Cholera was brewing up a storm.
It resulted in being one of the worst cholera outbreaks in all of history.(Paneth 1998) This was in part due to the city’s terrible sewage and sanitation systems as well as officials not knowing how the disease spread. Due to the lack of knowledge on situations like these the amount of people that became infected was much higher than if it were to occur today. This exploration will analyze the spread of cholera through the nation of the The Democratic Republic of the Congo with a urban population from the early 2000s. It will model this epidemic and the predict what could happen in the future.
The following table and graph display data on the number of cases and deaths in different African countries.(World Health Organization 2003) Table One:Weekly Epidemiological RecordCountry CasesDeaths Cameroon668Democratic Republic of the Congo31,6581979Liberia1,1150Mozambique24,375342Nigeria 5,429154Somalia 2,775159Uganda2,274133Zimbabwe3,125192Figure 1:Weekly Epidemiological Graph Typically to being with it is important to know the number of people infected as well as the number of people who were infected, but then recovered. These people now have an immunity and can no longer become infected. If one were to disregard immigration and emigration of people from the area then logically the rest of the population can potentially become infected. From this one can divide the DRC’s (Democratic Republic of the Congo) population (51,964,00 people) into three groups and for purposes of the investigation those who have died and can no longer affect others will be put with those who are immune. The infectedThe immune The susceptible To begin with, one must identify the independent and dependent variables. For this investigation time, in days, will be the independent. While the dependent will be separated into two sets.
The first one will be the number of people in the different groups as a function of t, time. S=S(t) which is the amount of susceptible peopleI=I(t) which is the amount of infected peopleAnd R=R(t) which is the amount of people who recovered(immune or dead)The second one will be the proportion of people in the different groups. Then N will be the total population. s(t)=S(t)N which is those who are susceptiblei(t)=I(t)N those who are infectedAnd r(t)=R(t)N those who have recoveredUsually in a perfect situation, it would be expected that those who are susceptible decrease overtime, those have recovered increase overtime, and those who are infected increase and then decrease overtime. And no matter at what point in time, s(t) + i(t) + r(t)= N. Keep in mind that this investigation is still choosing to ignore births and migration. So the only manner in which someone can leave one group is by becoming infected, dying, and or recovering. Something else that should considered is that both the proportion of susceptibles and infected depends on the amount of contact between each of them.
For the investigation one should suppose the infected people have a set number of contacts per day, b. And if the population is continuously interacting each infected person should create a set number of new infected individuals per day, which will be shown as bs(t). For terms of simplicity the investigation will also assume the number of infected who recover on any day is a fixed proportion, k. Now one can observe the derivatives of the dependent variables. The first equation will be the change in the amount of susceptibles equals the negative number of contacts multiplied by the proportion of susceptibles out of the total population multiplied by the amount of infected people. dSdt = -bs(t)I(t)The amount of infected was included in this equation because they directly affect those who can still get infected.
The negative sign was included because there shouldn’t be any contact. Then we can obtain the differential equation for the change in the susceptible proportion of people from the total population which should equal the negative number of contacts multiplied by the product of the susceptible and infected proportions. dsdt=-bs(t)i(t)The second equation is the change of the amount recovered should equal the fixed number of people infected that will recover multiplied by the proportion of infected from the total population. drdt=ki(t)This follows what the investigation has assumed as it includes the the fixed fraction of those that will recover and the proportion of infected from the population. The third equation is the change of the amount infected should equal the fixed fraction of contacts multiplied by the product of the proportion of susceptibles and infected subtracted by the set amount of people that will recover and then finally multiplied by the proportion infected. didt=bs(t)i(t)-ki(t)Now one can complete this model by placing the values indicated at the beginning of the investigation and stating the amount of people initially infected is 20 people. S(0)=51,964,000 I(0)=20 R(0)=0The time of days is zero since the “epidemic” has just begun. Then from these values the conditions becomes(t)=S(t)Ns(0)=S(0)Ns(0)=51,964,00051,964,000s(0)=1i(t)=I(t)Ni(0)=I(0)Ni(0)=20 51,964,000i(0)=.
00000038i(0)=3.810-7r(t)=R(t)Nr(0)=R(0)Nr(0)=051,964,000r(0)=0Which then converts to dsdt=-bs(t)i(t) s(0)=1didt=bs(t)i(t)-ki(t) i(0)=3.810-7drdt=ki(t) r(0)=0Since the values of b and k have not yet been set the investigation will provide an example:The estimated average period of infectiousness is 5 days so that would mean k=15, basically on average then one fifth of the currently infected population becomes noninfectious each day, either through death or recovery. Then if we say that each infected would make a possibly infecting contact every two days then b=12. The following graph shows the solution curves for these choices of b and k.
Now if we substitute these numbers into our equations we get dsdt=-1213.810-7 dsdt=-1.910-7didt=1213.810-7-153.810-7didt=6.4610-14drdt=153.810-7drdt=0.
7610-7ConclusionAlgebraically from this we can conclude that at the beginning of the epidemic there would be a relatively low level of infection due to the slow rate of b and that eventually at its peak it would not need to rise that much to a higher level due to the amount of days of infectivity. Like the level of infected the level of recovered will be low at the beginning because no one has been infected then as the days continue gradually increase reaching its max and leveling out. Finally the level of susceptibles will begin at a high level then as the recovered exponentially increase the susceptibles will exponentially decrease at the same time, until both functions reach their max and min and level out.Now we must consider that this was an experiment and that the degree of accuracy is relatively small thus one must include all the possible decimal places to allow for this close range of accuracy.From this model one could predict what might happen in future epidemics and establish situations to observe and possibly prepare for these situations.
One could extend this to any disease and just substitute the different values specific to the individual disease.