Mathematics Internal Assessment SIR Model to study the spread of infectious diseases in Nepal’s earthquake camps.
Table of contentIntroductionPage 3SIR modelPage 4SIR model and assumptions and limitationsPage 5 to 8Equations and CalculationsPage 9 and 10ConclusionIntroductionAs a child growing up I was that kid that caught every viral infection that was floating around. I was constantly sick, whether it be a stomach bug or a nasty flu.I didn’t particularly like being sick all the time of course so I have always been super cautious about being around anyone even remotely sick.This fear morphed into an irrational one pretty quick because I would sometimes worry about sitting next to someone who coughed even though I knew full well it wasn’t because they had anything infectious, it was because they smoked.Although irrational, I always took the approach of better safe than sorry and continued being extra careful anyway.When the earthquake happened in April 2015, hundreds of thousands of Nepalis were left homeless and temporary shelters were set up in fields, breeding ground for infectious diseases.
After the earthquake I volunteered for The National Center for Disease Prevention and Control and the Red Cross, and that is when I was exposed to the many ways in which the spread of diseases can be predicted and epidemics can be stopped.Among the Spatial and temporal models,Stochastic models and deterministic models, deterministic models stood out the most to me (probably because I didn’t even remotely understand the other ones), but I found the math interesting and for the first time I was curious to learn the math behind real everyday problems. The aim of this Internal assessment is to better understand the workings of the SIRS model along with its practical uses and limitations by applying it to a situation that I have seen personally in order to truly gauge its usefulness and effectiveness in real life post-disaster situations.
There are many models in place that Medical professionals use to predict the spread of infectious diseases in populations to better equip themselves to deal with the outbreak. One of the more simple Models is the SIRS Model.It can tell us vital information like how long the outbreak may last and how many people will be affected by it.(Make your Own SIR Model)The SIR model is also known as a compartmental system, with S,I,and R being 3 different compartments.Every person in the population (N) begins in one of the compartments and moves on to others as time progresses.In the model, the letters represent the following;S= Susceptible to being InfectedI= Infectious and Infected (got the infection from someone and is able to pass along the infection too.R= Recovered (moved from being infected to being recovered and cannot get or pass along the infection again.
)The assumption of this model is that all members of the population(N) are in one these 3 states, so S+ I+ R= NThe model also works on the assumption that the population in question is a closed population and so there are no people entering or leaving the sample population. It was first developed by Hermack and Mekendrick in 1927, making it the oldest and most common model used for modelling the spread of infectious diseases in the world.(SIR Model)Like any other model the SIRS model also works on a number of assumptions. Some of them are as follows:Homogenous mixing:The SIR Model works on the assumption that every person has equal probability of coming in contact with every other person. So in the case of the investigation of this IA, each person in a temporary shelter camp is equally likely to come in contact with any other person at the camp.Based on this we can say that every uninfected person in the camp has equal likelihood of being infected.Constant Sample Population:This model also assumes that the population size never changes.
This is important because 3 of the variables we will define will be fraction with the total population in the denominator, and these variables cannot change through the course of the calculations. So in the case of this IA investigation, we assume that no one enters or leaves the camp, and that the only possibility for the people in the camp is to be free of infection, to be infected or to have recovered. There is no taking into account the people that come and leave the camp, or any births or deaths that may occur inside the campImmediate effect:The final assumption of the SIR model is the effects are immediate. So in this case the infection and recovery happen in a split second. A person will get infected the very second they are exposed to the infected individual and that once the infection has lasted the number of days it is supposed to last, the person instantly becomes better,and can thereafter never be infected again. In short the 3 assumptions we must remember are:Susceptible people eventually become infected or remain disease free;Infected people go through the full period of infection and recover immediately once that period is over.
People that have passed through the infected period are never a part of the susceptible group again. The independent variable in this model is time. An independent variable is the variable whose variation does not depend on that of another (English Dictionary).
In this case it is time because it isn’t affected by any of the other factors that we measure to use in the SIR model.On the other hand the dependent variables are Susceptibility, Infection and Recovery.The dependent variable is one whose value depends on that of another (English Dictionary). This variable is also known as the response variable.
In this model S(t). I(t) and R(t) are all dependent on the number of days (t), hence they are the dependent variables.The first set of dependent variables counts people in each of the groups, each as a function of time.
Make your Own SIR Model) So, S(t) the number of individuals that are susceptible but have not yet been infected at any given time. Similarly, I(t) is the number of individuals that are currently infected and are contagious and at risk of spreading the disease at a given time, and R(t) is the number of individuals that are recovered that can neither get nor pass the disease again.Another set of dependent variables are also used int hsi model.
This set of variables are the number of people in each compartment S, I and R as a fraction of the total population N.Let F(s) be the susceptible fraction of the total sample population.F(s)= S(t) / N ;Let F(i) be the infected fraction of the total sample population.F(i)= I(t) / N ;And let F(r) be the removed or recovered fraction of the total sample population.F(r)= R(t) / N .
Now we must remember that each person is equally likely to run into another person in the camp. So keeping this in mind, the number of encounters between infectives and susceptibles is the product of S(t) and I(t), ie:S(t) x I(t)So to take an example if there are 100 people in a camp and 2 of them are infected first (so no one is part of the removed or recovered group yet) then the number of encounters would be 98 x 2 which is equal to 196.Another thing that is essential to know for this model to work is what the time for a cycle of infection is. So in the case of the common cold the average period o is 7 days.
(lasting in this case means a person has the infection and is at the risk of spreading it still even though they may not show all severe symptoms still.) So 7 days would be 1 t. 14 days would be 2t.However usually while measuring an epidemic, they usually set the time as 1 to make calculations uncomplicated. This means that usually the value of t can be ignored and is merely symbolic.Now based on that let us look at how we can measure the change in the 3 compartments.Just to recall, the three compartments are susceptible people , infected people , people who have recovered .
So in summary, the things we need to know to successfully run the SIRS model are the initial number of people that are infected and susceptible, the time it takes for recovery and the time for a cycle of infection (which again, is set as 1).When we are using this data, let us say that Ci are the number of people that go from being Susceptible to being infected in time t(1). Similarly, let us assume that Cr are the number of people everyday that go from being infected to being recovered.Based on these we have 3 formulas to find the number of people in each compartment at any given point in time(n).(Make your Own SIR Model)Sn=Sn-1 -(Sn-1/S)*(Ci*In-1)In=In-1 +(Sn-1/S)*(Ci*In-1)*-(Cr* In-1)Rn=Rn-1 + (Cr* In-1)(all equations are extracted from Make your Own SIR Model)While these may look very confusing once explained they make a lot of sense.
For example; Sn=Sn-1 -(Sn-1/S)*(Ci*In-1)The equation above simply means that the number of susceptible people today(Sn ) would be the number of susceptible people yesterday Sn-1 from which we would deduct the number of people that got infected yesterday and today.We would calculate the number of people infected yesterday by multiplying F(s) by the rate of infection. Similarly,In=In-1 +(Sn-1/S)*(Ci*In-1)*-(Cr* In-1)This simply means that the number of people that number of people infected today would be the number of people that were infected yesterday plus the number of people that got infected yesterday and today. Then you deduct the number of people that recovered yesterday. The final number is the number of people in the infected category today. Lastly, Rn=Rn-1 + (Cr* In-1)This equation simply means that the number of people recovered today are the number of people that were recovered yesterday as well as the number of infected people that recovered today.
Now to apply it to a scenario in an earthquake rehabilitation camp in Nepal. It is very unfortunate that there is no reliable data online for a particular camp in nepal which means we will just take a hypothetical sample number of people and do the calculations on that.After the earthquake 3.5 million people were left homeless. In one of the camps I worked at with the red cross, they told me they had closed the doors because they were at maximum capacity which was 2500 people. This is going to be out population N.
Now, in this population let’s say there are 5 people that have the flu. The infection fate of the flu is 0.47. The recovery rate of the flu is 0.21 per day.
And we are taking a time difference of a single day. So to calculate the number of Susceptible people on any given day let’s apply the formula with our sample numbers.Sn=Sn-1 -(Sn-1/S)*(Ci*In-1)When the numbers are plugged in we get:Sn= 2495-(2495/2500)*(0.47*5)That is 2492.7 people that are the new number of susceptible people. This makes sense as the days go on the number of susceptible people does need to decrease.Now for the number of infectious people for that day,In=In-1 +(Sn-1/S)*(Ci*In-1)*-(Cr* In-1)Replacing the values we getIn=5+(2495/2500)*(0.47*5)-(0.
21*5)That would be 6.29 people that would be infected the very next day.For the final Value;Rn=Rn-1 + (Cr* In-1)Replacing the values we haveRn=0+(0.
21*5)Which would result in 1.05Now adding all the new numbers we have;1.05+6.29+2492.
7 we get 2500.This is how we know that our calculations are accurate. Because we are working with a closed population, the total sum of the 3 categories on any given day has to be 2500.
When this example is applied for say 20 days you can get the numbers of people in each category on day 20.ConclusionThe SIR model is used in many different ways nowadays.By estimating the susceptible population and the kind of infections coming up people can calculate the vaccinations needed. Researching different populations using SIR model can tell us the most effective ways to approach and tackle public health issues in different demographics.This helps save important resources like money and time.However, because there are so many limitations and so many things are based on assumption, it is very possible to get completely inaccurate results as well.The SIR model is the one of the many that can be used to find the spread of epidemics in the world.Although it is widely used by scientists, the limitations are very large and hence the results are questionable.In real life situations it is much harder to track the spread of a disease because of the many variables involved.Overall I had a chance to explore many topics in math that were taught in class as well as topics that we have not yet studied, and I was able to satiate my curiosity through the completion of this IA.Bibliography”SIR Model.” From Wolfram MathWorld, mathworld.wolfram.com/SIRModel.html.Make Your Own SIR Model, www.pandemsim.com/data/index.php/make-your-own-sir-model/.English Dictionary, Thesaurus, & grammar help | Oxford Dictionaries.” Oxford Dictionaries | English, Oxford Dictionaries, en.oxforddictionaries.com/.