Epidemiology SEIR/SIR Models

Epidemiology: A Compartmental Analysis of

Recent Influenza Data Using Mathematica

Murphy, [email protected]

Sidze, [email protected]

Moreta, [email protected]

Velez, [email protected]

Foley, [email protected]

Figueroa, [email protected]

December 11, 2014

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 SIR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.1 SIR Model w/ Vaccinations . . . . . . . . . . . . . . . 84.2 SIR Model w/ Fatalities . . . . . . . . . . . . . . . . . 11

5 SEIR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Predicting Future Data . . . . . . . . . . . . . . . . . . . 19

7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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1 Introduction

In our society, some groups (non-governmental and governmental organiza-tions,) try to predict and plot various diseases in areas around the world.These areas are usually broken up into zones to allow for easier analysis andmanagement. The Center of Disease Control and Prevention (CDC) and theWorld Health Organization (WHO) do this annually with common diseases.The development of this project consisted of various compartmental modelsthat numerically exemplified the spread of a common virus, influenza.

[6] This project will mainly focus on two different models, SIR and SEIRmodels. To begin the models will be created and compared to the CDCsinfluenza data in order to obtain the best model after testing several pa-rameters in in Mathematica using several different parameters that weregathered based on the information from the CDC on the virus.

The first model tested was the SIR model which consist of susceptible,infected and resistant which are the essential variables in the model. AnSEIR model consists of an extra variable E which is considered exposed,this model is more accurate because it accounts for that extra variable. Theexposed variable makes the SEIR model more accurate because it accountsfor people who have been unprotected from the virus but are not infected.

[3] When including a term to compensate for vaccinations, the term al-lows the model to lower the amount of people who are infected becausethe susceptible can skip the infections and move forward to being resistant.Lowering the infections and increasing the amount of resistance. In orderto account for vaccinations we must adjust the original SIR equations andproperly add a new variable to account for the vaccines.

Over time the number of susceptible will asymptotically approach zero whilethe amount of infected should peak in the middle and then by the end, beginapproaching zero. The amount of resistant people eventually plateaus theoriginal amount of people susceptible to the virus.

2 Data Collection

Over the past couple of decades, the Center of Disease Control and Preven-tion (CDC) has acquired data on the influenza virus through many collab-

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orating laboratories. [1] The CDC receives its information from the Epi-demiology and Prevention Branch in Influenza Division. This branch col-lects, gathers and analyses information on the influenza virus in the UnitedStates, year round. This branch has different categories of surveillance totrack influenza activities, one being Virological Surveillance. In this type ofsurveillance, 85 U.S. World Health Organization (WHO) laboratories and 60National Respiratory and Enteric Virus Surveillance System (NREVSS) lab-oratories cooperate together. Weekly, they report to the CDC the numberof respiratory specimens tested and the number of those specimens whichconclude positive for the influenza virus types A and B. A majority of theU.S. WHO labs participating in this program also report the influenza Asubtypes H1 and H3, along with the age groups of the specimens collected.Another type of surveillance used by the Epidemiology and PreventionBranch in Influenza Division is Outpatient Illness Surveillance. This typeof surveillance includes a network, called the U.S. Outpatient Influenza-like Illness Surveillance Network (ILINet), which gathers information onpatient visits for influenza-like illness. More than 2,900 outpatient healthcare providers from all 50 states participate in ILINet. Roughly 1,800 out-patient healthcare providers around the country report the total number ofpatient visits and the number of those patients with influenza-like illnesses,by age group, to the CDC.

3 Models

[7] The SIR model examine three compartments for an infectious diseasewhich are; S for the susceptible to the disease. I for the infected by the dis-ease, and R for those who either recovered from the disease or were resistant.Each letter represent the number of people in each compartment as a func-tion time. Compartments have completely different behaviors. Initially, thewhole population is susceptible except for those who brought in the diseases,which are the only infected. That is, initially, the sum of the susceptible,infected, and the resistant/recovered equals the population - which shouldequal zero if no one is dying. However, and the end of the outbreak, It isassumed relatively no one is infected, nor susceptible. Therefore the wholepopulation is in the recovered compartment. The draw back for this modelis that it doesnt take into account time infected people become infectious.To improve the SIR model and get rid of it draw back, an E is introduced tothe model. The SEIR model introduce the function E which is the numberpeople who are infected but not yet infectious over time.

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4 SIR Model

[6] The first model that we used was an SIR model. This model is very basicand gives the general idea of how disease spreads over time. In this modelS is the number of people that are susceptible to the virus; I is the numberof people that are infected, and R is number of people that have recoveredfrom the virus. All of these variables are constants while and are variablesthat need to be found. In order to solve for these unknown we needed touse a mathematical program that could solve this many times.

dSdt = IS

dIdt = IS I

dRdt = I

[6] When we used this program to find the unknown variables we had to findwhich values worked the best. Finding what value makes the graph closestto the ideal graph is called the goodness of fit characteristic. The goodnessof fit describes how well a model fits a set of observations or sets of data.Typically, goodness of fit measures the discrepancy between expected andobserved values of a given model.

The equation below explains how to calculate the goodness of fit, whereOi is observed data and Ei is expected/theoretical data.

2 =ni=1

(Oi Ei)2

The smaller this value is the better the model becomes. To make this modelbecome more realistic we can add a vaccination constant as well as fatalitiesto these equations. This brings in the effects of what happens when peopletry to control the disease. When we add these we find out how many peopledie from this disease over time.

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Figure 1: SIR Model (2012-2013) - S,I,R vs. Time

Figure 2: SIR Model (2012-2013) - Model Infections vs. CDC Data

The SIR model above was calculated using incremented data in Mathe-matica. The table of values, along with their Goodness of fit characteristicswas generated over about 10 minutes of calculation time. Unfortunately,this method was very slow in determining an accurate group of parametersdue to being forced to recalculate with a more precise number/step. Listing1 below shows working code to generate the above outputs.

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The code below also shows how to calculate the Root Mean Squared Devia-tion (RMS) for the model. Percent error for each point of CDC data is alsocalculated.

Listing 1: SIR Model - Incremented Combinations

1 ClearAll["Global*"]2

3 CDCdata = {{0, 161}, {1, 197}, {2, 298}, {3, 339}, {4, 409}, {5,4 614}, {6, 848}, {7, 1359}, {8, 1985}, {9, 3346}, {10, 4590}, {11,5 5938}, {12, 6511}, {13, 6514}, {14, 6952}, {15, 5982}, {16,6 5167}, {17, 4077}, {18, 3126}, {19, 2457}, {20, 2136}, {21,7 1848}, {22, 1680}, {23, 1497}, {24, 1269}, {25, 1007}, {26,8 898}, {27, 644}, {28, 459}, {29, 380}};9

10 Squarediff[beta_,11 nu_] := (sol =12 First@NDSolve[{s[t] == -beta i[t] s[t],13 i[t] == beta i[t] s[t] - nu i[t], r[t] == nu i[t],14 i[0] == 161, s[0] == 20328, r[0] == 0}, {s, i, r}, {t, 0, 29}];15 Sum[(((CDCdata[[j, 2]] - i[CDCdata[[j, 1]]]) /. sol)2), {j, 1,16 Length[CDCdata]}])17

18 (*Creates table of values in the form {{beta,nu}, goodness}*)19 values =20 Flatten[Table[{{a, n}, Squarediff[a, n]}, {a, 0.000029, 0.000033,21 0.0000001}, {n, 0.2, 0.3, 0.001}], 1];22

23 (*Finds the smallest value of Goodness*)24 min = Min[ values[[All, 2]]];25 pos = Position[values, min, 2, 1];26 lowest = values[[pos[[1, 1]], 1]];27

28 (*Prints smallest values and saves as variables*)29 BetaSolved = lowest[[1]]30 NuSolved = lowest[[2]]31

32 *(Plug in solved values*)33 Solved = First@34 NDSolve[{s[t] == -(BetaSolved) i[t] s[t],35 i[t] == (BetaSolved) i[t] s[t] - (NuSolved) i[t],36 r[t] == (NuSolved) i[t], i[0] == 161, s[0] == 20328,37 r[0] == 0}, {s, i, r}, {t, 0, 29}];38

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39 (*Generate plot of all solutions as well as infected vs. CDC Data*)40 Show[Plot[{s[t] /. Solved, r[t] /. Solved, i[t] /. Solved}, {t, 0,41 32}], ListPlot[CDCdata]]42 Show[ListPlot[CDCdata], Plot[{i[t] /. Solved}, {t, 0, 32}]]43

44 (*Generate table of all errors for each point*)45 i = i /. Solved;46 ErrorFind[num_] :=47 Abs[(((i[num]) - (CDCdata[[(num + 1), 2]]))/i[num])*100];48 Error = Table[{{inc}, {ErrorFind[inc]}}, {inc,49 0, (Length[CDCdata] - 1) , 1}];50

51 (*Find maximum error for research purposes*)52 MaxError = Max[Error[[All, 2]]]53 Pos = Position[Error, MaxError];54 MaxErrorPoint = Pos[[1, 1]] - 155

56 (*Calculate Root Mean Squared of model*)57 RMS = Sqrt[min/Length[CDCdata]]

Results = 0.0000324 = 0.234MaxError = 51.4648%RMS = 602.036

By using a random number function within a range, it was easier to sethow many numbers were to be tested for each parameter, plus account for acertain number of decimal points. Almost every time, the random generatorwas more accurate at finding a better fit for the infected function againstthe CDCs data. Listing 2 below shows modified code to the values tableby using randomly generated combinations for and .

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Figure 3: SIR Model (2011-2012) - Randomly Generated Parameters

Listing 2: SIR Model - Random Combinations

1 (*Creates table of values in the form {beta, goodness}*)2 values = Flatten[3 Table[{{a, n}, Squarediff[a, n]}, {a,4 RandomReal[{0.00003, 0.000035}, {150}]}, {n,5 RandomReal[{0.22, 0.25}, {150}]}], 1];

Results = 0.0000324242 = 0.234369MaxError = 51.4867%RMS = 601.971

4.1 SIR Model w/ Vaccinations

Individuals get vaccinations either at birth or later in life to immune them-selves from an epidemic disease. According to dictionary.com vaccine is anypreparation used as a preventive inoculation to confer immunity against aspecific disease, usually employing an innocuous form of the disease agent,as killed or weakened bacteria or viruses, to stimulate antibody production.Therefore, the parameter p for vaccination must be added to the SIR model

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to account for the vaccinated individuals since the vaccinated people cantbecome infected.

dSdt = ISN SdIdt = ISN IdRdt = I + S

By using a new term to account for people getting a vaccine (in this case aflu shot,) the model accuracy was greatly improved compared to other mod-els. Seen below, the vaccination models more realistically aid in solving theSIR model considering how vaccinations effect real CDC disease/infectiondata.

Figure 4: SIR Model (2011-2012) - Including Vaccinations

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Figure 5: SIR Model (2011-2012) - Including Vaccinations

Listing 3: SIR Model - Vaccinations

1 Squarediff[beta_, nu_,2 gamma_] := (sol =3 First@NDSolve[{s[t] == -beta ((i[t] s[t])/population) -4 gamma s[t], i[t] == beta ((i[t] s[t])/population) - nu i[t],5 r[t] == nu i[t] + gamma s[t], i[0] == 161, s[0] == 20328,6 r[0] == 0}, {s, i, r}, {t, 0, 29}];7 Sum[(((CDCdata[[j, 2]] - i[CDCdata[[j, 1]]]) /. sol)2), {j, 1,8 Length[CDCdata]}])9

10 (*Creates table of values in the form {beta, goodness}*)11 values = Flatten[12 Table[{{b, n, g}, Squarediff[b, n, g]}, {b, 0.55, 0.65, 0.01}, {n,13 0.15, 0.2, 0.01}, {g, 0.0, 0.0001, 0.00001}], 2];

Results = 0.63 = 0.2 = 0.0001MaxError = 57.6483%RMS = 843.905

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4.2 SIR Model w/ Fatalities

Above we assumed constant population to find parameter that closely rep-resent our data. This assumption is unrealistic and doesnt reflect the realworld; since all populations are bound to have a certain degree of dynam-icity. Birth rate and death rate in a community can never be equal for aduration of 30 weeks. In an attempt to make the SIR model more accu-rate, the death term was added to the original model. One should note thatthe flu is usually not fatal, but this was included to possibly account foranomalies in data and/or just for experimentation. Below is a new set ofdifferential equation including the death term.

dSdt = N S INdIdt =

IN S ( + )I

dRdt = I R

[2] It can be noticed that the terms N S, I R were added.

Definition of TermsN: Number of birthsS: Death of susceptibleI: Death of infectedR: Death of recovered

Modeling the effects of a fatal disease on a population is extremely fas-cinating and helpful. Below are several models of the effect of a death rateamong the infected for a disease. As shown in the different sets of graphs,a smaller death rate doesnt always result in the fewest amount dead. Un-fortunately, for most models that include fatalities, there is somewhat of asweet spot where the amount of deaths is maximized. Surprisingly, if 50%of infected beings died, it would be less detrimental to the population as awhole since the infection would less-widely spread.

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Figure 6: SIR Model (2011-2012) - Death Rate (0.1%)

Figure 7: SIR Model (2011-2012) - Death Rate Population (0.1%)

Below are the models for the maximum killed with a death rate of 18.5%of the infected. It is interesting to note that this happens when the resistantand susceptible functions are asymptotic at the same y-axis value. Thisphenomena is still unexplained.

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Figure 8: SIR Model (2011-2012) - Death Rate (Max - 18.5%)

Figure 9: SIR Model (2011-2012) - Death Rate Population (Max - 18.5%)

Being that the death rate can be a bit more complicated, the full codebelow for Mathematica is below. Manipulate is used to allow for playingwith values. Note that and are specified - these are values from solvingthe SIR model in Listing 1.

Listing 4: SIR Model - Death Rate w/ Manipulation

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3 CDCdata = {{0, 161}, {1, 197}, {2, 298}, {3, 339}, {4, 409}, {5,4 614}, {6, 848}, {7, 1359}, {8, 1985}, {9, 3346}, {10, 4590}, {11,5 5938}, {12, 6511}, {13, 6514}, {14, 6952}, {15, 5982}, {16,6 5167}, {17, 4077}, {18, 3126}, {19, 2457}, {20, 2136}, {21,7 1848}, {22, 1680}, {23, 1497}, {24, 1269}, {25, 1007}, {26,8 898}, {27, 644}, {28, 459}, {29, 380}};9

10 p = Interpolation[CDCdata];11 Population = 20328;12

13 BetaSolved = 0.0000309;14 NuSolved = 0.2134;15

16 sir = ParametricNDSolveValue[{s[t] == -BetaSolved i[t] s[t],17 i[t] == BetaSolved i[t] s[t] - NuSolved i[t] - Death i[t],18 r[t] == NuSolved i[t], i[0] == 161, s[0] == 20328,19 r[0] == 0}, {s, i, r}, {t, 0, 50}, {Death}];20

21 Manipulate[22 Module[{s, i, r}, {s = sir[a][[1]], i = sir[a][[2]], r = sir[a][[3]]};23 Grid[{{Plot[{i[t], r[t], s[t]}, {t, 0, 50}, PlotRange -> {0, 20000},24 ImageSize -> Medium],25 Plot[s[t] + i[t] + r[t], {t, 0, 50},26 ImageSize -> Medium]}}]], {{a, 0.001, "Death Rate"}, 0.0, 1,27 Appearance -> "Labeled"}]

5 SEIR Model

An SEIR model is used to find out how different types of diseases react in areal life situation. This model can be used to predict how many people aregoing to die from the disease or be infected from it. This model is also usedto determine how many people are going to be recovered after they becomeinfected from the disease. [5] The SEIR model consists of the variables S,E, I, and R. The variables represent people that are Susceptible, Exposed,Infected, and Recovered. The variable S is going to be used to representthe number of people that are not yet infected but have the chance to besusceptible to the disease. E is the variable that we are going to use torepresent the people that are going to be exposed to the disease but are notnecessarily going to be infected towards it. [4] The people exposed have thechance to potentially be immune to the disease. The variable I is going to

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be used to represent the number of people that are infected by the disease.And the R is the variable that is going to be used to represent the numberof people that are recovered from the disease. The equations that were usedfor the model are:

dSdt = N S IN SdEdt =

IN S (+ )E

dIdt = E ( + I

dRdt = I R

The variables N, , , , and are also included in this model. N is thetotal population, and was used as the transmission rate of disease. Thevariable was used as the natural death rate of the sub-population. List-ing 5 below includes full Mathematica code for solving an SEIR model withincremented combinations for all variables.

Listing 5: SEIR Model - Incremented Combinations


3 CDCdata = {{0, 161}, {1, 197}, {2, 298}, {3, 339}, {4, 409}, {5,4 614}, {6, 848}, {7, 1359}, {8, 1985}, {9, 3346}, {10, 4590}, {11,5 5938}, {12, 6511}, {13, 6514}, {14, 6952}, {15, 5982}, {16,6 5167}, {17, 4077}, {18, 3126}, {19, 2457}, {20, 2136}, {21,7 1848}, {22, 1680}, {23, 1497}, {24, 1269}, {25, 1007}, {26,8 898}, {27, 644}, {28, 459}, {29, 380}};9 Population = 20328;

10 Squarediff[mu_, beta_, alpha_,11 nu_] := (sol =12 First@NDSolve[{s[t] ==13 mu*Population - mu*s[t] - beta*(i[t]/Population)*s[t],14 e[t] == beta*(i[t]/Population)*s[t] - (mu + alpha) e[t],15 i[t] == alpha*(e[t]) - (nu + mu)*(i[t]),16 r[t] == nu*i[t] - mu*r[t],17 i[0] == 161,18 s[0] == 20328,19 r[0] == 0, e[0] == 0},20 {s, e, i, r}, {t, 0, 29}];21 Sum[(((CDCdata[[j, 2]] - i[CDCdata[[j, 1]]]) /. sol)2), {j, 1,22 Length[CDCdata]}])23

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24 values = Flatten[25 Table[{{a, b, c, d}, Squarediff[a, b, c, d]}, {a, 0, 0.001,26 0.001}, {b, 1, 1.1, 0.001}, {c, 1, 1.1, 0.001}, {d, 0.24, 0.25,27 0.001}], 3];28

29 min = Min[ values[[All, 2]]];30 pos = Position[values, min, 2, 1];31 lowest = values[[pos[[1, 1]], 1]];32

33 MuSolved = lowest[[1]]34 BetaSolved = lowest[[2]]35 AlphaSolved = lowest[[3]]36 NuSolved = lowest[[4]]37

38 Solved = First@39 NDSolve[{s[t] == (MuSolved)*Population - MuSolved*s[t] -40 BetaSolved*(i[t]/Population)*s[t],41 e[t] ==42 BetaSolved*(i[t]/Population)*43 s[t] - (MuSolved + AlphaSolved) e[t],44 i[t] == AlphaSolved*(e[t]) - (NuSolved + MuSolved)*(i[t]),45 r[t] == NuSolved*i[t] - MuSolved*r[t],46 i[0] == 161,47 s[0] == 20328,48 r[0] == 0, e[0] == 0},49 {s, e, i, r}, {t, 0, 29}];50

51 Show[Plot[{s[t] /. Solved, e[t] /. Solved, r[t] /. Solved,52 i[t] /. Solved}, {t, 0, 32}], ListPlot[CDCdata]]53 Show[ListPlot[CDCdata], Plot[{i[t] /. Solved}, {t, 0, 32}]]

Results = 0.0 = 1.0 = 1.02 = 0.25MaxError = 40.3743%RMS = 401.864

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Figure 10: SEIR Model (2011-2012) - S,E,I,R vs. Time

Figure 11: SEIR Model (2011-2012) - Model Infections vs. CDC Data

As mentioned above, random values allow for quicker calculations and covera larger spectrum of numbers without causing a large amount of calcula-tion time. This also allows a group of numbers that would have never beenplaced in combination using incremented values to be solved together.

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Figure 12: SEIR Model (2011-2012) - Randomly Generated Parameters

The variables N, , , , and are also included in this model. N is thetotal population, and was used as the transmission rate of disease. Thevariable was used as the natural death rate of the sub-population. Listing6 below includes manipulated code for randomly generated values to solvean SEIR model using Mathematica.

Listing 6: SEIR Model - Random Combinations

1 Population = 20328;2 Squarediff[mu_, beta_, alpha_,3 nu_] := (sol =4 First@NDSolve[{s[t] ==5 mu*Population - mu*s[t] - beta*(i[t]/Population)*s[t],6 e[t] == beta*(i[t]/Population)*s[t] - (mu + alpha) e[t],7 i[t] == alpha*(e[t]) - (nu + mu)*(i[t]),8 r[t] == nu*i[t] - mu*r[t],9 i[0] == 161,

10 s[0] == 20328,11 r[0] == 0, e[0] == 0},12 {s, e, i, r}, {t, 0, 29}];13 Sum[(((CDCdata[[j, 2]] - i[CDCdata[[j, 1]]]) /. sol)2), {j, 1,14 Length[CDCdata]}])15

16 values = Flatten[17 Table[{{a, b, c, d}, Squarediff[a, b, c, d]}, {a,18 RandomReal[{0, 0.00001}, {1}]}, {b,

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19 RandomReal[{1, 1.1}, {15}]}, {c, RandomReal[{1, 1.1}, {15}]}, {d,20 RandomReal[{0.23, 0.245}, {15}]}], 3];

Results = 6.17903 106 = 1.0105 = 1.00591 = 0.244964MaxError = 42.314%RMS = 438.685

6 Predicting Future Data

Predicting future outbreaks and their results is a large reason why epidemi-ology models exist, making them one of the most useful tools to disease con-trol/healthcare organizations. Solving the model from a given years data canlead to constants allowing the prediction of the consecutive year with justthe use of initial conditions. By gathering a small amount of data from thebeginning of the outbreak, the predictions of the lifespan of the disease canhelp predict how much vaccine needs to be produced, etc. Unfortunately,a small sample size could potentially be detrimental to results compared tothe year before or after.

We had initially solved the 2012-2013 general model for the influenza bythe CDC. The 2011-2012 data was then manipulated and a model was alsogenerated. To our surprise, the data took a massive spike in the middle ofthe outbreak, creating a large amount of error throughout the rest of themodel. Comparing our parameters for both 2012-2013 and 2011-2012, theyproved to be quite different. Below is our results for the 2011-2012 influenzamodel.

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Figure 13: SIR Model (2011-2012)

Listing 7: SEIR Model - Random Combinations


3 CDCdata = {{0, 75}, {1, 73}, {2, 106}, {3, 129}, {4, 192}, {5,4 175}, {6, 177}, {7, 229}, {8, 312}, {9, 466}, {10, 658}, {11,5 797}, {12, 1050}, {13, 1386}, {14, 1928}, {15, 2282}, {16,6 2646}, {17, 2171}, {18, 1561}, {19, 1446}, {20, 1254}, {21,7 1157}, {22, 851}, {23, 720}, {24, 614}, {25, 510}, {26, 422}, {27,8 371}, {28, 304}, {29, 303}};9

10 p = Interpolation[CDCdata];11

12 Population = 8295;13

14 sir = ParametricNDSolveValue[{s[t] == -beta i[t] s[t],15 i[t] == beta i[t] s[t] - nu i[t], r[t] == nu i[t], i[0] == 33,16 s[0] == 8295, r[0] == 0}, {s, i, r}, {t, 0, 29}, {beta, nu}];17

18 Manipulate[Module[{s, i, r}, {s == sir[a, b][[1]], i = sir[a, b][[2]]};19 Plot[{i[t], p[t]}, {t, 0, 29}]], {{a, 0.00001, \[Mu]}, 0.00001,20 0.00015, Appearance -> "Labeled"}, {{b, 0, \[Nu]}, 0., 0.25,21 Appearance -> "Labeled"}]

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The code above allows CDC data from 2011-2012 to be solved by manipu-lating values for and .

7 Conclusion

In this project we modeled what the influenza virus would look like in a smallcountry of 21,000 people. We showed how well this model was by seeing howlow we can make the goodness of fit characteristics. This shows how closeour model was to the ideal model. Determined by the goodness of fit, ourdata was tightly knit to the CDCs model of the virus. The SIR model wasenhanced by the SEIR model with the account of the extra variable E whichsymbolized the people exposed to the virus. After including this variablein the model we get a more realistic model of this epidemic. This variablerepresents the number of people that do not have a defense against the virusbut are yet to be infected. Adjusting the equations to consist of vaccinationswhich creates a more practical model. Over the course of this pandemic thenumber of people that can catch this virus will decrease to zero while thenumber infected will grow to the total population except for the people thathave the vaccine. The people that are resistant. As time increases the ini-tial value of susceptible people will closely resemble the amount of resistantpeople by the end of the pandemic, representing either the natural processof the human immune system, or the use of modern medicine.

Future WorkIn the future, since all above work has been evaluated heavily, our intentionswould be to predict the beginning of the 2013-2014 outbreak of influenza.Another possibility would be to monitor the spread of other diseases thatact similarly.

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References

[1] Centers for Disease Control and Prevention. Overview of InfluenzaSurveillance in the United States. Centers for Disease Control and Pre-vention, 2013. [Online; accessed 9-November-2013].

[2] Youjin Lee. Stochastic Modeling of Vaccine-Derived Poliomyelitis. 2008.[Online; accessed 11-November-2013].

[3] Hanz Nesse. SIR Model. [Online; accessed 11-November-2013].

[4] Raul et al. Nistal. Limit Periodic Solutions of a SEIR MathematicalModel for Non-lethal Infectious Disease. 2012. [Online; accessed 11-November-2013].

[5] Wikipedia. Epidemic Model - Wikipedia, the free encyclopedia. 2012.[Online; accessed 13-November-2013].

[6] Wikipedia. Compartmental models in epidemiology - Wikipedia, the freeencyclopedia. 2013. [Online; accessed 13-November-2013].

[7] Yongqing Yuan. The Collision Regions Between Two. 2012. [Online;accessed 11-November-2013].

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IntroductionData CollectionModelsSIR ModelSIR Model w/ VaccinationsSIR Model w/ Fatalities

SEIR ModelPredicting Future DataConclusionReferences

Epidemiology SEIR/SIR Models

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