-
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
1
-
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-
2
-
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.
3
-
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.
4
-
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.
5
-
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
6
-
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 .
7
-
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
8
-
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
9
-
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
10
-
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.
11
-
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.
12
-
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
13
-
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 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
14
-
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
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 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
15
-
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
16
-
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.
17
-
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,
18
-
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.
19
-
Figure 13: SIR Model (2011-2012)
Listing 7: SEIR Model - Random Combinations
1 ClearAll["Global*"]2
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"}]
20
-
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.
21
-
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].
22
IntroductionData CollectionModelsSIR ModelSIR Model w/
VaccinationsSIR Model w/ Fatalities
SEIR ModelPredicting Future DataConclusionReferences