Project Help Notebook by Keith Stroyan

Help NoteBook Copyright 1998 by Academic Press, Inc. - All rights reserved.
Modify this Notebook to prepare your project report.

Project 4: The Endemic Limit

by A. Student and B. Pupil

with technical help from Da LabMonitor and Help Notebook

My Project Introduction

Write this section last.  Begin with the basic question and state your main finding clearly, but with few technical details.  You might even modify the title of this section to make it more interesting.  Following is a rough start.

An "S-I-S" disease is one that does NOT confir immunity, so when you recover from the disease, you return to susceptible status and could contract it again.  These diseases can become "endemic" in a population.  This means that various people pass the disease around from one to the other and back in such a way that some people are always sick.

This report derives a new model similar to the S-I-R differential equation model and uses it to show that after a long time,  __ % of the population will always be infected, where  __  is the measured value of __.  If  __  is less than  __, the disease dies out.

S-I-S Equations

This section of our report derives the "equations of change" for an S-I-S disease.

Derivation of the model

Assumptions

We make some basic assumptions in formulating a model for the change of susceptible and infectious people.

ASSUMPTION 1:
S-I-S: We study only diseases that do NOT confir immunity.  This means...

ASSUMPTION 2:
Fixed population:  Since a flu epidemic takes place in __ days, the number of births, deaths, and immigrations bla, bla, blah...

ASSUMPTION 3:
Well mixed bla, bla, blah...  this assumption is used...

ASSUMPTION 4:
Large population bla, bla, blah...  this assumption is used...

Variables

Our model uses the following variables

t = time in days
	s = susceptible fraction  = number susceptible/total population
	i = infectious fraction = number infectious/total population

You may wish to use auxillary variables as follows:

S = number of susceptible people = s*n (see n below in parameters)
	I = number of infectious people = i*n

Parameters

b = 1/the number of days a person is infectious
	c = the number of contacts an infectious person makes that could transmit the disease
	a = b c, the daily rate of making contacts
	n = the fixed total population, n = S + I (see Assumption 2 above.)

Bla, bla, blah.

We can measure the paremeters  b  and  c for a disease by ....

Daily rate of recovery

If the population consists of  n people total and the fraction infectious is  i,  i*n people are infectious.  The average number of people who recover per day is b* i * n, because bla, bla, blah.

Daily rate of transmission

The daily rate of making "close contacts" is given by the measured parameters b and c,  a = b*c.  If the infectious fraction is  i and the total population is  n,  the total number of contacts is  Bla, bla, blah.

daily rate of close contacts = a * i * n = a I

The fraction  s  of these contacts is made with susceptible people, resulting in new cases.

daily rate of new cases = a * i * n * s = a I (S/n) = a I s

Therefore, the rate of of the fraction of the population

You need some additional explanation here to arrive at the differential equations:

The S-I-S differential equations

(ds/dt) = ?? ??

(di/dt) = a s i - b i

Beginning Explorations or Computer Experiments

HINT 4.4

We used the program from the class notebook SIRsolver to make predicitons with our model differential equations. (You will want to make additional modifications an leave the computations in your report, and then remove this remark.)

c = 1/10

Needs["Graphics`Colors`"];

Clear[s, i, t, soln, sRate, iRate]; sRate[s_, i_] := (b - a s ) i;
iRate[s_, i_] := (a s - b) i;
b  =  N[1] ;    
c  =  N[0.1];		  
a  =  b c ;		  

startUchange={s[0] == 3/4,i[0] == 1/4,s'[t]==sRate[s[t], i[t]],i'[t]==iRate[s[t], i[t]]};
finalTime = 20;
soln = NDSolve[startUchange, {s[t], i[t]}, {t, 0, finalTime}];
Print["Contact number, c = ", c, 
        ", Infectious period 1/b =", (1/b)];
combinedPlot  =  
        Plot[{s[t] /. soln, i[t] /. soln}, {t, 0, finalTime}, 
          PlotStyle -> {Red, Blue}, 
          AxesLabel -> {"t", "s-Red,i-Blue"}, PlotRange -> {0, 1}];
lastS = s[t] /. soln;
Print["s[", finalTime, "] = ", lastS /. t -> finalTime]
        ;

The graph above shows an S-I-S epidemic for a disease with  b=1 and c=1/10.  Notice that in the long term (as t ->∞) s[t] -> ??  and  i[t] -> ??

c = 10

Clear[s, i, t, soln, sRate, iRate]; sRate[s_, i_] := (b - a s ) i;
iRate[s_, i_] := (a s - b) i;
b  =  N[1] ;    
c  =  N[10];		  
a  =  b c ;		  

startUchange={s[0] == 3/4,i[0] == 1/4,s'[t]==sRate[s[t], i[t]],i'[t]==iRate[s[t], i[t]]};
finalTime = 20;
soln = NDSolve[startUchange, {s[t], i[t]}, {t, 0, finalTime}];
Print["Contact number, c = ", c, 
        ", Infectious period 1/b =", (1/b)];
combinedPlot  =  
        Plot[{s[t] /. soln, i[t] /. soln}, {t, 0, finalTime}, 
          PlotStyle -> {Red, Blue}, 
          AxesLabel -> {"t", "s-Red,i-Blue"}, PlotRange -> {0, 1}];
lastS = s[t] /. soln;
Print["s[", finalTime, "] = ", lastS /. t -> finalTime]
        ;

The graph above shows an S-I-S epidemic for a disease with  b=1 and c=10.  Notice that in the long term (as t ->∞) s[t] -> ??  and  i[t] -> ??

c = 2

Clear[s, i, t, soln, sRate, iRate]; sRate[s_, i_] := (b - a s ) i;
iRate[s_, i_] := (a s - b) i;
b  =  N[1] ;    
c  =  N[2];		  
a  =  b c ;		  

startUchange={s[0] == 3/4,i[0] == 1/4,s'[t]==sRate[s[t], i[t]],i'[t]==iRate[s[t], i[t]]};
finalTime = 20;
soln = NDSolve[startUchange, {s[t], i[t]}, {t, 0, finalTime}];
Print["Contact number, c = ", c, 
        ", Infectious period 1/b =", (1/b)];
combinedPlot  =  
        Plot[{s[t] /. soln, i[t] /. soln}, {t, 0, finalTime}, 
          PlotStyle -> {Red, Blue}, 
          AxesLabel -> {"t", "s-Red,i-Blue"}, PlotRange -> {0, 1}];
lastS = s[t] /. soln;
Print["s[", finalTime, "] = ", lastS /. t -> finalTime]
        ;

The graph above shows an S-I-S epidemic for a disease with  b=1 and c=2.  Notice that in the long term (as t ->∞) s[t] -> ??  and  i[t] -> ??

More experiments

Cut & paste the computation above and do organized numerical experiments until you can make a conjecture about the way the long-term limits of  s[t]  and  i[t] depend on the disease parameters.

HINTs 4.5 & 4.6: The Importance of the Contact Number c

Read section 4.4 of the project and modify the sample computations above to formulate a conjecture about the endemic limit.

"Large" c = 2, 3, 4

"Small" c = 1/2, 1/3, 1/4

HINT 4.7

Read section 4.5 of the project and give general mathematical arguments to show when

\(TraditionalForm
      \`lim\+\(t -> ∞\)i[t]\  ≠ \ 0\)

As it says in section 4.5, there is a simpler formula for

(lim_(t -> ∞))s[t]  =  ??

at least in certain cases.

"Large" c

"Small" c