A Model for Epidemics - The Logistic Model

Ken Levasseur , UMass Lowell

This webMathematica page illustrates the logistic model which, among other things, is used to model the spread of a disease. Given a population of M individials, we assume that one individual becomes infected with a disease on Day 1. In each successive day, every infected person comes into contact with exactly one random person, who may or may not be already infected. A healthy person is infected if contacted by an infected person. This script simulates the process with a random number generator

Run the simulation several times and make observations. How would you describe a function resembles the total infected individuals as a function of time? Can you find an analytic function that resembles that function? How is the derivative of the function related to the total number of individuals infected?

Do this in Mathematica

Here is the Mathematica code that is used in this script, adapted to WebMathematica.



contacts := Map[{Random[Integer, {1, M}], #} &, Range[M]];

newsicks := (addtosick = {}; 
        Map[If[MemberQ[well, First[#]] && MemberQ[sick, Last[#]], 
              addtosick = Union[addtosick, {First[#]}]] &, #];
        {well, sick} = {Complement[well, addtosick], Union[sick, addtosick]}; 
        Length[sick]) &;

M = Input["Enter the population - no more than 1000 is suggested"];
{well, sick} = {Range[M - 1], {M}};

spreaddata = 
    Table[newsicks[contacts], {M}] //. {{a___, M, M} -> {a, M}} // 
      Prepend[#, 1] &;

ListPlot[spreaddata, PlotRange -> {0, M}, PlotJoined -> True, 
  Ticks -> {{0, Length[spreaddata]}, {0, M}}, PlotStyle -> RGBColor[0, 1, 0], 
  Epilog -> {PointSize[0.02], RGBColor[1, 0, 0], 
      Map[Point[{#, spreaddata[[#]]}] &, Range[Length[spreaddata]]]}]