an introduction to stochastic epidemic models-part i

AN INTRODUCTION TO STOCHASTIC EPIDEMICMODELS-PART I

Linda J. S. AllenDepartment of Mathematics and Statistics

Texas Tech UniversityLubbock, Texas U.S.A.

2008 Summer School onMathematical Modeling of Infectious Diseases

University of AlbertaMay 1-11, 2008

L.J.S. Allen, TTU Stochastic Epidemic Models - I

Outline of Presentation-PART I

I. What is a Stochastic Model and What is the Difference Between aDeterministic and Stochastic Model?

II. Some Stochastic Models are Illustrated Through Study of an SIS EpidemicModel.

(a) Discrete Time Markov Chain – DTMC

(b) Continuous Time Markov Chain – CTMC

(c) Diffusion Process and Stochastic Differential Equations – SDE

III. Some Differences Between the Stochastic SIS and SIR Epidemic Models.


I. What is a Stochastic Model?

A stochastic model is formulated in terms of a stochastic process.

A stochastic process is a collection of random variables

{Xt(s)|t ∈ T, s ∈ S},

where T is the index set and S is a common sample space. The index set oftenrepresents time, such as

T = {0, 1, 2, . . .} or T = [0,∞)

Time can be discrete or continuous.

The study of stochastic processes is based on probability theory.


How do Stochastic Epidemic Models Differ fromDeterministic Epidemic Models?

• A deterministic model is often formulated in terms of a system ofdifferential equations or difference equations.

• A stochastic model is formulated as a stochastic process with a collectionof random variables.

• A solution of a deterministic model is a function of time or space and isgenerally uniquely dependent on the initial data.

• A solution of a stochastic model is a probability distribution for each ofthe random variables. One sample path over time or space is one realizationfrom this distribution.

• Stochastic models are often used to show the variability inherent dueto the demographics or environment variablility are particularly important whenquantities in the processes are small- small population size or initial number ofinfectives.


The Following Graphs Illustrate the Dynamics of aDeterministic versus a Stochastic Epidemic Model

0 500 1000 1500 20000

5

10

15

20

25

30

35

Time Steps

Num

ber

of In

fect

ives

, I(t

)


Whether the Random Variables Associated with TheStochastic Process are Discrete or Continuous Distinguishes

Some of the Different Types of Stochastic Models.

A random variable X(t) of a stochastic process assigns a real value to eachoutcome A ⊂ S in the sample space and a probability,

Prob{X(t) ∈ A} ∈ [0, 1].

The values of the random variable constitute the state space, X(t;S). Forexample, the number of cases associated with a disease may have the followingdiscrete or continuous set of values for its state space:

{0, 1, 2, . . .} or [0, N ].

The state space can be discrete or continuous and correspondingly, therandom variable is discrete or continuous. For simplicity, the sample spacenotation is suppressed and X(t) is used to denote a random variable indexedby time t. The stochastic process is completely defined when the set of randomvariables {X(t)} are related by a set of rules.


We will Study Stochastic Processes that have the MarkovProperty.

A stochastic process with the Markov property is one where the future stateof the process depends only on the current state, and not on the past. That is,for a discrete-time stochastic process, a d

Prob{X(t + ∆t)|X(t),X(t − ∆t), . . . , X(0)} = Prob{X(t + ∆t)|X(t)}.

At a fixed time t, each random variable X(t) has an associated probabilitydistribution.

Discrete: Prob{X(t) = i} = pi(t), i ∈ {0, 1, 2 . . .}

Continuous: Prob{X(t) ∈ [a, b]} =∫ b

ap(x, t)dx


The Choice of Discrete or Continuous Random Variableswith a Discrete or Continuous Index Set Defines the Type

of Stochastic Model.

Discrete Time Markov Chain (DTMC): t ∈ {0, ∆t, 2∆t, . . .} X(t) is a discreterandom variable.

X(t) ∈ {0, 1, 2, . . . , N}The term chain implies that the random variable is discrete.

Continuous Time Markov Chain (CTMC): t ∈ [0,∞), X(t) is a discreterandom variable.

Xt ∈ {0, 1, 2, . . . , N}

Diffusion Process, Stochastic Differential Equation (SDE): t ∈ [0,∞), X(t) isa continuous random variable.

X(t) ∈ [0, N ]


II. Before we Formulate the Stochastic SIS EpidemicModels, we Review the Dynamics of the Deterministic SIS

Epidemic Model.

Deterministic SIS:

S I

dS

dt= − β

NSI + (b + γ)I

dI

dt=

β

NSI − (b + γ)I

where β > 0, γ > 0, N > 0 and b ≥ 0, S(t) + I(t) = N .


The Dynamics of the Deterministic SIS Epidemic ModelDepend on the Basic Reproduction Number.

The parameter values represent

β = transmission rate

b = birth rate = death rate

γ = recovery rate

N = total population size = constant.

Basic Reproduction Number:

R0 =β

b + γ

If R0 ≤ 1, then limt→∞ I(t) = 0.

If R0 > 1, then limt→∞ I(t) = N(

1 − 1R0

)

.


We will Formulate the Three Types of Stochastic SISEpidemic Models by Defining Relationships Among the

Random Variables Assuming the Markov Property Holds.

S(t) = random variable for the number of susceptible individuals.

I(t) = random variable for the number of infected individuals.

S(t) + I(t) = N = maximum population size.

Discrete Time Markov Chain (DTMC): t ∈ {0, ∆t, 2∆t, . . .}, I(t) is a discreterandom variable,

I(t) ∈ {0, 1, 2, . . . , N}

Continuous Time Markov Chain (CTMC): t ∈ [0,∞), I(t) is a discrete randomvariable.

I(t) ∈ {0, 1, 2, . . . , N}

Diffusion Process, SDEs: t ∈ [0,∞), I(t) is a continuous random variable.

I(t) ∈ [0, N ]


First, We Formulate a DTMC SIS Epidemic Model.

Let I(t) denote the discrete random variable for the number of infected (andinfectious) individuals with associated probability function

pi(t) = Prob{I(t) = i}

where i = 0, 1, 2, . . . , N is the total number infected at time t. The probabilitydistribution is

p(t) = (p0(t), p1(t), . . . , pN(t))T

for t = 0, ∆t, 2∆t, . . . . Now we relate the random variables {I(t)} indexed bytime t by defining the probability of a transition from state i to state j, i → j,in time ∆t as

pji(∆t) = Prob{I(t + ∆t) = j|I(t) = i}.


Assume that ∆t is Sufficiently Small, Such that the Numberof Infectives Changes by at Most One in Time ∆t.

That is,i → i + 1, i → i − 1 or i → i.

Either there is a new infection, birth, death, or a recovery. Therefore, thetransition probabilities are

pji(∆t) =

βi(N − i)/N∆t = b(i)∆t, j = i + 1(b + γ)i∆t = d(i)∆t, j = i − 11 − [βi(N − i)/N + (b + γ)i]∆t =

1 − [b(i) + d(i)]∆t, j = i0, j 6= i + 1, i, i − 1,


The Probability Distribution Associated with the EpidemicProcess Over Time is Found by Repeated Multiplication of

the Transition Matrix.

Matrix P (∆t) = (pji(∆t)) is known as the transition matrix:

p(t + ∆t) = P (∆t)p(t),

where p(t) = (p0(t), . . . , pN(t))T is the probability distribution and P (∆t) is

1 d(1)∆t 0 · · · 00 1 − [b(1) + d(1)]∆t d(2)∆t · · · 00 b(1)∆t 1 − [b(2) + d(2)]∆t · · · 00 0 b(2)∆t · · · 0... ... ... ... ...0 0 0 · · · d(N)∆t0 0 0 · · · 1 − d(N)∆t

.

Matrix P (∆t) is stochastic, the column sums equal one.


The Stochastic Process for the DTMC SIS Model is knownas a Finite State Markov Chain with the Following

Properties.

• The stochastic process {I(t)} for t ∈ {0, ∆t, 2∆t, . . .} is time-homogeneous (transition probabilities do not depend on time) and has theMarkov property.

• The probability of no infections p0 is an absorbing state.

0 1 2 N

• For any initial distribution p(0) = (p0(0), . . . , pN(0))T , zero through atotal of N infections

limt→∞

p(t) = (1, 0, . . . , 0)T limt→∞

p0(t) = 1.


Three Sample Paths of the DTMC SIS Model areCompared to the Solution of the Deterministic Model.

A sample path or stochastic realization of a stochastic process {I(t)} fort ∈ {0, ∆t, 2∆t, . . .} is an assignment of a possible value to I(t) for each valueof t.

R0 = 2.

0 5 10 15 20 250

10

20

30

40

50

60

70

Time

Num

ber

of In

fect

ives

∆t = 0.01, N = 100, β = 1, b = 0.25, γ = 0.25, I(0) = 2, and p2(0) = 1.The

MATLAB program is in the Appendix.


The Probability Distribution p(t) for the Number of InfectedIndividuals in the DTMC SIS Model can be Approximated.

0

50

100

0

1000

2000

0

0.25

0.5

0.75

1

StateTime, n

Pro

babi

lity

Probability distribution for the DTMC SIS model, ∆t = 0.01, N = 100, β = 1, b = 0.25,

γ = 0.25, R0 = 2, I(0) = 2 and p2(0) = 1. MATLAB program is in the Appendix.

Note: Asymptotically, limt→∞ p0(t) = 1, the epidemic ends with probabilityone. But it may take a long time before p0 ≈ 1, if N and I(0) are large. In thisexample,

p0(t) ≈

„

1

R0

«I(0)

=

„

1

2

«2

=1

4.


Next, We Formulate a CTMC SIS Model.

This type of model is most often used to study stochastic epidemic processes,time is continuous, but the random variable for number of infected individualsis discrete. The discrete random variable I(t), t ∈ [0,∞) has an associatedprobability function

pi(t) = Prob{I(t) = i}

The probability of a transition for small ∆t satisfies

pji(∆t) =

βi(N − i)/N∆t + o(∆t) = b(i)∆t + o(∆t), j = i + 1(b + γ)i∆t + o(∆t) = d(i)∆t + o(∆t), j = i − 11 − [βi(N − i)/N + (b + γ)i]∆t + o(∆t)

= 1 − [b(i) + d(i)]∆t + o(∆t), j = io(∆t), otherwise,

where o(∆t) → 0 as ∆t → 0.

i → i + 1, i → i − 1, or i → i.


A System of Differential Equations for the Probabilities Canbe Derived Based on the Transition Probabilities.

For small ∆t,

pi(t + ∆t) = pi−1(t)[b(i − 1)∆t] + pi+1(t)[d(i + 1)∆t]

pi(t)[1 − (b(i) + d(i))∆t] + o(∆t)

Subtracting pi(t), dividing by ∆t, and letting ∆t → 0,

dpi

dt= pi−1b(i − 1) + pi+1d(i + 1) − pi[b(i) + d(i)]

dp0

dt= p1d(1)

for i = 1, 2, . . . , N, where

b(i) = βi(N − i)/N, d(i) = (b + γ)i.

These differential equations are known as the forward Kolmogorov differentialequations.


The Epidemic Process is Captured by a System ofDifferential Equations Expressed in Matrix Form.

In matrix notation,

dp

dt= Qp,

where p(t) = (p0(t), . . . , pN(t))T and Q is known as the generator matrix:

Q =

0

B

B

B

B

B

B

B

B

B

@

0 d(1) 0 · · · 0

0 −[b(1) + d(1)] d(2) · · · 0

0 b(1) −[b(2) + d(2)] · · · 0

0 0 b(2) · · · 0... ... ... ... ...

0 0 0 · · · d(N)

0 0 0 · · · −d(N)

1

C

C

C

C

C

C

C

C

C

A

.

b(i) = βi(N − i)/N and d(i) = (b + γ)i


The DTMC Transition Matrix and CTMC DifferentialEquations are Closely Related when ∆t is Small.

In the DTMC Model,p(t + ∆t) = P (∆t)p(t),

where P (∆t) is the transition matrix. Letting ∆t → 0, we obtain theKolmogorov differential equations for the CTMC model,

p(t + ∆t) − p(t)

∆t=

P (∆t) − I

∆tp(t)

dp

dt= Qp

where

Q = lim∆t→0

P (∆t) − I

∆t.

The Discrete-Time Process can be used to Approximate the Continuous-Time Process.


Because of the Markov Property, the Inter-Event Time in aCTMC Model Has an Exponential Distribution.

The exponential distribution has what is known as the memoryless property.Let I(t) = n and Tn denote the inter-event time, a continuous random variablefor the time to the next event. Take the sum of all the probabilities of allpossible events where there is a change in state, i → i + 1, i → i − 1:

∞∑

j=0,j 6=n

pjn(∆t) = a(n)∆t + o(∆t)

andpnn(∆t) = 1 − a(n)∆t + o(∆t).

Then the interevent time has an exponential distribution with parameter a(n),

Tn ∼ E(a(n))

Prob{Tn ≤ t} = 1 − exp(−a(n)t).


For the SIS Epidemic Model, with I(t) = n,

∞∑

j=0,j 6=n

pjn(∆t) = [b(n) + d(n)]∆t + o(∆t)

= [βn(N − n)/N + (b + γ)n]∆t + o(∆t)

a(n) =β

Nn(N − n) + (b + γ)n


To Numerically Simulate the Inter-Event Time in a CTMCModel, We Use a Uniform Random Variable.

The inter-event time, waiting time until an event occurs, can be numericallycomputed using a uniform random variable and the cumulative distribution forTn. Let U be uniform random variable on [0, 1] and Fn(t) the cumulativedistribution for Tn

Fn(t) = Prob{Tn ≤ t} = 1 − exp(a(n)t).

Then

Prob{F−1n (U) ≤ t} = Prob{Fn(F

−1n (U)) ≤ Fn(t)}

= Prob{U ≤ Fn(t)}

= Fn(t)

The inter-event time Tn, given I(t) = n satisfies

Tn = F−1n (U) = −ln(1 − U)

a(n)= −ln(U)

a(n).


Three Sample Paths of the CTMC SIS Model areCompared to the Deterministic Solution.

R0 = 2

0 5 10 15 20 250

10

20

30

40

50

60

70

80

Time

Num

ber

of In

fect

ives

b = 0.25, γ = 0.25, β = 1, N = 100, I(0) = 2, R0 = 2.

For ∆t small, the dynamics of the DTMC and the CTMC Models are Similar.The DTMC model can be used as an approximation for the CTMC model.MATLAB program is in the Appendix.


Next, We Formulate the Third Type of Stochastic Model, aSDE Model.

The number of infectives, I(t), is continuous random variable and the time,t ∈ [0,∞), is also continuous. The random variable I(t) has an associatedprobability density function (pdf), p(x, t),

Prob{I(t) ∈ [a, b]} =

∫ b

a

p(x, t)dx.

We can derive a system of differential equations satisfied by the pdf. This systemof equations is also known as the forward Kolmogorov differential equations:

∂p

∂t= −∂ {[βx(N − x)/N − (b + γ)x]p}

∂x

+1

2

∂2 {[βx(N − x)/N + (b + γ)x]p}∂x2

,

x ∈ [0, N ], t ∈ [0,∞). The first term is known as the drift and second termdiffusion.


The Stochastic Differential Equation (SDE) Depends on theDrift and Diffusion Terms.

The Stochastic Differential Equation (SDE) follows directly from the differentialequations,

dI

dt=

β

NI(N − I) − (b + γ)I +

r

β

NI(N − I) + (b + γ)I

dW

dt,

where W (t) is a Wiener process (white noise), normally distributed, with meanzero and variance t:

W (t) ∼ Normal(0, t), W (t + ∆t) − W (t) ∼ Normal(0, ∆t).

Sample paths for a Wiener process are continuous but not differentiable.

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

Time

W(t

)


The SDE Depends on Relationship Between Births andDeaths and Drift and Diffusion

Let b(I) =births (new infection or birth) and d(I) =deaths(recovery or death). Then the probability density p(x, t), where

Prob{I(t) ∈ [a, b]} =∫ b

ap(x, t)dx satisfies the differential equation

∂p(x, t)

∂t= −∂([b(x) − d(x)]p(x, t))

∂x+

1

2

∂2([b(x) + d(x)]p(x, t))

∂x2

and the stochastic differential equation (SDE) satisfies

dI

dt= b(I) − d(I) +

√

b(I) + d(I)dW

dt= drift + diffusion


The Drift and Diffusion Terms Determine the Change inNumber of Infections Over Time

SDE:dI

dt= b(I) − d(I) +

√

b(I) + d(I)dW

dt

∆I(t) is approximately normally distributed with mean b(I) − d(I)]∆t andvariance, b(I) + d(I)]∆t :

∆I(t) = I(t + ∆t) − I(t) ∼ Normal([b(I) − d(I)]∆t, [b(I) + d(I)]∆t).

The Wiener process ∆W (t) (white noise) is normally distributed with mean0 and variance ∆t:

∆W (t) = W (t + ∆t) − W (t) =√

∆t η ∼ Normal(0, ∆t).


In General, the SDE is Expressed in Terms of theParameters for Recovery, Transmission and Birth.

SDE:

dI

dt=

β

NI(N − I) − (b + γ)I +

√

β

NI(N − I) + (b + γ)I

dW

dt,

where W (t) is a Wiener process (white noise), normally distributed, with meanzero and variance t:

W (t) ∼ Normal(0, t), W (t + ∆t) − W (t) ∼ Normal(0, ∆t).


Three Sample Paths for the SDE SIS Model are ComputedNumerically and Compared to the Deterministic Solution.

R0 = 2

0 5 10 15 20 250

10

20

30

40

50

60

70

80

Time

Num

ber

of In

fect

ives

b = 0.25, γ = 0.25, β = 1, N = 100, I(0) = 2. MATLAB program is in the Appendix.

Note: For large N and I(0), then the SDE model is a good approximationto the CTMC model. However, for small N or I(0), the CTMC model is abetter model.


Some Advantages of the Stochastic Models Over theDeterministic Model for the SIS Epidemic Model:

The SIS Deterministic Model Does Not capture

(i) The Variability Inherent in the Transmission, Recovery, Birth, and DeathProcesses

(ii) The Probability of No Epidemic Occurrence when R0 > 1.

The Stochastic Models Do Capture these Features.


III. The SIS is a Simple Epidemic Model Because theDynamics Reduce to a Single Variable. This is not the Case

for the SIR Epidemic Model.

First, we review the dynamics of the deterministic SIR Epidemic model.Then we will illustrate some of the differences in the stochastic formulation forthe SIS versus the SIR epidemic model.

Deterministic SIR:

S I R


Deterministic SIR: S(t) + I(t) + R(t) = N

dS

dt= − β

NSI + b(I + R)

dI

dt=

β

NSI − (b + γ)I

dR

dt= γI − bR

Basic Reproduction Number:

R0 =β

b + γ

If R0 > 1 and b > 0, then limt→∞ I(t) = I > 0.If R0 > 1 and b = 0, then limt→∞ I(t) = 0.

There is an epidemic if R0S(0)

N> 1.

If R0 ≤ 1, then limt→∞ I(t) = 0.


Formulation of a DTMC SIR Epidemic Model Results In AMultivariate Process.

S(t) + I(t) + R(t) = N = maximum population size.

Let S(t) and I(t) denote discrete random variables for the number of susceptibleand infected individuals, respectively. These two variables have a jointprobability function

p(s,i)(t) = Prob{S(t) = s, I(t) = i}

where R(t) = N −S(t)− I(t). For this stochastic process, we define transitionprobabilities as follows:

p(s+k,i+j),(s,i)(∆t) = Prob{(∆S, ∆I) = (k, j)|(S(t), I(t)) = (s, i)}

=

8

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

:

βi(N − i)∆t/N, (k, j) = (−1, 1)

γi∆t, (k, j) = (0,−1)

bi∆t, (k, j) = (1,−1)

b(N − s − i)∆t, (k, j) = (1, 0)

1 − [βi(N − i)/N + γi + b(N − s)]∆t, (k, j) = (0, 0)

0, otherwise


Three Sample Paths of the DTMC SIR Epidemic Model areCompared to the Solution of the Deterministic Model.

R0 = 2, b = 0

R0S(0)

N= 1.96.

0 500 1000 1500 20000

5

10

15

20

25

30

35

Time Steps

Num

ber

of In

fect

ives

, I(t

)

∆t = 0.01, N = 100, β = 1, b = 0, γ = 0.5, S(0) = 98, and I(0) = 2.


The SDE Model for the SIR Epidemic is a System of TwoIto SDEs.

For example, in the case with b = 0,

dS

dt= − β

NSI + B11

dW1

dt+ B12

dW2

dtdI

dt=

β

NSI − γI + B21

dW1

dt+ B22

dW2

dt

where W1 and W2 are two independent Wiener processes and B = (Bij) is the

square root of the following covariance matrix, B =√

Σ,

Σ =

(

βSI/N −βSI/N−βSI/N βSI/N + γI

)

.

Notice that matrix V is positive definite and thus, has a unique positive definitesquare root,

√Σ = B.


Three Stochastic Sample Paths of the SDE SIR EpidemicModel Are Compared to the Deterministic Solution.

R0 = 2, b = 0

R0S(0)

N= 1.96.

0 5 10 15 200

5

10

15

20

25

30

35

Time

Num

ber

of In

fect

ives

∆t = 0.01, N = 100, β = 1, b = 0, γ = 0.5, I(0) = 2.


To Summarize the Main Points:

• Stochastic epidemic models capture the variability inherent in thetransmission, recovery, birth and death processes. Here we did not considerenvironmental variability.

• For small population sizes or small number of infected individuals, CTMCor DTMC models with discrete random variables more accurately capture thevariability in the epidemic process than deterministic models.

• The DTMC model may be used to approximate the CTMC model whenthe time interval ∆t is small.

• The SDE model may be used to approximate the CTMC model when thepopulation size and initial values are large.


(Part II) Stochastic SIS and SIR Epidemic Models areUseful for Quantifying the Following:

(a) Probability of No Epidemic

(b) Stationary or Quasistationary Distribution

(c) Final Size of an Epidemic

(e) Expected Duration of an Epidemic


References and MATLAB programs:

1. Allen, L. J. S. 2003. An Introduction to Stochastic Processes with Applications to Biology.

Prentice Hall, Upper Saddle River, N.J.

2. Allen, L. J. S. and A. Burgin. 2000. Comparison of deterministic and stochastic SIS and

SIR models in discrete time. Mathematical Biosciences. 163: 1-33.

3. Andersson, H. and T. Britton. 2000. Stochastic Epidemic Models and Their Statistical

Analysis. Lecture Notes in Statistics. Springer-Verlag, New York, Inc.

4. Daley, D. J. and J. Gani. 1999. Epidemic Modelling An Introduction. Cambridge Studies

in Mathematical Biology, Vol. 15. Cambridge University Press, Cambridge.

5. Gard, T. C. 1988. Introduction to Stochastic Differential Equations. Marcel Dekker, Inc.,

New York and Basel.

6. Mode, C. J. and C. K. Sleeman. 2000. Stochastic Processes in Epidemiology. HIV/AIDS,

Other Infectious Diseases and Computers. World Scientific, Singapore, New Jersey.


MATLAB Programs For:(1) Three Sample Paths for DTMC SIS Model(2) Probability Distribution for the DTMC SIS Model(3) Three Sample Paths for CTMC SIS Model(4) Three Sample Paths for the SDE SIS Model.

(1)

% Matlab Program

% DTMC SIS Epidemic Model

% Three Sample Paths

clear

set(0,’DefaultAxesFontSize’, 18)

beta=1;

g=0.25;

b=0.25;

N=100;

init=2;

dt=0.01;

time=25;

sim=3;

for j=1:sim


i(1)=init;

for t=1:time/dt

r=rand;

birth=beta*i(t)*(N-i(t))/N*dt;

death=(b+g)*i(t)*dt;

if r<=birth

i(t+1)=i(t)+1;

elseif r>birth & r<=birth+death

i(t+1)=i(t)-1;

else

i(t+1)=i(t);

end

end

if j==1

plot([0:dt:time],i,’r-’,’LineWidth’,2);

hold on

elseif j==2

plot([0:dt:time],i,’g-’,’LineWidth’,2);

else

plot([0:dt:time],i,’b-’,’LineWidth’,2 end

end


% Euler’s Method for Deterministic SIS Model

y(1)=init;

for k=1:time/dt

y(k+1)=y(k)+dt*(beta*(N-y(k))*y(k)/N-(b+g)*y(k));

end

plot([0:dt:time],y,’k--’,’LineWidth’,2);

hold off

xlabel(’Time’);

ylabel(’Number of Infectives’);

0 5 10 15 20 250

10

20

30

40

50

60

70

Time

Num

ber

of In

fect

ives


(2)

% Matlab Program

% Discrete Time Markov Chain

% Stochastic SIS epidemic model

% Transition matrix and Graph of Probability Distribution

clear all

set(gca,’FontSize’,18);

set(0,’DefaultAxesFontSize’,18);

time=2000;

dtt=0.01; % Time step

beta=1*dtt;

b=0.25*dtt;

gama=0.25*dtt;

N=100; % Total population size

en=50; % plot every enth time interval

T=zeros(N+1,N+1); % T is the transition matrix, defined below

v=linspace(0,N,N+1);

p=zeros(time+1,N+1);

p(1,3)=1; % Two individuals initially infected

bt=beta*v.*(N-v)/N;

dt=(b+gama)*v;

for i=2:N % Define the transition matrix

T(i,i)=1-bt(i)-dt(i); % diagonal entries

T(i,i+1)=dt(i+1); % superdiagonal entries


T(i+1,i)=bt(i); % subdiagonal entries

end

T(1,1)=1;

T(1,2)=dt(2);

T(N+1,N+1)=1-dt(N+1);

for t=1:time

y=T*p(t,:)’;

p(t+1,:)=y’;

end

pm(1,:)=p(1,:);

for t=1:time/en;

pm(t+1,:)=p(en*t,:);

end

ti=linspace(0,time,time/en+1);

st=linspace(0,N,N+1);

mesh(st,ti,pm);

xlabel(’number infected’);

ylabel(’time steps’);

zlabel(’probability of infection’);

view(140,30);

axis([0,N,0,time,0,1]);


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(3)

% Matlab Program

% Continuous Time Markov Chain

% SIS Epidemic Model

% Three Sample Paths Compared to the Deterministic Model

clear

set(0,’DefaultAxesFontSize’, 18);

set(gca,’fontsize’,18);

beta=1;

b=0.25;

gam=0.25;

N=100;

init=2;

time=25;

sim=3;

for k=1:sim

clear t s i

t(1)=0;

i(1)=init;

s(1)=N-init;

j=1;

while i(j)>0 & t(j)<time

u1=rand;

u2=rand;


a=(beta/N)*i(j)*s(j)+(b+gam)*i(j);

probi=(beta*s(j)/N)/(beta*s(j)/N+b+gam);

t(j+1)=t(j)-log(u1)/a;

if u2 <= probi

i(j+1)=i(j)+1;

s(j+1)=s(j)-1;

else

i(j+1)=i(j)-1;

s(j+1)=s(j)+1;

end

j=j+1;

end

plot(t,i,’r-’,’LineWidth’,2)

hold on

end

% Euler’s Method Applied to the Deterministic SIS Epidemic Model

dt=0.01;

x(1)=N-init;

y(1)=init;

for k=1:time/dt

x(k+1)=x(k)+dt*(-beta*x(k)*y(k)/N+(b+gam)*y(k));

y(k+1)=y(k)+dt*(beta*x(k)*y(k)/N-(b+gam)*y(k));

end



axis([0,time,0,80]);

xlabel(’Time’);


hold off

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(4)

% Matlab Program

% SDE SIS Epidemic Model

% Three Sample Paths using Euler’s Method

clear

beta=1;

b=0.25;

gam=0.25;

N=100;

init=2;

dt=0.01;

time=25;

sim=3;

for k=1:sim

clear i, t

j=1;

i(j)=init;

t(j)=dt;

while i(j)>0 & t(j)<25

mu=beta*i(j)*(N-i(j))/N-(b+gam)*i(j);

sigma=sqrt(beta*i(j)*(N-i(j))/N+(b+gam)*i(j));

rn=randn;

i(j+1)=i(j)+mu*dt+sigma*sqrt(dt)*rn;

t(j+1)=t(j)+dt;


j=j+1;

end

plot(t,i,’r-’,’Linewidth’,2);

hold on

end

% Euler’s method applied to the deterministic SIS epidemic model.

y(1)=init;

for k=1:time/dt

y(k+1)=y(k)+dt*(beta*(N-y(k))*y(k)/N-(b+gam)*y(k));

end


axis([0,time,0,80]);

xlabel(’Time’);


hold off


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an introduction to stochastic epidemic models-part i

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