The Markov Chains & S.I.R epidemic model

BY

WRITWIK MANDAL

M.SC BIO-STATISTICS

SEM 4

What is a Random Process?

A random process is a collection of random variables indexed by some set I, taking values in some set S.

† I is the index set, usually time, e.g. Z+, R, R+.

† S is the state space, e.g. Z+, {1, 2, . . . , n}, {a, b, c}.

Basics of Markov ChainsA random process is called a Markov Process if, conditional on the current state

of the process, its future is independent of its past.

If simply put , it is a mathematical model of a random phenomenon evolving with time in a way that the past affects the future only through the present. The “time” can be discrete (i.e. the integers), continuous (i.e. the real numbers).

x | X x X x x X | X x X nnnnnnnn 111111 PrPr

X1 X2 X3 X4 X5

There are three items involved: to specify a Markov chain State space S :-

S is a finite or countable set of states , that is, values that the random variables Xi may take on. Let us label the states as follows

S = {1, 2…..N } for some finite N.

Initial distribution п0 :-

This is the probability distribution of the Markov chain at time 0. For each state

i є S, we denote by п0(i) the probability [ P{X0 = i} ] that the Markov chain starts out in state i. Formally, п0 is a function taking S into the interval [0,1] such that

п0(i) ≥ 0 for all i є S and =1.

Probability transition rule :-

This is specified by giving a matrix P = (Pij ). If S is the finite set {1…N}, say, then P is an N х N matrix. “Pij” is the conditional probability, given that the chain or system is in state i at time n, say, that the chain jumps to the state j at time n + 1 and it is independent of n. That is,

The pij’s are often referred to as the transition probabilities for the Markov chain.

This equation implies that the probability law relating the next period’s state to the current state does not change over time.

It is often called the Stationary Assumption and any Markov chain that satisfies it is called a stationary Markov chain.

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û

ù

êêêê

ë

é

=

NNNN

N

N

ppp

ppp

ppp

P

K

MMM

K

K

2 1

22221

11211

We also know that each entry in the P matrix must be nonnegative.Hence, all entries in the transition probability matrix are nonnegative, and the entries in each row must sum to 1.

k-Step Transition Probabilities A question of interest when studying a Markov chain is: If a Markov chain is in a state i at time

n, what is the probability that k periods later the Markov chain will be in state j ?

This probability will be independent of n (time homogenous markov chain), so we may write

P(Xn+k =j|Xn = i) = P(Xk =j|X0 = i) = Pij(k)

where Pij(k) is called the k-step probability of a transition from state i to state j.

For k > 1, Pij(k) = ijth element of P k

Pij(2) is the (i, j)th element of matrix P2 = P1 P1

Pij(k) is the (i, j)th element of matrix Pk = P1 Pk-1

Note that Pij is transition probability from state i to state j that is, an element of the transition matrix

The S.I.R Model

The spread of a disease caused by a microorganism through a population can be modeled mathematically using differential equations. The SIR model presented here combines disease spread person-to-person and are familiar to students, such as measles, smallpox, and influenza.

Members of a population are categorized into one of three groups:-

Susceptibles Infective Recovered

Assumptions:-

This model is an appropriate one to use under the following assumptions :-

1) The population is fixed.

2) The only way a person can leave the susceptible group is to become infected.

The only way a person can leave the infected group is to recover from the

disease. Once a person has recovered, the person receives immunity.

3) Age, sex, social status, and race do not affect the probability of being infected.

4) There is no inherited immunity.

5) The member of the population mix homogeneously (have the same

interactions with one another to the same degree).

6) Demographic changes are not considered when the epidemic lasts not for long.

The two fundamental parameters of the model , β (the daily infection rate, ≥ 0) and r (the recovery rate that maybe ≥ 0), act as rate constants that control how fast members progress into the I & R groups, respectively.The scheme can also be translated into a set of differential equations:

Using this model, we will consider a mild, short-lived epidemic, e.g. influenza, in a closedpopulation. Closed means that there is no immigration or emigration. Moreover, given the time scale of influenza epidemics, we will not consider demographic turnover (birth or death), and all infections are assumed to end with recovery. The size of the population (S +I +R) is therefore constant and equal to the initial population size, which we denote with the parameter N.

1

Greenwood assumption

Let the number of susceptibles and (new) infectives of generation t be S(t) and I(t), respectively. It is assumed that the number of infectives of generation t + 1 is a binomial random variable with parameters S(t) and p(I(t)), the latter being the probability that an existing susceptible will become infected when the number of infectives is I(t).

Thus,

for k = 0,1,. . . ,x. In the Greenwood model, p(i) = p is a constant not depending on the number of infectives . Time is discrete, with epochs t = 0,1,2, . . . The natural unit for the duration of an epoch is one day.

SIR Model with Greenwood assumption

S I R

Iβ*S r*I

rIdt

dRrISI

dt

dISI

dt

dS

Transition matrix Markov chain

To

S I R

S PSS PSI PSR

From I PIS PII PIR

R PRS PRI PRR

P = probability to go from a state at time t to a state at time t+1

Markov chain modeling

To

S I R

S PSS PSI PSR

From I PIS PII PIR

R PRS PRI PRR

R

I

S

Starting vector*

* number of S, I and R at the start of the modeling

Example:- Markov chain modeling

To

S I R

S 0.90 0.10 0.00

From I 0.00 0.80 0.20

R 0.00 0.00 1.00

0

1

99

Starting vector*

number of S, I and R at the start of the modeling

Results of Markov chain model

Time step(k)

S I R

0 99 1 0

1 =99*0.9+1*0+0*0=

89.1

=99*0.1+1*0.8+0*0=

10.7

99*0+1*0.2+0*1=

0.2

Example :- Markov chain modeling

To

S I R

S 0.90 0.10 0.00

From I 0.00 0.80 0.20

R 0.00 0.00 1.00

2.0

7.10

1.89

Starting vector *

* number of S, I and R at the end of time step 1

Results of Markov chain model

Time step S I R

0 99 1 0

1 99*0.9+1*0+0*0=

89.199*0.1+1*0.8+0*0=10.7

99*0+1*0.2+0*1=

0.2

2 89.1*0.9+10.7*0+0.2

*0=80.289.1*0.1+10.7*0.9+

0.2*0=17.589.1*0+10.7*0.2+0.2

*1=2.3

Course of number of S, I and R animals in a closed population (Greenwood assumption)

0 1 2 3 4 5 6 7 8 9 10

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0

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100S I R

Number of time steps

Num

ber

of

anim

als

Drawback of the Greenwood assumption

The major significance of the model at the time of its rst publication was a mathematical demonstration that even with a major outbreak of a disease satisfying the simple model, not all susceptibles would necessarily be infected.

Primarily these models have been applied to small epidemics, in particular epidemics within households where an initial infected individuals spreads the infection to other household members and not like deterministic models that are appropriate for large populations. Evolution of epidemics is deterministic in the sense that no randomness is allowed.

References :- Courtsey “Massachusetts Institute of Technology” open courseware.

“Markov Chains” by J.Chang,March 30, 1999.

“Intensity matrix” by Anders Andersson

Introductory lecture notes on “Markov Chains and Random Walks” by Takis Konstantopoulos

http://demonstrations.wolfram.com/SIREpidemicDynamics/

SIR models of epidemics by Florence Debarre from Institute of Integrative Biology ETH Zürich,Germany.

“Some properties of a simple stochastic epidemic model of SIR type” by Henry C. Tuckwell (Max Planck Institute for Mathematics in the Sciences ) & Ruth J. Williams (Department of Mathematics, University of California San Diego)

Thank you for your attention

When the Infectious period is long--

Moreover, the SIR model is based on assumption that each infected individual can transmit the infection and later he recovers fully immune. When the infectious period is long, the assumption should be modified and a new parameter E(t), number of infected but not infectious individuals at time t, should be introduced.

γ is the expected latent period. Also the assumption that the demographic changes are none is false in this condition hence death rate μ is incorporated signifying death due to unrelated causes

REASON to the equation in slide 11 .

Suppose the population size is n, a constant and let the numbers of susceptibles and (new) infectives of generation t be S(t) and I(t), respectively.

Then the initial condition is S(0) + I(0) = n and

S(t + 1) + I(t + 1) = S(t),

where t =0, 1, 2, . . ., as the infectives and susceptibles of generation t + 1 are drawn from the susceptibles of generation t.

Thus, ‘’ where ‘j=0 to t’ and is the total number infected up to and including generation ‘t’. It is assumed that the number of infectives of generation t + 1 is a

binomial random variable with parameters S(t) and p(I(t)) so hence the conditional probability (Markov property)

Properties to equation 1 slide 10

This system is non-linear.

This follows that

►

This means there will be a proper epidemic outbreak with an increase of the number of the infectious and vice versa if the above expression is less than 0.