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MBW:Comparison of Deterministic and Stochastic SIS and SIR Models in Discrete Time

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Executive Summary

In the article “Comparison of deterministic and stochastic SIS and SIR models in discrete time” by Linda J.S. Allen and Amy M. Burgin several three different infectious disease models are analyzed using both deterministic and stochastic methods and results are compared. The three models are a SIS model with constant population, a SIR model with constant population, and a SIS model with variable population. All three are discrete-time models. For all three deterministic models it is shown that the extinction or persistence of the disease is dependent upon the basic reproductive number R0. In the cases of disease persistence the non-trivial equilibrium is dependent upon the form of the force of infection. For the stochastic models, regardless of the value of the basic reproductive number, the probability of disease extinction approaches 1. For the SIS model with variable population the elapsed time until absorption is observed to increase as population increases. For the analysis of the stochastic models probability distributions and quasi-stationary distributions are utilized as well as difference equations for the mean and the quasi-stationary mean. These discrete-time stochastic models can be considered approximations to continuous-time Markov jump processes, and have restrictions put upon the time step size so that true probability distributions may be obtained. Ultimately a significant difference in the disease characteristics of the SIS stochastic model was found when comparing the constant population model to the variable population model.

Context/Biological Phenomenon

All of the discrete-time models discussed in this paper are directly applicable to specific infectious diseases, and can also be used as approximations of the continuous-time SIS and SIR disease models. Two types of infection dynamics are considered. SIS models are used to study infections for which a permanent immunity is not built up by the infected population. An individual within the population is initially susceptible. If they become sick and recover they return to being susceptible again. As a result of this there may exist an endemic equilibrium where a proportion of the population is always sick. This differs from SIR models in that after an individual leaves the infected population they are unable to become infected again and remain in the recovered population.


Model

Deterministic SIS Model with Constant Population

The first infectious disease model considered in this paper was a discrete-time deterministic SIS model with the total population held constant. It takes the following form.

S1.jpg

S2.jpg

These two equations define how the susceptible and the infected populations change. The variables are defined to be:

S3.jpg

This SIS model has three parameters. They are defined to be:

S4.jpg

S5.jpg

This model assumes that α > 0, β > 0, and that γ > 0. Since a constant population is being considered the relationship S(t) + I(t) = N is true for all values of t. It must also be true that the number of births is equal to the number of deaths in this population as the population must remain constant. Individuals in the population who recover from the infection immediately become susceptible again, and it is assumed that all newborns are susceptible. As a result of these assumptions this model is equivalent to an SIS model which does not have any births or deaths, as desired. The following assumptions are made regarding the force of infection:

S6.jpg

Two more useful definitions are defined bellow:

S7.jpg

From the definition of average number of infection per susceptible individual per unit time it becomes possible to define a function for the probability of k successful encounters resulting in a susceptible individual becoming infected in one unit of time. This is defined assuming that the probability follows a Poisson distribution.

S8.jpg

The probability that a susceptible individual does not become infected is S10.jpg S11.jpg From this it can be said that the number of individuals who do not become infected during the time interval Δt is equal to S12.jpg. The number of individuals who do become infected during the time interval Δt is equal to S13.jpg. The basic reproductive ratio is what determines the asymptotic behavior of the model.

S9.jpg

Conclusions of Model

When the basic reproductive ratio is less than or equal to 1 then the infection becomes extinct. When the basic reproductive ratio is greater than 1, solutions approach a unique and positive equilibrium solution.

Stochastic SIS model with Constant Population

This model is a Markov chain with finite state space. For this model it is assumed that there is a maximum of one event which may occur at any given time interval. Which event that is depends upon the values of the state variables. The events are an infection, a birth, a death, or a recovery, however in this case as the population size must remain constant a birth and a death must occur together.

Se12.jpg

Note that I is integer-valued. The state probabilities are defined as

Se13.jpg

The probability of there being a new infective in during time interval Δt is defined as

Se1.14.jpg

Note that if i takes on a value that is less than zero or greater than N, then the probability of there being a new infective is equal to zero. The following are other definitions:

Se14.jpg

The following are the transition probabilities for this SIS model:

Se15.jpg

The following difference equations are satisfied by the probabilities:

Se16.jpg

For the above difference equations i = 1, … , N and pi = 0 for values of i that are not within the integer set {0, 1, … , N}. The transition probabilities defined here will approximate the transition probabilities of the continuous time Markov jump process. The discrete time process makes the assumption that if the time interval is sufficiently small, then the transition probability is (β + γ)iΔt.

Se17.jpg

The matrix T defined above is the N + 1 x N + 1 transition matrix. It is the expression of the difference equations for this discrete time model.

Se18.jpg

Now the relationship between p(t + Δt) and p(t) can easily be defined.

Se19.jpg

Note that all of the elements of T should be probabilities. To ensure that this is true no element of T is allowed to be greater than 1 or less than 0. The following conditions will guarantee that the elements of T remain within the correct range.

Se20.jpg

Conclusions of Model

From the theory of Markov chains, the limit of p0(t) as t approaches infinity is equal to 1. This result is due to the fact that the matrix T is a stochastic matrix which has a single absorbing state. In this case that state is the zero state. An absorbing state is defined as a state from which it is impossible to leave. An absorbing state may be recognized by examining the matrix T. For a given column i in the matrix if there is a 1 in the aii position of the column and zeros filling all other positions of the column then state i is an absorbing state. Simply by observing the matrix T it can be seen that these conditions are only met with the first column. T is an N + 1 x N + 1 matrix. The values of i are the integers {0, 1, … , N}, which makes the first column correspond to the zero state. This means that eventually there will be no infected individuals in the population independent of the value of R0. Although eventually the infectious disease will become extinct, for different values of R0 extinction will occur at different rates.

Deterministic SIR with Constant Population

The deterministic SIR model has the following form:

Se21.jpg

The following are the variable and parameter definitions for this model:

Se22.jpg

Se23.jpg

The conditions S(0) > 0, I(0) > 0, and R(0) ≥ 0 are assumed. As the population is being held constant the relationship S(t) + I(t) + R(t) = N must be true for all t. The same assumptions are made regarding the force of infection in this model as in the deterministic SIS model. The basic reproductive number R0 determines the asymptotic behavior of the model as it did in the deterministic model. The proof of this is not reproduced here, but can be seen in the original paper.

Conclusions of Model

If less strict assumptions were made for the force of infection it would be possible for there to exist periodic solutions. For R0 ≤ 1 solutions to the model will approach the disease free equilibrium state. For R0 > 1 then an endemic equilibrium exists. For the force of infection defined as it is in these models, the endemic equilibrium is locally asymptotically stable. This conclusion is dependent upon the assumption that β > 0. If β = 0 then the limit of the infected population would approach zero as t approached infinity, and the disease would become extinct even though the basic reproductive ratio is greater than 1. This intuitively makes sense, as individuals develop immunity after becoming infected, and never rejoin the susceptible population.

Stochastic SIR model with Constant Population

This model is a Markov chain with finite state space. This model is dependent upon the following two random variables:

Se24.jpg

The model has a joint probability function, as opposed to the stochastic SIS model which had a probability function dependent only upon the random variable for the number of infected individuals.

Se25.jpg

Se26.jpg

The probability that a new individual becomes infected in time interval Δt is defined as:

Se27.jpg

The following are the other three transition probabilities for this model; the probabilities fore each of the events which might take place during time interval Δt:

Se28.jpg

To maintain the constant population size, whenever there is a death there is also a birth. A death of either an infected or an immune individual will be accompanied by the birth of a susceptible person. The above probability equations satisfy the following difference equations:

Se29.jpg

For these difference equations again we have i, r = 0, 1, … , N and 0 ≤ i + r ≤ N. Also, if i or r are less than zero or greater than N then pir(t) = 0. It is necessary that all of the transition probabilities are positive, as a negative probability is meaningless, and that they are bounded by 1. To ensure that these conditions are satisfied, the inequality Se30.jpg must always be true. This can be guaranteed if Δt is kept small enough.

Conclusions of the Model

As this discrete-time model is a Markov chain there does exist a transition matrix which expresses the difference equations of the model, however it is not possible to express it in a simple form. It is therefore not expressed explicitly. Just as in the stochastic SIS model, there is a single absorbing state for the stochastic SIR model. In this case it occurs when I and R are both equal to zero. For this model, just as with the deterministic model, it is seen that the infection is endemic within the population. This is a result of the influx of new susceptible due to a positive birth rate.

Deterministic SIS with Variable Population

For this model it is assumed that the size of the population, N(t), is a function of time. The relationship N(t) = S(t) + I(t) is true for all t. This relationship results in there existing two independent dynamic variables in the model. The growth of the population is described with the following difference equation:

Se31.jpg

The following conditions are satisfied by the functions f and g:

Se32.jpg

This implies that F(K) = 1 and that f(K) = 0 and that g(N) > N for N ϵ (0, K). The conditions placed upon the population growth difference equation imply that N(t) is monotonically increasing to its limit, K. There are various different functions F(N) which will satisfy the necessary conditions described above. Two examples are:

Se33.jpg

The deterministic SIS model with variable population size can now be expressed with the following equations:

Se34.jpg

The following is a list of definitions for this model:

Se35.jpg

For this model the conditions S(t) > 0, I(t) > 0, and N(0) = S(0) + I(0) < K must be true. The following conditions are assumed to be true:

Se36.jpg

Se37.jpg

Mass action incidence rate is no longer possible as it was in the previous models. This is due to the fact that now population size is variable.

Conclusions of the Model

For this model the asymptotic behavior is determined by the basic reproductive ratio. When the basic reproductive ratio, R0, is less than 1 the limit of the infected population is zero as time goes to infinity, and the limit of the susceptible population is equal to the carrying capacity as time goes to infinity. When the basic reproductive ratio is greater than or equal to 1 and when the function Se38.jpg has a unique and positive fixed point x* where 0 < x* < xM with h(xM) being the maximum value when x is within [0, 1] there are endemic equilibrium solutions to the model. In this case the limit of the infected population will be equal to x*K as time goes to infinity and the limit of the susceptible population, as time goes to infinity, will be greater than zero and equal to the K minus the infected population limit. The proof of this result is written out in full in the original paper, however it will not be reproduced here.

Stochastic SIS Model with Variable Population Size

This model is formulated as a Markov chain. There are two random variables used, and they represent the number of infected individuals and the total number of individuals.

S14.jpg

This model also uses a joint probability function. This will be defined as follows:

S15.jpg

The following are the transition probabilities for this model along with their corresponding event definitions: ΠinΔt = λ(i/n)(n – i) = probability that a new individual becomes infected in time interval Δt, where i/n = the proportion of infectives

S16.jpg

For these probabilities, i ≤ n, i, n = 0, 1, …, M, and f(n) is as it was defined for the deterministic model with variable population. Notice that in the stochastic SIS model with constant population the transition probabilities were very similar to these, however there were only four a new birth was associated with every death. In this model since population is not being held constant the probability of a birth has its own transition probability. The following conditions from the deterministic SIS model with variable population size are assumed to be true for this model:

S17.jpg

And two new conditions are added to the assumptions:

S18.jpg

These last two conditions are imposed upon the model because the population size should be bounded. Without these conditions the population size would be able to increase above the carrying capacity, K. The addition of the last two conditions causes the probability of the death of an individual to be greater than the probability of a new birth when the total population is greater than K and less than or equal to M. Therefore M is the upper bound on the population size. The following are the difference equations for pin(t):

S19.jpg

For i ≤ n ≤ M, i, n = 0, 1, … , M and pin = 0 for i, n not in [0, M]. The probabilities must also satisfy S20.jpg

Conclusions of Model

For this model there is only one absorbing state, and it is the state when I = 0 and N = 0. As seen in the previous two stochastic models considered, this indicates that regardless of the value of R0 the infection will eventually go extinct. The duration of the infection will be determined by the value of R0 and upon the value of the carrying capacity, K. If the basic reproductive ration or the carrying capacity is very large it could cause the infection to be present in the population for a long period of time.

Numerical Analysis

From the Original Paper

The following are plots of the numerical examples from the original paper.

SIS Model with Constant Population Size

S21.jpg

S22.jpg

Stochastic SIR Model with Constant Population Size

S23.jpg

Stochastic SIS Model with Variable Population Size

S24.jpg

Reproduction of Results

The numerical analysis for the deterministic and stochastic SIS models was reproduced using Matlab.

S25.jpg

Matlab Code

S26.jpg

Matlab Code

S27.jpg

Matlab Code

S28.jpg

Matlab Code

S29.jpg

Matlab Code

S30.jpg

Matlab Code

S31.jpg

Matlab Code

Results

For the deterministic SIS model, the numerical analysis is in agreement with the findings that the solutions are dependent upon R0 as was determined within the original paper. When the basic reproductive ration was less than one, the infection became extinct. When the basic reproductive ratio was greater than one, the infection remained endemic within the population. For the stochastic SIS model, the numerical analysis also is consistent with the findings of the paper. The infection went extinct when the basic reproductive ratio was both less than and greater than zero. Simulations of 100, 500, and 1000 were run for the stochastic SIS model, all of which show the infection becoming extinct very rapidly, less than 30 units of time. The probability function of the stochastic SIS model was also plotted. For this graph the axis only extends out 10 units of time, and the probability of there being zero infected individuals can be seen to be approaching 1.

Conclusion

Models of three different types of infectious diseases were developed and analyzed using both deterministic and stochastic modeling techniques in “Comparison of deterministic and stochastic SIS and SIR models in discrete time” by Linda J. S. Allen and Amy M. Burgin. All three of the stochastic models were formulated as Markov chains which could be considered approximations to their corresponding continuous time Markov jump processes. The first type of disease model looked at was an SIS model with constant population size. It was found that for the deterministic model, the disease dynamics were dependent upon the value of the basic reproductive ratio. When it was less than one the infection was not sustained within the population. When it was greater than one the infection was able to become endemic within the population. For the stochastic model the number of infected individuals always went to zero. The duration of the epidemic was found to be dependent upon the initial conditions. Numerical examples were developed, and are consistent with the findings of the analytical findings of the original article, and with the numerical examples provided in the article.

The second type of disease model looked at was an SIR model with constant population size. For the deterministic model it was found that the dynamics of the infection were dependent on the value for the basic reproductive ratio. When it was less than one, the infected population went to zero, and when it was greater than one an endemic equilibrium was found. This is possible only because the model allowed for there to be both births and deaths within the population. Then number of births was equal to the number of deaths so that the total population was kept constant. A newborn in the population was immediately added to the susceptible population, and so there was a constant influx of new susceptible. If this was not included in the model then the infection would have died out regardless of the value of the basic reproductive ratio. For the stochastic model, the infection was shown to become extinct from the population regardless of the basic reproductive number, although this was not verified globally within the paper.

The third type of disease model examined in this paper was an SIS model with a variable population size. For the deterministic model, once again, it was shown that when the basic reproductive ratio was less than one the infectious population went to zero, and when the basic reproductive ratio was greater than one there existed a positive equilibrium solution to the model. For the stochastic model eventually the infectious population approached zero, but the time in which the infection was present within the population varied depending upon the size of the population.

Reference

Allen, Linda L.S, and Amy M. Burgin. "Comparison of Deterministic and Stochastic SIS and SIR Models in Discrete Time." Mathematical Biosciences 163.1 (2000): 1-33. SciVerse. Web. Apr. 2013.