Expected Time to Extinction of SIS Epidemic Model Using Quasy Stationary Distribution

03/16/2018 ∙ by Kurnia Susvitasari, et al. ∙ 0

We study that the breakdown of epidemic depends on some parameters, that is expressed in epidemic reproduction ratio number. It is noted that when R_0 exceeds 1, the stochastic model have two different results. But, eventually the extinction will be reached even though the major epidemic occurs. The question is how long this process will reach extinction. In this paper, we will focus on the Markovian process of SIS model when major epidemic occurs. Using the approximation of quasi--stationary distribution, the expected mean time of extinction only occurs when the process is one step away from being extinct. Combining the theorm from Ethier and Kurtz, we use CLT to find the approximation of this quasi distribution and successfully determine the asymptotic mean time to extinction of SIS model without demography.

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1 Introduction

According to the threshold theorem, epidemic can only occur if the initial number of susceptibles is larger than some critical values, which depends on the parameters under cosideration [4] . Usually, it is expressed in terms of epidemic reproductive ratio number, .

Susvitasari [22] showed that both deterministic and stochastic models performed similar results when , i.e. the disease–free stage in the epidemic. But then, when , the deterministic and stochastic models had different interpretations. In the deterministic model, both SIS and SIR showed an outbreak of the disease, and after some time , the disease persisted and reached equilibrium stage, i.e. endemic. The stochastic model, on the other hand, had different interpretation. There are two possible outcomes of this approach. First, the infection may die out in the first cycle. If it did, it would happen very quickly since the time of the disease removed must less than the time of infectee–susceptible contact. Second, if it survives the first cycle, the outbreak was likely to occur, but after some time , it would reach equilibrium as the deterministic version. In fact, the stochastic model would mimic the deterministic path and scattered randomly around its equilibrium. Furthermore, by letting population size be large and ignoring the initial value of infectious individuals, as , the empirical distribution of , where is the number of infectious individuals at time with

population size followed normal distribution in endemic phase as seen in Figure

2.

In this paper, we will focus on the endemic stage, the stage where and the outbreak occurs in SIS epidemic model without demography to determine the expected mean time to extinction of this model.

2 The Birth and Death Process in SIS

Suppose that denotes the Markovian process that represents the number of infected individuals in the population at time . Then we define as the state space of the process, where state zero is an absorbing state. Figure 1 is the illustration of this process.

Figure 1: Birth–death of SIS without demography transition process

Consider the case where and the outbreak occurs. Eventually, given a necessary large and some time , the process will not visit state zero. So, let us redefine the birth–death process before by only considering the transient states. Let be the modified process of with state space . The intuitive explaination is that whenever the SIS process goes extinct, it immediately will restart again in state 1. It is not hard to show that the process

is time–reversible Markov chain. Therefore, there exist colections of equilibrium distributions

for all that sums to unity and satisfy the detailed–balanced conditions:

  1. ,

  2.   ,

  3. ,

with constraint . Thus the solution is

(1)

Therefore,

(2)

Now, let denote the time to extinction in SIS epidemic model. Recall that the process is time-reversible on state . Then, the time to extinction is almost surely finite since the state is visited finitely often.

Suppose that is in equilibrium, then the mean time to extinction has intensity times the rate of the process getting absorbed to state .

(3)

As when we set . Ball et al. [6] showed the approximation of mean time extinction when and .

In this paper, we will focus on the time to extinction when the process reaches endemic–stage, i.e. when and the process survives the first cycle of extinction using quasi– statioary distribution and property of central limit theorm.

3 Expected Time of Extinction of SIS in Endemic–Stage

Before we start this section, we will introduce first the quasi–stationary distribution.

3.1 Quasi–Stationary Distribution

Recall the process with finite state space , where 0 is an absorbing state. For , let

(4)

Since we concerns on the distribution of the process in endemic stage, we wish to find .

We define and as sets of absorbing and transient state. Suppose

where is transition rate matrix, defined as . Let . Since is symmetry, according to spectral decomposition theorem,

where

are eigenvalues of

and

are corresponding orthonormal vectors of right-eigenvectors. Therefore,

(5)

where and we can see in Appendix ……… that . Recall that forward Kolmogorov equation, must satisfy , where with solution . So, for ,

(6)

According to Darroch and Seneta [12] there exists a simple, real eigenvalue such that all other eigenvalues of have real part less than . Therefore, eq. (4) becomes

(7)

Consequently, as .

Note that and it turns out that is left eigenvector of (see Appendix B). In the stationary scenario, when

, each row of probability transition matrix

has nearly similar entries. So, it follows that the quasi-distribution is given by the left-eigenvector of corresponding to eigenvalue with constraint .

3.2 Expected Time to Extinction by Quasi–Stationary Distribution

Suppose that and let . Since two events (infection and recovery) in the epidemic model follows the Poisson process, there is zero probability that two or more events occur at the same time. So, the extinction only occurs when the process is one step away from being extinct. Therefore,

(8)

Recall that is the left-eigenvector of , corresponding to the eigenvalue . So,

Hence, the equation (8) becomes

(9)

Since the process is Poisson process, then the inter arrival time must follow exponential distribution. Therefore, in eq. (

9), follows exponential distribustion with intensity . But, recall the memoryless property of the Poisson process and that the extinction only occurs if the process is one step away from being extinct. Intuitively, it also means that the extinction happens with intensity times the rate of the process to jump into absorption state , which is . On the other hand, (note that ).

Proof.

Using the fact that is the left-eigenvector of ,

Therefore, the mean time to extinction of the SIS process is

(10)

where .

The quasi–stationary is a very powerful approximation when we let population size large enough. Unfortunately, consider that by using this method, we need to capture all the possible transition rate for matrix . Considering the large , the size of matrix will also big and leads to inefficient computation. Therefore, we should approximate the distribution of .

Now, suppose that we only concern on the size of infectious individuals in population when the outbreak occurs, i.e. . By ignoring the initial value of infectives, the empirical distribution of follows normal distribution with mean converges to equilibrium point in deterministic model as seen in Figure 2. It turns out that by applying Central Limit Theorem, we could approximate the quantity of .

4 Central Limit Theorem

We define to be the CTMC on with finite state space and finite number of possible transition in such a way that for all . Then let as the drift function of the process .

Suppose that is a continuous function and differentiable in lattice such that exists and is continuous and let be matrix function defined as

Further, let , where , be the solution of the matrix differential equation

(11)

where .

Theorem 4.1.

(Ethier and Kurtz) Suppose that

  1. is a constant, where ,

  2. and are continuous functions.

Then, for ,

as , where is a zero–mean Gaussian process with covariance function given as

(12)

The functions and are as defined previously.

The proof of this theorem can be seen in Britton and Andersson [9] or in Ethier and Kurtz [14].

Note that the covarian function in equation (12

) is time-dependent. It implies that the variance function is also time-dependent. Using equation (

12), the variance function of is

(13)

where must satisfy equation (11). By differentiating equation (13) with respect to ,

(14)

where .

Note that in the epidemic case, when we let , tends to zero because if the epidemic survives and takes off, it will reach the endemic stage and the infection process converges to a certain distribution. Otherwise, if the epidemic dies out, we will see in the later section that the stable stage is reached in the disease-free stage. Both scenarios are in equilibrium level. Therefore, once one of these two cases is attained, definitely goes to zero as .

Now let and be the variance-covariance matrix and the stationary infection point when the epidemic reaches endemic stage. Hence, as ,

(15)
(a)
(b)
(c)
(d)
Figure 2: Empirical distribution of SIS endemic level with CLT approximation

Figure 2 illustrates the distribution of .

4.1 Expected Mean Time to Extinction by CLT

Suppose that and is in quasi– equilibrium (meaning that enters endemic level). Then, the process will definitely mimic the deterministic model and with equilibrium point .

Now, recall a drift function , where we define in such a way that . In SIS, represents the possible transitions of the process, i.e., for infection to occur and for removal to occur. Therefore,

(16)

Furthermore,

(17)

and

(18)

Thus since we assume that the process is in quasi–equilibrium, for all . Therefore,

(19)
(20)

Since (19) is continuous, then according to Theorem 4.1, for any ,

(21)

as and must satisfy

(22)

with initial value . Therefore, the solution of equation (22), for , is

(23)

The eq.(21) is equivalent to

(24)

for large and large .

Suppose that and for simpler notations. Let and denote the pdf and cdf of . Then using a continuity correction, we can approximate quasi-stationary distribution in (10) as follows

(25)

where and are pdf and cdf of standardised normal distribution. Therefore, the approximate mean time to extinction of the SIS epidemic model by applying Theorem 4.1 is

(26)

for large and large .

5 Conclusion

We have successfully determined the approximation of expected mean time to extinction of SIS model using quasi–stationary distribution . An interesting result is that the empirical distribution of the SIS in endemic–stage followed normal distribution as the population size went to infinity. By using the CLT of Ethier and Kurtz, it turned out that the mean parameters of the SIS is the equilibrium points derived from solving deterministic’s ODE. This result was not surprising since we have showed in previous work that the stochastic models moved randomly around the deterministic model’s equilibrium point. Furthermore, the variances of the models converged to some positive constants since intuitively, as the population size went sufficiently large, the stochastic paths moved randomly around its endemic–equilibrium point.

6 Appendix A

First, we need to show that for ,

Proof.

Note that since is orthonormal, then

(27)

since

And secondly, we need to show that .

Proof.

Note that,

(28)

Note that . Then, for any ,

7 Appendix B

Notice . Then,

Hence, is the left-eigenvector of .

8 References

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