SIS Epidemic Model: Birth-and-Death Markov Chain Approach
SSIS EPIDEMIC MODELBIRTH-AND-DEATH MARKOV CHAIN APPROACH
AUBAIN NZOKEM Department of Mathematics & Statistics, York University, Toronto A BSTRACT . We are interested in describing the infected size of the SIS Epidemic model usingBirth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined withina population of constant size M ; the size is kept constant by replacing each death with a newbornhealthy individual. The life span of each individual in the population is modelled by an exponen-tial distribution with parameter α ; and the disease spreads within the population is modelled by aPoisson process with a rate λ I . λ I = β I ( − IM ) is similar to the instantaneous rate in the logisticpopulation growth model. The analysis is focused on the disease outbreak, where the reproduc-tion number R = βα is greater than one. As methodology, we use both numerical and analyticalapproaches. The analysis relies on the stationary distribution for Birth and Death Markov process.The numerical approach creates sample path simulations into order show the infected size dynam-ics, and the relationship between infected size and R . As M becomes large, some stable statisticalcharacteristics of the infected size distribution can be deduced. And the infected size is shownanalytically to follow a normal distribution with mean ( − R ) M and Variance MR .Deterministic Model, Stochastic Model, Birth - Death Markov Chain, Irreducible MarkovChain (IMC), Jensen Inequality, Epidemic Model
1. I
NTRODUCTION
The Birth and Death Markov Chain is a special class of the continuous stochastic process. Theimportance of such class arises from the fact that it is generated by combining two standardprocesses (Birth process and Death process). The stationary distribution of such process at theequilibrium was studied in the Mathematics literature [1]. The findings are useful as one of theinterests in studying stochastic process is to describe the behaviour of the stochastic process inthe long run; that is, how the process is distributed when the time becomes large.The Susceptible-Infected-Susceptible (SIS) model is one of the simplest and most paradigmaticmodels in mathematical epidemiology. The stochastic version of the SIS model was studied byN˚asell [2, 3], who is among the pioneers to report the normal distribution nature of the quasi-stationary distribution when the reproduction number is greater than one and the population size( M ) is large. The major critic was his methodology. Ovaskainen [4] argues that the methodologywas heuristic. In general, the literature review [3–8] offers many approximations of the quasi-stationary distribution of the SIS model, which may reflect the difference in methodologies orin parametrisations. Some studies [8] consider transmission parameter ( β ) as a function of thepopulation size ( M ).In our study, the Birth and Death Markov chain is used to describe the dynamics of the SIS model [email protected]. a r X i v : . [ q - b i o . P E ] F e b and its features; the parameter are fixed, and the reproduction number ( R ) is greater than one,only the size of the population ( M ) can change. In the next section, we will formulate the diseasespreading parameters and analyze some infected size sample paths and how they are impactedthe reproduction number. And the last section will focus on the distribution of the infected size.Both numerical and analytical approaches will be used to analyse the distribution nature of theasymptotic infected size.2. E PIDEMIC S PREADING : D
EATH AND I NFECTION P ROCESS M ODELLING
In SIS model, the population is divided at each time t into susceptible individuals ( S ( t ) ) andinfective individuals ( I ( t ) ). The evolution of these quantities is usually described in Epidemiologyby the following deterministic differential equations (2.1) ( a ) : ( a ) ( b ) dSdt = − β M SI + α I SM = αβ = RdIdt = β M SI − α I IM = − αβ = − R (2.1)The parameters β and α are respectively the transmission rate and the rate of death and birth.To have population size ( M = S + I ) constant over time, each individual who dies is replaced bya susceptible individual. The threshold value R = βα , which is a basic Reproduction Number, isan indicator that determines whether we will have extinction of the disease (0 < R <
1) or anoutbreak of the disease ( R > ( b ) of the system (2.1) ( a ) in the long run when R >
1. At the equilibrium(2.1) ( b ) , the portion infected is constant (1 − R ).The deterministic version of the SIS model was introduced by Kermack and McKendrick and hasbeen fully analysed. For related deterministic work of interest, see [5, 9, 10].The stochastic version, called the stochastic logistic epidemic model, is usually modelled as Birthand Death Markov Chain where the transition probability is defined as follows for I(t) taking valueon { , , . . . , M } P I , J ( t , t + h ) = λ I h + o ( h ) if J = I + µ I h + o ( h ) if J = I − − ( λ I + µ I ) h + o ( h ) if J = I (2.2)with lim h → o ( h ) =
0; lim h → o ( h ) =
0; lim h → o ( h ) = β = c θ can be written as a product of contact rate ( c ) andthe probability of infection ( θ ) . The transition probability ( . ) can be fully determined by thefollowing set of rules [7]: (a) each individual gets into contact with another individual after anelapsed time, which follows an independent and identically distribution (iid). The elapsed time isexponentially distributed with parameter c . And if the contact involves a susceptible individualand infected individual, the probability of infection is θ . The transmission rate is β = c θ . (b) the infected lifetime is also an exponentially distributed with parameter α , because of the memory-less property of the lifetime.For J = I + ( . ) , all I infected individuals get into contact with another individual accordingto a Poisson process with parameter cI , since ( − IM ) is the probability to meet susceptibleindividuals and θ is the probability of infection, by thinning the Poisson process, we concludethat the contacts between infected and susceptible individuals that will end up with infectionfollows a Poisson process with parameter λ I = β I ( − IM ) .For ( J = I − ) in ( . ) , the number of infected individuals that becomes susceptible individualsfollows a Poisson process with parameter µ I = α I . Based on the assumptions and parameters developed previously, a MATLAB program with M individuals was created, and the main variables were age, health status, cumulative elapsed oftime between events. At the death of an individual, a healthy individual and his life span areintroduced in the program code. By controlling the age, we can focus on the infection processover the time.For R = α = .
3, we have the infected size from two variables: heathy status and cumulativeelapsed of time between infections. Two sample paths are presented in Fig 1 with only oneinfected initially and with 95% infected initially.
Cumulative elapsed time between events I n f e c t ed s i z e R=2
Only One infected initially95% infected initially F IGURE
1. Deterministic versus Stochastic equilibrium of the infected sizeIn the case of only one infected initially, the infected size grows at an increasing rate before fluc-tuating around the deterministic equilibrium M ∗ ( − R ) developed in (2.1) ( b ) . In the secondcase of 95% infected initially, as shown in Fig 1, the infected size decreases rapidly before fluc-tuating around the deterministic equilibrium. We have illustrated in Fig 2 how the sample path reacts with respect to the reproduction number(R). In fact, when there is only one infected initially and the reproduction number is greaterthan 1, the infected size increases rapidly, before fluctuating around deterministic equilibrium M ∗ ( − R ) , straight line in black color in Fig 2. The same pattern is observed when there is 95%infected initially, the fluctuation follows a rapidly decreasing. The stability is also shown in Fig2 for R = R =
4, whereas for R =
1, the process is unstable, and the disease will eventuallydie out. F IGURE
2. Impact of the Reproduction Number ( R ) on the equilibrium of thestochastic infected sizeIn addition, for R ≥
1, there is a positive relation between the infected size and the reproductionnumber ( R ). As illustrated in Fig 2, the infected size increases when R increases. Based on the parameters developed in ( . ) , we have a Birth and Death process on the state space { , , . . . , M } with transition rates: λ k = β (cid:18) M − kM (cid:19) k ( k → k + ) µ k = α k ( k → k − ) · · · · · · M − M µ λ µ λ µ λ µ λ M − µ M − λ M − µ M However, λ = β ( M − ) M = {
1, 2, ...., M } and { } . In order to have an irreducible Markov Process, we introduce an external source of disease through a small nonnegative parameter ( ε >
0) in theinfection transition rate. We have a new Process P M , ε on the state space { , , . . . , M } . λ [ ε ] i = β ( M − i ) M i + ε ( i → i + ) µ i = α i ( i → i − ) According to Karlin et al [1], we define the following quantities: θ = θ [ ε ] i = λ [ ε ] λ [ ε ] · · · λ [ ε ] i − µ µ · · · µ i for i = , . . . , M . And therefore, we have the stationary distribution of P M , ε [1]. π [ ε ] M ( i ) = θ θ + ∑ Mj = θ [ ε ] j i = θ [ ε ] i θ + ∑ Mj = θ [ ε ] j ( i = , , . . . , M ) In order to appreciate the shape of the stationary distribution, we look at four cases with Repro-duction Number (R): R = R = . R = R = .
5. For small ε = − , M = R =
1, the distribution is mostly concentrated at state 0, but not only at state 0 as shown in Fig 3a. d i s t r i bu t i on R=1 & Eps=10 -3 ( A ) Distribution concentrates at 0 R=1.5 & Eps=10 -3 ( B ) Bell-shaped distribution around 66 R=2 & Eps=10 -3 ( C ) Bell-shaped distribution around 100 R=2.5 & Eps=10 -3 ( D ) Bell-shaped distribution around 120 F IGURE
3. Distribution ( P M , ε ) : M = α = . β = R α When R greater than 1, the distribution is continuous and well-spread. For R = .
5, the distri-bution has a bell-shape and spreads on the left side around 66 infections in Fig 3b. For R = P M , ε ), we can look at some measures of central tendency and dispersion. InFig 4, four statistical indicators (mean, variance, skewness and kurtosis) are used to summarizethe characteristics of the distribution. As illustrated in Fig 4a, the mean of infected size is afunction of the population size ( M ). In fact, when the population size ( M ) is small ( M < M becomes greater than100. At this stage, the mean continues to increase but at a constant rate (1 − R ). In Fig 4b,the variance is also a function of the population size ( M ) and is unstable when M is not large( M < M becomes large ( M > R ). Population size(M) m ean s / M ( A ) Mean/M converges to 1 − R Population size(M) v a r i an c e / M ( B ) Variance/M converges to R Population size(M) -2-1.5-1-0.500.511.522.53 sk e w ne ss ( C ) Skewness converges to 0 Population size(M) k u r t o s i s ( D ) Kurtosis converges to 3 F IGURE
4. Central and dispersion characteristics of the distribution ( P M , ε ) : R = ε = − and α = . R ). The Skewness is an indicator of lack of symmetry, that is, both left and ¯ x = ∑ Mj = π [ ε ] M ( j ) j σ = ∑ Mj = π [ ε ] M ( j )( j − ¯ x ) Sk = ∑ Mj = π [ ε ] M ( j )( j − ¯ x σ ) right sides of the distribution ( P M , ε ) are unequal with respect to the mean. In Fig 4c, the Skew-ness as a function of M shows that the distribution of the system lacks symmetry when M is notlarge enough ( M < M become large enough, the Skewness converges to 0; and thesymmetric natures of the distribution appears.The Kurtosis is a measure of how heavy-tailed or light-tailed the distribution ( P M , ε ) is relative toa normal distribution. In Fig 4d, the Kurtosis as a function of M shows that the SIS model alter-nates between heavy-tailed and light-tailed before reaching a stable value of 3, when M becomeslarge enough. It is important to point out that the normal distribution has kurtosis equal to 3.Figs 3 and 4 provide evidence that the infected size follows a normal distribution when the pop-ulation size(M) reaches a certain threshold.3. I NFECTED S IZE D ISTRIBUTION : A
NALYTICAL R ESULTS
We revise the continuous-time Markov chain on the state space { , , . . . , M − } with transitionrates. λ k = (cid:26) β (cid:0) M − kM (cid:1) k for k = . . . M − ε for k = ( k → k + ) µ k = α k ( k → k − ) In this revision version, the external factor ε > λ . Here α and β are strictlypositive parameters, and ε is a non-negative parameter. We define θ , θ [ ε ] , . . . , θ [ ε ] M by θ = θ [ ε ] i = λ λ · · · λ i − µ µ · · · µ i for i = , . . . , M . The equilibrium distribution of P M , ε is derived as follows: π [ ε ] M ( i ) = θ θ + ∑ M − j = θ [ ε ]( M − j ) i = θ [ ε ] i θ + ∑ M − j = θ [ ε ]( M − j ) ( i = , , . . . , M ) Lemma 3.1
Assume β > , α > , ε > , M > and R = βα . We have : θ = θ [ ε ] M − k = ε R α ( M − k ) (cid:18) RM (cid:19) M − k M ! k ! for k = , . . . , M − K = ∑ Mj = π [ ε ] M ( j )( j − ¯ x σ ) Proof: θ [ ε ] M − k = λ λ · · · λ ( M − k − ) µ µ · · · µ ( M − k ) = λ µ ( M − k ) M − k − ∏ j = λ j µ j = εα ( M − k ) M − k − ∏ j = [ βα (cid:18) M − jM (cid:19) ]= εα ( M − k ) M − k − ∏ j = R (cid:18) M − jM (cid:19) = ε R α ( M − k ) (cid:18) RM (cid:19) M − k M ! k ! (cid:3) Some properties [11] of Poisson distribution will be stated with proof and the results will beapplied in the next subsection.
Lemma 3.2
Suppose X follows a Poisson distribution with parameter λ and µ ( d ) = E [ X | X ≤ d ] ∀ d ∈ N ∗ Then: µ ( d ) = λ g ( d − ) g ( d ) where g ( d ) = d ∑ i = λ i i ! and lim d → ∞ g ( d − ) g ( d ) = Proof:
Let us define the following function p ( x , λ , d ) = P ( X = x | X ≤ d ) for x = , . . . , dp ( x , λ , d ) = P ( X = x | X ≤ d ) = p ( X = x ) p ( X ≤ d ) = λ x x ! ∑ di = λ i i ! = λ x x ! g ( d ) = g ( d − ) g ( d ) p ( x , λ , d − ) with g ( d − ) g ( d ) = ∑ d − i = λ ii ! ∑ di = λ ii ! = − λ dd ! ∑ di = λ ii ! and lim d → ∞ g ( d − ) g ( d ) = − lim d → ∞ λ dd ! lim d → ∞ ∑ di = λ ii ! = λ dgd λ ( d ) = ∑ dj = j λ j j ! = λ g ( d − ) and the result follows µ ( d ) = E [ X | X ≤ d ] = d ∑ j = j p ( j , λ , d ) = ∑ dj = j λ j j ! g ( d ) = λ dgd λ ( d ) g ( d ) = λ g ( d − ) g ( d ) (cid:3) Corollary 3.3
Assume R > , M > , and X follows a Poisson distribution with parameter MR .Then: E [ XM | X < M ] = R g ( M − ) g ( M − ) with lim M → ∞ g ( M − ) g ( M − ) = Proof:
From lemma 3.2, λ = MR and d = M − E [ XM | X < M ] = E [ XM | X ≤ M − ] = M E [ X | X ≤ M − ] = M µ ( M − ) = R g ( M − ) g ( M − ) and E [ XM | X < M ] = R g ( M − ) g ( M − ) (cid:3) Lemma 3.4
Assume X follows a Poisson distribution with parameter λ and let a > λ .We have: P ( X > a ) ≤ e − λ + a − a log ( a λ ) Proof: M ( θ ) = e λ ( e θ − ) is the moment generating function of the Poisson distribution. M ( θ ) = E ( e θ X ) = ∑ k → ∞ i = e k θ P ( X = k ) > e a θ P ( X > a ) and P ( X > a ) < e λ ( e θ − ) − a θ ∀ θ ∈ R Therefore, P ( X > a ) ≤ inf θ ∈ R { e λ ( e θ − ) − a θ } = e − λ + a − a log ( a λ ) . The function ψ ( θ ) = e λ ( e θ − ) − a θ reaches its minimum at θ ∗ = log ( a λ ) (cid:3) Lemma 3.5
Assume X follows a Poisson distribution with parameter λ = MR .Then P ( X > MR + δ M ) ≤ e φ ( δ R ) R M ∀ δ > where φ () is a function and φ ( δ R ) < Proof:
Let us define φ ( x ) = x − ( + x ) log ( + x ) and it can be shown that : φ ( x ) < ∀ x > x = δ R , we have φ ( δ R ) = δ R − ( + δ R ) log ( + δ R ) < a = MR + δ M and λ = MR , we apply Lemma 3.4 P ( X > MR + δ M ) ≤ e − λ + MR + δ M − ( MR + δ M ) log ( MR + δ M λ ) = e − MR + MR + δ M − ( MR + δ M ) log ( MR + δ MMR ) ≤ e MR ( δ R − ( + δ R ) log ( + δ R )) ≤ e φ ( δ R ) R M (cid:3) Assume R > , M > , X follows a Poisson distribution with parameter λ = MR .Then lim sup M → ∞ E ( MM − X | X < M ) ≤ − R Proof: I ( X < M ) is an indicator function, E [ MM − X I ( X < M )] = E [ MM − X I ( X ≤ M ∗ )] + E [ MM − X I ( M ∗ < X < M )] where M ∗ = MR + δ M and δ > M ∗ < M , which is equivalent to 0 < δ < R − R E [ MM − X I ( X ≤ M ∗ )] ≤ − R − δ P ( X ≤ MR + δ M ) (3.1) E [ MM − X I ( M ∗ < X < M )] ≤ MP ( MR + δ M < X ) (3.2) By applying Lemma 3.5, E [ MM − X I ( M ∗ < X < M )] ≤ M e φ ( δ R ) R M where φ ( δ R ) < E [ M ( M − X ) | X < M ] = E [ MM − X I ( X < M )] P ( X < M ) ≤ P ( X ≤ MR + δ M ) P ( X < M ) − R − δ + P ( X < M ) M e φ ( δ R ) R M ≤ − R − δ + P ( X < M ) M e φ ( δ R ) R M lim M → ∞ P ( X < M ) M e φ ( δ R ) R M = δ → lim sup M → ∞ E [ M ( M − X ) | X < M ] ≤ lim δ → − R − δ for 0 < δ < R − R lim sup M → ∞ E [ M ( M − X ) | X < M ] ≤ − R (cid:3) Lemma 3.7
Assume R > , M > , X follows a Poisson distribution with parameter λ = MR .Then lim inf M → ∞ E ( MM − X | X < M ) ≥ − R Proof:
The function f M ( x ) = MM − x is convex over 0 ≤ x < M . Using the Jensen Inequality property, f M ( E [ X | X < M ]) ≤ E [ f M ( X ) | X < M ] (3.3)From corollary 3.3, E [ X | X < M ] = MR g ( M − ) g ( M − ) From (3.3), we have E [ MM − X | X < M ] ≥ MM − E [ X | X < M ] = MM − MR (cid:16) g ( M − ) g ( M − ) (cid:17) We take the limit lim inf M → ∞ E [ MM − X | X < M ] ≥ lim inf M → ∞ − R (cid:16) g ( M − ) g ( M − ) (cid:17) = − R From lemma 3.6 and lemma 3.7, we have:lim M → ∞ E [ MM − X | X < M ] = − R (cid:3) (3.4) P M , ε .Lemma 3.8 Assume β > , α > , ε > , M > with R = βα > . M ∑ k = θ [ ε ]( M − k ) = + M − ∑ k = θ [ ε ]( M − k ) ∼ C ( M ) RR − ( MR ) As M → ∞ and C ( M ) = M ! ε MR α (cid:0) RM (cid:1) M Proof: M − ∑ k = θ [ ε ] M − k = M − ∑ k = ε R α ( M − k ) (cid:18) RM (cid:19) M − k M ! k ! ( θ [ ε ] M − k from Lemma 3.1) = C ( M ) e ( MR ) M − ∑ k = MM − k k ! (cid:18) MR (cid:19) k e ( − MR ) with C ( M ) = M ! ε MR α (cid:0) RM (cid:1) M = C ( M ) e ( MR ) M − ∑ k = MM − k P [ X = k ] with X ∼ Poisson ( MR )= C ( M ) e ( MR ) E [ MM − X I ( X < M )] ( I ( X < M ) indicator function)From the conditional expectation: E [ M ( M − X ) I ( X < M )] = P [ X < M ] E [ MM − X | X < M ] (3.5)Previously, we show that M − ∑ k = θ [ ε ]( M − k ) = C ( M ) e ( MR ) E [ MM − X I ( X < M )] We deduce the following relation1 + ∑ M − k = θ [ ε ]( M − k ) C ( M ) e ( MR )( RR − ) = ( − R ) E [ MM − X I ( X < M )] + C ( M ) e ( MR )( RR − ) (3.6)From the results (3.5) and (3.4), we havelim M → ∞ ( − R ) E [ MM − X I ( X < M )] = M → ∞ C ( M ) e ( MR )( RR − ) = ∀ R > M → ∞ + ∑ M − k = θ [ ε ]( M − k ) C ( M ) e ( MR )( RR − ) = (cid:3) Theorem 3.9
Assume β > , α > , ε > , M > with R = βα > . The infected size has thefollowing equilibrium distribution. π [ ε ] M ( k ) ∼ π ( k ) As M → ∞ with π ( k ) = R − R M ( M − k ) ! k (cid:18) MR (cid:19) M − k e ( − MR ) ( k = , , . . . , M ) Proof:
For k = , , . . . , M π [ ε ] M ( k ) = θ [ ε ] k + ∑ M − j = θ [ ε ]( M − j ) = + ∑ M − j = θ [ ε ]( M − j ) C ( M )( RR − ) e ( MR ) θ [ ε ] k C ( M )( RR − ) e ( MR )= + ∑ M − j = θ [ ε ]( M − j ) C ( M )( RR − ) e ( MR ) R − R M ( M − k ) ! k (cid:18) MR (cid:19) M − k e − ( MR )= + ∑ M − j = θ [ ε ]( M − j ) C ( M )( RR − ) e ( MR ) π ( k ) We have the following quotient: π [ ε ] M ( k ) π ( k ) = + ∑ M − j = θ [ ε ]( M − j ) C ( M )( RR − ) e ( MR ) The result follows from lemma 3.8 (cid:3)
Theorem 3.10
Assume β > , α > , M >> with R = βα > . The infected size followsasymptotically a normal distribution with mean µ = ( − R ) M and variance σ = MR . Proof:
For k fixed, We know that ( M − k ) ! ∼ √ π e − ( M − k ) ( M − k ) ( M − k )+ for M → ∞ We have the following equivalence when M → ∞ π ( k ) = R − R M ( M − k ) ! k (cid:18) MR (cid:19) M − k e − ( MR ) ∼ R − R √ π Mk ( M − k ) (cid:18) M ( M − k ) R (cid:19) M − k e − ( MR ) +( M − k ) = ψ ( k ) And we have lim M → ∞ ψ ( k ) π ( k ) = x = k = ( − R ) M ( + δ ) . By applying the second order Taylor’s expansion techniques,we have the following expression [5, 9] As M → ∞ log ( ψ ( k )) = log ( R − R √ π ) + log ( Mk ) −
12 log ( M − k ) + ( M − k ) log ( M ( M − k ) R ) − MR + ( M − k )= −
12 log ( π MR ) − ( R − ) R M δ + [ ( R − ) δ + (( R − ) + ) δ + O ( δ )]+ [ − ( R − ) R M δ − ( − ( R − ) δ R M + ) O ( δ )] We have a simple expression ψ ( k ) = (cid:113) π MR e (cid:26) − ( R − ) R M δ (cid:27) e (cid:110) ( R − ) δ + (( R − ) + ) δ + O ( δ ) (cid:111) e (cid:26) − ( R − ) R M δ − ( − ( R − ) δ R M + ) O ( δ ) (cid:27) (3.9)Now, we can prove the Local Central Limit Theorem. For k integer, to get a convenient limit, wewill choose m M , σ M and k as a function of M ( k = k ( M ) ) that satisfy the following properties:lim M → ∞ k − m M σ M = s for some real number s . andPr ( X M = k ( M )) ∼ √ π σ M e − s / as M → ∞ In our case (theorem 3.10), we will choose m M = (cid:0) − R (cid:1) M and σ M = (cid:112) M / R . Fix a real number s , we choose the sequence k ( M ) such aslim M → ∞ k ( M ) − m M σ M = s (3.10) k ( M ) is provided by the previous Taylor’s expansion condition k ( M ) = (cid:0) − R (cid:1) M ( + δ ) with Min ( R − , ) > δ .The limit (3.10) holds if δ = δ ( M ) = s √ R ( R − ) √ M . In addition, we have the following propertylim M → ∞ δ = M → ∞ M δ = lim M → ∞ s R ( R − ) √ M = M → ∞ , we have: π ( k ) ∼ √ πσ M e { − s } (cid:3)
4. C
ONCLUSION
The dynamics of the infected size of the SIS epidemic model and its distribution at the equilib-rium were our main interests in this article. The stability and the equilibrium convergence ofthe resulting infected size was shown through the sample path simulations. In addition to thedynamics, the stochastic simulations show that when the reproduction number ( R ) increases for >
1, the infected size sample path increases as a whole. These results are not different from thedeterministic findings. The distribution of infected size is based on the Birth and Death Markovprocess. It results from the analysis that the distribution of the infected size is a symmetric-bellshaped curve, the mean and the variance are functions of the reproduction number ( R ). An in-depth analysis of the distribution shows that asymptotic distribution of the infected size follows aNormal distribution with mean ( − R ) M and variance MR . ACKNOWLEDGEMENT
I would like to express my special thanks to Prof. Neal Madras for providing advice and feedbackon this article. R EFERENCES [1] Samuel Karlin and Howard M Taylor.
A First Course in Stochastic Processes , volume 1. New York : AcademicPress, 1975.[2] Ingemar N˚asell. On the quasi-stationary distribution of the stochastic logistic epidemic.
Mathematical Bio-sciences , 156(1-2):21–40, 1999.[3] Ingemar N˚asell. The quasi-stationary distribution of the closed endemic SIS model.
Advances in Applied Proba-bility , 28(3):895–932, 1996.[4] Otso Ovaskainen. The quasistationary distribution of the stochastic logistic model.
Journal of Applied Probabil-ity , 38(4):898–907, 2001.[5] Aubain Hilaire Nzokem.
Stochastic and Renewal Methods Applied to Epidemic Models . PhD thesis, York Uni-versity , YorkSpace institutional repository, 2020.[6] Linda JS Allen. An introduction to stochastic epidemic models. In
Mathematical Epidemiology , pages 81–130.Springer, 2008.[7] Damian Clancy and Sang Taphou Mendy. Approximating the quasi-stationary distribution of the SIS model forendemic infection.
Methodology and Computing in Applied Probability , 13(3):603–618, 2011.[8] John C Wierman and David J Marchette. Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction.
Computational Statistics & Data Analysis , 45(1):3–23, 2004.[9] A. Nzokem and N. Madras. Epidemic dynamics and adaptive vaccination strategy: Renewal equation approach.
Bulletin of Mathematical Biology , 82 9:122, 2020.[10] A. Nzokem and N. Madras. Age-structured epidemic with adaptive vaccination strategy: Scalar-renewal equationapproach. In
Recent Developments in Mathematical, Statistical and Computational Sciences . Springer, 2021.[11] Jeremy. J. Collis-Bird.
The Poisson Distribution . PhD thesis, Doctoral dissertation, McGill University Libraries,1963.. PhD thesis, Doctoral dissertation, McGill University Libraries,1963.