Discrete-Time Stochastic Epidemic Models and Their Statistical Inference

There is a wide range of models, both stochastic and deterministic, for the spread of an epidemic. Usually, when the population is constituted of a large number of individuals, a deterministic model is useful as a first approximation, and random variations can be neglected. As an alternative, a stochastic model could be more appropriate for describing the epidemic, but it is less tractable and its mathematical analysis is usually possible only when the population size is very small. However, most populations are not large enough to neglect the effect of statistical fluctuations, nor are they small enough to avoid cumbersome mathematical calculations in the stochastic model. In these cases, it uses to be convenient to take into account both types of models and their relationship. The interplay between ordinary differential equations and Markovian counting processes has been widely investigated in the literature. Major references on this subject can be found in [11, 20–22]. Concerning the deterministic epidemic models, those using ordinary differential equations in their formulation have received special attention and a great number of epidemics is modeled by means of Markovian counting processes. For example, some epidemic models known as SIR, SI, SIS, and others derived from these ones, use differential equations andMarkovian counting processes in their formulations. Furthermore, stochastic models based onMarkovian counting processes and differential equations are mainly used to carry out the statistical analysis of the model parameters. The Mathematical Theory of Infectious Diseases by Bailey [2] represents a classical reference containing a presentation and analysis of these models. However a more recent book by Andersson and Britton [1] entitled Stochastic Epidemic Models and their Statistical Analysis is a more appropriate reference according to the point of view of this chapter. The spread of these epidemics is developed in a closed population, which is divided into three individual compartments, i.e. susceptible, infective and removed cases; different types of transitions can occur among these three groups of individuals. These models include the stochastic and deterministic versions of the Kermack and McKendrick model [19] and the SIS epidemic model, among others. Moreover, a number of variations of these models has been widely studied. Modeling of epidemics by continuous-time Markov chains has a long history; thus, it seems pertinent to cite the works by [4–6, 18, 24, 28].

Also, an epidemic can be modeled by means of discrete-time. This is the case of the classical Reed-Frost model, which is a Markovian discrete-time SIR epidemic model. However, this modeling has two differences with the corresponding one based on counting processes. First, its latent period is assumed constant and equal to the time unit. Secondly, there are no deterministic counterpart based on differential equations as it is the case of an epidemic modeled by means of a Markovian counting process. Another type of population modeling, which is applied to metapopulations, has been introduced by some authors such [9,10,27] and other references therein. These researchers derive an approximation that preserves the discrete time structure and reduces the complexity of the models. Probably, these results could be applied to epidemic models and asymptotic inference on the parameters of these models, could be carried out.
This chapter is a compendium of two works by the author, whose references are [12,13]. A wide class of discrete-time stochastic epidemic models is introduced and analyzed from a statistical point of view. Just as some models based on ordinary differential equations involve a natural alternative through Markovian counting processes, this class includes a counterpart based on differential equations. Unlike those epidemic models where transitions occur at random times, our proposal involves the advantage of being suitable for epidemics that cannot be observed for a long period of time, as in some epidemics where observations are done at previously determined times. This is the main reason for preferentially considering these kind of stochastic models instead of those based on continuous time. It is expected the smaller the periods of time between transitions and the bigger the population, the more similar the stochastic and deterministic models would become. Indeed, one of the main aims of this paper is to prove such a similarity. As a second aim, we are highly interested in carry out statistical analysis on the parameters of the modeling. For this purpose, martingale estimators for the parameters involve in the modeling are derived and their asymptotic normality is proved.
Since the results stated here do not assume a distribution for the process modeling the epidemic, it is not possible to derive a likelihood ratio and hence maximum likelihood estimators cannot be obtained. Even, in many cases when the process representing the model is Markovian, the maximum likelihood estimators cannot be obtained in a closed form, which makes difficult to carry out statistical inference on the parameter of the model. As pointed out in [7], likelihood functions corresponding to epidemic data are often very complicated. In these cases, parameter estimation based on martingale estimators use to be an appropriate alternative to work out this difficulty. This method arises as a natural way of estimation when no distribution in the model is assumed or, when the maximum likelihood estimators cannot be obtained in a closed form.
This chapter is organized as follows. The general form of the model and two preliminary lemmas are introduced in Section 2. Section 3 contains brief definitions of some typical models included in the biomathematical literature. The deterministic counterpart of the general model along with its relationships with it is presented in Section 4. Indeed, the convergence of the stochastic model to the deterministic one and the asymptotic behavior of the corresponding fluctuations are proved. Moreover, in Section 4 a version of the SIS epidemic model is presented and numerical simulations are carried out. The parameter estimators are defined in Section 5, and their asymptotic normality is proved. The General Epidemic Model along with the statistical analysis on the parameters is stated in Section 6.
Results of Section 5 are applied here to some hypothesis tests. Section 7 is devoted to some numerical simulations. Finally, Section 8 contains some conclusions.

Discrete-time modeling
The model which we introduce here is defined as follows: a community of individuals divided into three different compartments is considered, namely, susceptible, infective and removed individuals.
Suppose the size population is n, and for each t ≥ 0, S n (t), I n (t) and R n (t) represent, respectively, the number of susceptible, infective and removed individuals at time t. Since it is assumed the population size is constant, then for each t ≥ 0, should be S n (t) + I n (t) + R n (t) = n. These processes are observed at discrete-time instants which are defined by the sequence {t n k } k∈N , where for each k ∈ N, t n k = kΔ/n, (Δ > 0), i.e. each time subinterval has length Δ/n. Let (Ω, F , IP) be a probability space. In the sequel, all stochastic processes and random variables are defined on this probability space and for a stochastic process Z, we denote Let M denote the transpose of a matrix M and X n = (S n , I n , R n ) . Transitions of individuals among the three compartments are determined by m increasing stochastic processes Z n 1 , . . . , Z n m , and the number of individuals in each compartment is obtained, for each t ≥ 0, by means of where Z n (t) = (Z n 1 (t), . . . , Z n m (t)) and A is a 3 × m-incidence matrix. It is assumed Z n 1 , . . . , Z n m take values in the set of non-negative integer numbers, have right-continuous trajectories and start at zero, i.e. Z n 1 (0) = · · · = Z n m (0) = 0. Let F n k be the σ-field σ(Z n (t n 1 ), . . . , Z n (t n k )) generated by Z n (t n 1 ), . . . , Z n (t n k ). The stochastic processes Z n 1 , . . . , Z n m increase according to m density dependent transition rates, which are defined by means of m non-negative functions a 1 , . . . , a m , respectively. The domain of these functions is and open set of R 3 containing the 3-simplex E = {(u, v, w) ∈ [0, 1] 3 : u + v + w = 1} and it is assumed the following condition holds: (C) For each k ∈ N, ΔZ n 1 (t n k ), . . . , ΔZ n m (t n k ) are F n k−1 −conditionally independent and satisfy where χ n (t) = (σ n (t), ι n (t), ρ n (t)) , σ n (t) = S n (t)/n, ι n (t) = I n (t)/n and ρ n (t) = R n (t)/n.
A wide variety of stochastic models for epidemics satisfy condition (C). It is important to point out this condition does not determine the law or distribution of χ n , i.e. there could be two or more processes satisfying this condition, though they have different transition probabilities. This fact enables this condition to be applied to a wide class of models, since in order to verify condition (C), the distribution of the process need not be known. Actually, a stochastic process satisfying condition (C) need not be Markovian. Nevertheless, some Markov chains, having density dependent transition rates, satisfy condition (C) and hence they may be included in our setting.

Two preliminary lemmas
In what follows, [a] stands for the integer part of a real number a and for each i = 1, . . . , m, we denote L n i (t) = ∑ [nt] k=1 ξ n k (i)Δt n and L n (t) = (L n 1 (t), . . . , L n m (t)) , (t ≥ 0), where ξ n k (i) = ΔZ n i (t n k ) − a i (χ n (t n k−1 )) and Δt n = 1/n. From Condition (C), it is obtained that, by defining Through this chapter, for each i = 1, . . . , m, v n i (t) and L n i stand for the random variable v n and the predictable quadratic variation of L n i , respectively. Lemma 2.1. For each t ≥ 0, L n 1 (t), . . . , L n m (t) are G n t− -conditionally independent random variables and, the predictable quadratic variation matrix of L n is given by Proof. For t ≥ 0, the G n t− -conditional independence of L n 1 (t), . . . , L n m (t) follows from Assumption (C) and it is clear that for each i = 1, . . . , m, the predictable quadratic variation of L n i is given by L n , which concludes the proof.
In the sequel, for each d ∈ N, · stands for the Euclidean vector norm in R d .

Lemma 2.2.
Let T > 0 and suppose for each i = 1, . . . , m, { 1 n v n i (T)} n∈N converges in probability to zero, as n goes to ∞. Then, {sup 0≤t≤T L n (t) } n∈N converges in probability to zero.
For each T > 0 and each i = 1, . . . , m, { 1 n v n i (T)} n∈N converges in probability to zero, as n goes to ∞.
Proof. From Theorem 1 in [26], for any , η > 0 we have and hence, Lemma 2.1 implies By assumption this lemma follows.

Some epidemic models
Most of the typical models involved in the biomathematical literature have a version which belongs to the model class defined in this approach. Some of them are included below.

The general epidemic model
A deterministic version of this model based on differential equations, was introduced by [19], while the article by [18] was a pioneer in the stochastic version based on counting processes. This model has received the most attention in the literature, and its analysis can be found in [2], [1] and [23]. Two type of transitions are possible for any individual, from susceptible to infected and from infected to removed individuals. Thus under the perspective of this work, the modeling of this epidemic must satisfy (1) with Transitions from susceptible to infected and from infected to removed cases are described by Z n 1 and Z n 2 , respectively, and the functions defining their transition rates are given by a 1 (u, v, w) = βuv and a 2 (u, v, w) = γv, where β and γ are two parameters denoting the infection and removal rates. This model is also known as SIR model.

The SIRS Model
This is a slight modification of the preceding model. Besides the transitions determined by Z n 1 and Z n 2 in the SIR model, a transition from removed to susceptible case is allowed and determined by an increasing stochastic process Z n 3 , where its transition rate is defined by a 3 (u, v, w) = δw. In this case, some of the removed cases may become susceptible to be infected again. The incidence matrix defining this model is given by

The SIS Model
One of the simplest epidemic models is the SIS model, which uses to be suitable for infections resulting from bacteria such as gonorrhea, malaria, etc. In this case, only transitions from susceptible to infected individuals are allowed, as well as transitions from infected to susceptible individuals. According to the approach of this study, two transitions are allowed, and determined by Z n 1 and Z n 2 with transition rates defined by a 1 (u, v, w) = βuv and a 2 (u, v, w) = γv, respectively. In this case, the incidence matrix is Notice the compartment corresponding to removed cases is considered having no individuals.

The modified SIR model
This is a modification of the general epidemic model and aims to AIDS modeling. As in the general epidemic model, two transitions Z n 1 and Z n 2 define the model with transition rates given by a 1 (u, v, w) = βuv/(u + v) and a 2 (u, v, w) = γv, respectively. As before, β and γ correspond to the model parameters. Some references concerning the deterministic version of this model are, for instance, [16,17], while the stochastic version based on Markovian counting processes was introduced by [4]. The incidence matrix is defined as in the SIR model, i.e.
In general, as much in the SIR as the modified model, the transition rate due to infection is proportional to the susceptible density and the fraction of infected individuals with respect to individuals in circulation. Consequently, in the SIR model this fraction is v/(u + v + w) = v, while in an epidemic where to be removed is equivalent to be dead or out of circulation, this fraction is v/(u + v). In modeling AIDS, removed cases are presumed to be so ill with AIDS that they no longer take part in transmission.

The deterministic counterpart
In this section, we examine the relationship between the model we are introducing here and an associated ordinary differential equation, which we call its deterministic counterpart.
In order to obtain existence and uniqueness of the solution to (2), it will be assumed the following usual Lipchitz condition holds: for all x, y ∈ E, where K is a positive constant.

Comparison between the stochastic and deterministic models
The theorem below stated the consistency of the stochastic model with respect to the deterministic one and we will use it to study the asymptotic behavior of the estimators for the parameters of the model.  Then, {χ n } n∈N converges in probability uniformly over compact subsets of R + to χ, i.e. for each T > 0, {sup 0≤t≤T χ n (t) − χ(t) } n∈N converges in probability to zero, as n goes to ∞. and Hence where From (3) and (L), for each t ≥ 0, we have where α n = g n (0) + sup 0≤t≤T AL n (t) + O(1/n). Since {g n (0)} n∈N converges in probability to zero and sup 0≤t≤T AL n (t) ≤ |||A||| sup 0≤t≤T L n (t) , Lemma 2.1 and Gronwall's inequality imply {g n (T)} n∈N converges in probability to zero. This completes the proof.
The following result aims to the problem of finding approximate confident bands for the solution to (2). Before stating it, for each In the sequel, {Y n } n∈N is the sequence defined as Y n = √ n(χ n − χ).

Theorem 4.2.
Let {Y n } n∈N be the sequence defined as Y n = √ n(χ n − χ) and suppose the following three conditions hold:

x) exists and it is continuous at x in an open
set containing E.
in probability to zero.
Then, {Y n } n∈N converges in law to the solution Y satisfying the following stochastic differential equation: where M is a continuous martingale with predictable quadratic variation given by the matrix Proof. Let M n = √ nAL n . By making use of Corollary 12, Chapter II in [29], (4.2.2) and (4.2.4) imply that { √ nL n } n∈N converges in law to a continuous martingale Q with predictable quadratic variation Q given by Q (t) = Diag(v 1 (t), . . . , v m (t)). Consequently, {M n } n∈N converges in law to a continuous martingale M = AQ with predictable quadratic variation M given, for each t ≥ 0, by where θ n (s) is between χ n (s) and χ(s), and U n (t) = √ n n (t) = (nt−[nt]) √ n F(χ n (t)).
Consequently, from a standard application of the Gronwall inequality, we obtain Since {Y n (0)} n∈N and {sup 0≤t≤1 M n (t) } n∈N are sequences converging in distribution, Theorem 6.2 in [8] implies lim and, from (18), for each > 0 and any t ≥ 0, In order to prove the convergence in law of {Y n } n∈N , fix T > 0 and let us define the modulus of continuity where the infimum extends over the finite sets {t i } of points satisfying Here, D([0, T], R 3 ) stands for the Skorohod space of all functions from [0, T] into R 3 , which are right continuous and left-hand limited.
From (6) we have  (9) and (10), for each > 0, we have Conditions (8) and (11) Moreover, since {Y n (0)} n∈N converges in distribution to η and Y(0) equals η in distribution, we have Y is a solution to (5). Finally, uniqueness of solutions to (5) implies {Y n } n∈N converges in distribution to Y, which concludes the proof.

Remark 4.2. By Itô's rule, the unique solution to (5) is given by
where Ψ is the unique solution to the matrix differential equation The stochastic process {Y(t); t ≥ 0} allows us to give confidence bands for the deterministic model defined by the solution χ to (2). In Section 6 such band is constructed for the SIS epidemic model, where some simulations are carried out.

Numerical simulations for the SIS epidemic model
In this section, our attention is focused on the SIS epidemic model. As explained in Subsection 3.3, it is assumed that only susceptible and infective individuals are in a closed homogeneously mixing population. Let σ(t) and ι(t) be the densities of susceptible and infective individuals, respectively. The deterministic model is defined by the following system of ordinary differential equations: Since for each t ≥ 0, σ(t) + ι(t) = 1, this model is completely determined by the ordinary differential equation: which, given ι(0) = ι 0 ∈]0, 1[, has the unique solution The relative removal-rate, see for instance [2] and [3], is defined as τ = γ/β, where γ and β represent the removal and infection rates, respectively. We note that is called the endemic level of the process.
Let σ n (t) and ι n (t) be the densities of susceptible and infective individuals in the stochastic version of the SIS epidemic model. According to our setting, σ n and ι n are defined as and IE(ΔZ n 2 (t n k )) = γι(t n k−1 ). Let us assume nι n (0) = [nι 0 ]. From Theorem 4.1, {ι n } n∈N converges uniformly in probability to ι, over compact subsets of R + , whenever ι(0) = ι 0 . It is worth noting 1 − τ is an asymptotically stable equilibrium state for the deterministic model and not for the stochastic model. However, it is expected ι n (t) is close to this value due to Theorem 4.1, for a large enough n. In Figure 1, the deterministic and stochastic models, for β = 3, γ = 1, ι 0 = (1 − τ)/2 and n = 1, 000, are compared. Let y n = √ n(ι n − ι). It follows from Theorem 4.2 that {y n } n∈N converges in law to y, the solution to the following Langevin equation: where W is a one dimensional standard Brownian motion. From Remark 4.2, the solution to this equation is where Hence, y(t) has a normal distribution with mean zero and variance Var(y(t)) = t 0 b(u) 2 e 2 t u a(s) ds du.
In Figure 2, a simulation of ι n is carried out for β = 3, γ = 1 and n = 10, and we note in this case random fluctuations are important to be neglected.
1 − Φ(w α/2 ) = α/2 and Φ is the cumulative distribution function of a standard normal random variable. The random bounds given by (14) allow to construct (nonuniform) confidence bands for the solution ι given by (12). In turn, by defining u ± α (t) = ι(t) ± w α/2 Var(y(t))/n, we have ι n (t) ∈ [u − α (t), u + α (t)] with an approximate probability 1 − α for large values of n. In Figure 3, another simulation of ι n is carried out for β = 3 and γ = 1. However, in order to appreciate the convergence of ι n to the equilibrium 1 − τ for the deterministic model, a population of size n = 5, 000 is now considered. The bounds u − α and u + α are pictured with dash lines for α = .05, and hence, in this case w α/2 = 1.96.

Estimators and their asymptotic behavior
The main results of this article are stated in this section; however their proofs are deferred to the last section.

Preliminaries
Let us suppose for each i = 1, . . . , m, a i splits as a i = β i b i , where β i is a parameter of the model and b i is a non-negative continuous function defined on an open set containing E.
Since for each i = 1, . . . , m, L n i is a martingale and by observing the epidemic through the time interval [0, T], (T > 0), we have is a martingale estimator of β i .

Asymptotic normality of estimators
In this subsection we state two asymptotic normality results for the martingale estimators for β.   Then, for each T > 0, { √ n( β n (T) − β)} n∈N converges in distribution, as n goes to ∞, to a normal random vector N(0, Σ) having mean zero and variance-covariance matrix Σ = {σ ij (T)} 1≤i,j≤m satisfying . . . , m). From Lemma 2.1, for each t ≥ 0, Q n 1 (t), . . . , Q n m (t) are G n t− -conditionally independent random variables and for each i = 1, . . . , m, the predictable quadratic variation of the martingale Q n i is given by Q n i = v n i . Condition (5.1.3) indicates for each i = 1, . . . , m, { Q n i (t)} n∈N converges in probability to v i (t). This fact and Condition (5.1.2) enable to conclude that the hypotheses of Corollary 12 in Chapter II in (Rebolledo, 1979) hold. Consequently, {Q n (T)} n∈N converges in distribution to a normal random vector Q(T) = (Q 1 (T), . . . , Q m (T)) with mean zero and satisfying is the diagonal random matrix with entries d n 1 , . . . , d n m in its diagonal given by k=1 b i (χ n (t n k−1 ))Δt n } n∈N converges in probability to T 0 b i (χ(t)) dt and Slutzky's theorem (see Theorem 5.1.6 in [25], for instance) enables us to conclude {D n Q n (T)} n∈N converges in distribution to a normal random vector with mean zero and variance-covariance matrix Σ = {σ ij } 1≤i,j≤m , which satisfies This concludes the proof of Theorem 5.1.
converges in probability to zero, as n goes to ∞. In particular, this condition is satisfied when the double sequence {ξ n k (i) 2 ; 0 ≤ k ≤ n, n ∈ N} is uniformly integrable.

The general epidemic model
In order to carry out asymptotical inference for a great number of epidemic models, results stated in Section 5 can be applied. In this subsection, statistical inference for the General Epidemic Model is developed.

The model
As mentioned in Subsection 3.1, this model contains two parameters β and γ denoting the infection and removal rate, respectively, and it is defined by two increasing integer valued processes, which we denote by A n and B n , so that From (15) and the definition of this model given in Subsection 3.1, the martingale estimators of β and γ in [0, T], are respectively given by In order to verify β n (T) and γ n (T) are martingale estimators, Condition (C) has to be hold. By taking into account some heuristic considerations, which are related to the infection spreading, the distribution of the process can be determined. This fact is sufficient, although not necessary as mentioned previously, to obtain Condition (C). Let β denote the average of effective contacts per time unit between an infected person and any other individual in the population. This constant is known as the contact rate, cf. [15]. Hence, β/n is the average number of effective contacts per time unit per capita of an infected, and it is natural to assume the probability of a susceptible individual to become infective in a time interval ]t n k−1 , t n k ] is (β/n)ι n (t n k−1 ). On the other hand, since the total number of adequate contacts in t n k−1 that may produce an infection in t n k equals the susceptible number S n (t n k−1 ), it is assumed that ΔA n (t n k ) and ΔB n (t n k ) conditionally on F n k−1 have independent Binomial distribution with parameters (S n (t n k−1 ), (β/n)ι n (t n k−1 )) and (I n (t n k−1 ), γΔt n ), respectively. Note that it satisfies IE(ΔA n (t n k )|F n k−1 ) = a 1 (χ n (t n k−1 )) and IE(ΔB n (t n k )|F n k−1 ) = a 2 (χ n (t n k−1 )), where a 1 (u, v, w) = βuv and a 2 (u, v, w) = γv. Hence, conditions (C) and (L) hold and, β n (T) and γ n (T) are martingales estimators. Also, in the next subsection, we see that this approach satisfies the hypotheses of Theorem 5.1 and hence, asymptotic normality of β n (T) and γ n (T) is obtained.
Regarding the initial state of the epidemic, two natural assumptions can be made and both of them satisfy Condition (5.1.1) in Theorem 5.1. The first consists in assuming {χ n (0)} n∈N is a deterministic sequence converging to χ(0) in E. This assumption is quite reasonable whenever a good knowledge about the initial numbers of susceptible, infected and removed individuals is involved. For instance, this assumption can be done when the population is small and the proportion of individuals belonging to each compartment can be observed and calculated at time zero. The second possible assumption consists in assuming (S n (0), I n (0), R n (0)) has multinomial distribution with parameters n, p 1 , p 2 and 1 − p 1 − p 2 , (0 < p 1 + p 2 < 1), i.e. for each (s, i) ∈ {0, . . . , n} × {0, . . . , n} such that s + i ≤ n, This second assumption will be held from now on, and although the parameters of this distribution can be estimated by taking a sample at time zero, in the sequel it will be assumed these parameters are known.

Asymptotic normality
As a consequence of Theorem 5.1, the following proposition is stated, which shows the parameter estimators for the SIR epidemic model are asymptotically normal. In this subsection all notations and facts given on the preceding subsection will be maintained.

)} n∈N converges in distribution to a normal random bivariate vector with mean zero and variance-covariance matrix
Proof. From the Kolmogorov Law of Large Numbers, {χ n (0)} n∈N converges almost sure to χ 0 = (p 1 , p 2 , 1 − p 1 − p 2 ) and hence, Condition (5.1.1) holds.
Let ξ n k (1) = ΔA n (t n k ) − βσ n (t n k−1 )ι n (t n k−1 ) and ξ n k (2) = ΔB n (t n k ) − γι n (t n k−1 ). In order to verify Condition (5.1.2), it suffices to prove for each i = 1, . . . , m, the double sequence {ξ n k (i) 2 ; 0 ≤ k ≤ n, n ∈ N} is uniformly integrable. For this purpose, we note that, for a random variable X with Binomial distribution, the following inequality holds: Consequently, according to our approach, for each i = 1, 2, we have Hence, {ξ n k (i) 2 } n,k∈N is uniformly integrable and Condition (5.1.2) holds. Let v n Thus, as a consequence of Theorem 5.1, we have that { √ n( γ n (T) − γ)} n∈N converges in distribution to a normal random bivariate vector having mean zero and variance-covariance matrix From (6) and this concludes the proof.
Since Proposition 6.1 involves a limit distribution depending on the solution of (17), this proposition may be of limited utility. However, Proposition 4.1 and Slutsky's theorem have, as a consequence, the following corollary, which is an alternative to this possible difficulty.
Then, {Q n (T)} n∈N converges in distribution to a normal random bivariate vector with mean zero and covariance matrix I = β 2 0 0 γ 2 .

Hypothesis test for the infection rate
In this section, we are interested in proving whether the parameter β of this SIR epidemic model belongs to a subset of the parametric set. This analysis, which is known as hypothesis test, will intend to define whether the infection rate is bigger than a fixed value β 0 > 0, i.e. the null hypothesis is stated as H 0 : β = β 0 against the one-side alternative H 1 : β > β 0 . Since under H 0 , { β n (T)} n∈N converges in probability to β 0 , H 0 shall be rejected in favor of H 1 when β n (T) is too large, i.e. when β n (T) > u n , being u n determined by IP( β n (T) > u n |H 0 ) ≤ α. Here, α is the preassigned level of significance which controls the probability of falsely rejecting H 0 when H 0 is true.
By means of Corollary 6.1, it is possible to obtain an approximate value u n when n is large. For this purpose, IP( β n (T) > u n |H 0 ) ≤ α is replaced by the following weaker requirement: From the Proposition 6.1, under H 0 , { √ n(σ(0) − σ(T)) 1/2 ( β n (T) − β 0 )/β 0 } n∈N converges in distribution to a random variable having mean zero and variance one. Therefore, with an asymptotic level of significance α, Here, Φ is the cumulative distribution function of a standard normal variable. Hence, u n = β 0 + β 0 t α / n(σ(0) − σ(T)) + o(1/n). In particular, we can choose u n = β 0 + β 0 t α / n(σ(0) − σ(T)) and therefore, with an asymptotic level of significance α, a critical region for the test is Under H 0 , σ(T) is known but it can not be explicitly obtained due to the fact that (6) does not admit a closed-form solution. However, we can take advantage of the fact that {σ n (T)} n∈N converges in probability to σ(T), and by Slutzky's theorem, it can be obtained that is a critical region for the test with an asymptotic level of significance α.
Let Δ > 0 and let us consider the alternative hypothesis In this case, under H 1 the power of the test π n (β) = IP(R(n)|H 1 ), where R(n) = R 1 (n) or ).

Relative removal-rate and basic reproduction number
As pointed out in Subsection 4.2, a fundamental concept resulting from the mathematical theory of the general deterministic epidemic model is the relative removal-rate, see for instance [2] and [3]. This number is defined as τ = γ/β, where γ and β represent the removal and infection rates, respectively, and it plays a crucial part in determining the probable occurrence of an epidemic outbreak. Actually, due to the fact that in the deterministic version of this model, we have (see (17)) by assuming ι(0) > 0, one has the derivative of ι at zero is bigger than zero, if and only if, τ < σ(0). Consequently, unless the initial density of susceptible individuals is bigger than the relative removal-rate, no epidemics can start. It is worth pointing out that this parameter is connected with the basic reproduction number, which is defined as the number of cases generated by one infective over the period of infectivity when that infective was introduced into a large population of susceptible individuals. See for instance, [16], or, [14]. Actually, it becomes R 0 = σ(0)/τ. Notice σ(0) ≈ 1 when a large population of susceptible individuals is considered. Hence, an epidemic can start with a positive probability, if R 0 > 1, and it will die out quickly if R 0 ≤ 1. The knowledge of the basic reproduction number allows the establishment of vaccination policies when necessary, in order to reduce the number of susceptible individuals in a population to such a level that R 0 is brought below the unity threshold. Moreover, it is useful to carry out the following hypothesis test: A natural critical region for the test is { R n 0 (T) > u n }, where R n 0 (T) = σ(0) β n (T)/ γ n (T) and u n is a constant which should be determined by IP( R n 0 (T) > u n |H 0 ) ≤ α, being α a preassigned level of significance controlling the probability of falsely rejecting H 0 when H 0 is true. The following proposition allows us to carry out the mentioned hypothesis test.
From Propositions 5.1 and 6.1, {E n } n∈N converges in probability to zero. Hence, it follows from Proposition 6.1 that U n (T) is asymptotically normal with mean Δ and variance v 2 0 (T). Therefore, this proof is complete.
The constant u n can be chosen as In particular, we can choose u n = 1 + t α v 0 (T)/ √ n, and therefore, with an asymptotic level of significance α, a critical region for the test is { U n (T)/v 0 (T) > t α }. Note also that, under H Δ , the power of the test π n (Δ) = IP( U n (T)/v 0 (T) > t α |H Δ ) converges to π(Δ) = Φ(Δ/v 0 (T) − t α ). Remark 6.1. An application of the preceding proposition is to calculate the approximate power of the test, with a level of significance α, relative to We interpret Δ in H Δ as √ n(r − 1) and approximate the power of the test by means of π = IP(N > rv 0 (T) and variance 1. Consequently, the power of the test can be approximated by

Numerical simulations for the general epidemic model
In order to carry out asymptotical inference for a great number of epidemic models, results stated in Section 5 can be applied. In this subsection, statistical inference for the General Epidemic Model is developed. In this section we maintain notations used in Section 6.

Validating the population size
In this subsection, some numerical simulations are carried out in order to validate the appropriate population size under which the results contained in this work are applicable. Let According to Proposition 6.1, the square Euclidean norm of X n (T), namely X n (T) 2 , has asymptotically χ 2 (2) distribution. The positive part of the real straight line is partitioned in m subintervals, which are determined by 0 = t 0 < t 1 < · · · < t m−1 < ∞. Let F denote the χ 2 (2) distribution function. For β = 2, γ = 1 and T = 10, X n (T) 2 is simulated 1, 000 times with m = 10, where t 1 , . . . , t 9 have been chosen in such a way that F(t i ) = i/10, i.e., t 1 = 0.21, t 2 = 0.45, t 3 = 0.71, t 4 = 1.02, t 5 = 1.39, t 6 = 1.83, t 7 = 2.41, t 8 = 3.22 and t 9 = 4.61. In order to F(t 10 ) = 1, t 10 is defined as ∞. For different population sizes (n = 20, 50, 100, 300, 500, 1000), percentages obtained into the corresponding subintervals are presented in Table 1. From Proposition 6.1, better approximations correspond to percentages close to 10%. These closeness are measured by means of the standard deviation (SD) of the observations corresponding to the different population sizes, which are indicated at the last column of Table 1 below.  Table 1. Percentages of observations of X n (T) 2 for the indicated population size, β = 2, γ = 1 and T = 10.
By carrying out five simulations of a random variable χ 2 (2), 1, 000 times each, the percentages of values that resulted in the corresponding subintervals for each of the five simulations, with the corresponding SD at the last column, are showed in Table 2 below. From Tables 1 and  Table 2. Percentages of χ 2 (2) observations into each subintervals for 5 series of 1, 000 trials each.
2, we can conclude that, for a population size over 300, the distribution of X n (T) 2 is quite approximate to the χ 2 (2) distribution. In Table 3, estimates of β, γ and R 0 have been simulated for different population size. Table 3 corroborates the convergence in probability demands n β n (T) γ n (T) R n 0 (T) 100 Table 3. Estimates for β n (T), γ n (T) and R n 0 (T) for the indicated population size, β = 2, γ = 1 and T = 10. bigger population size in order to obtain a suitable approximation of the parameters. We appreciate, appropriate estimates for β, γ and R 0 are obtained for population sizes over n = 5, 000.
For β = 2, γ = 1 and T = 10, X n (T) is simulated 1, 000 times again. Let X n,1 (T) and X n,2 (T) the first and second row of X n (T), i.e. X n (T) = (X n,1 (T), X n,2 (T)) . In Figure 4 below, histograms of X n,1 (T) and X n,2 (T) are showed for n = 300. It is observed that their corresponding frequencies of simulated values are quite close to the standard normal density. Let U n (T) and v 0 (T) be as in Proposition 6.2 and denote X n,3 (T) = U n (T)/(R 0 v 0 (T)). By observing in Figure 5 the histogram of X n,3 (10), we notice the graphical frequency of its simulated values is also quite close to the standard normal density.

The power function of the test for the basic reproduction number
According to Proposition 6.2, the power function π n (Δ) converges to the asymptotic power function (apf) π(Δ) = Φ(Δ/v 0 (T) − t α ). The level of significance of the test is assumed to be α = 0.05 and consequently, t α = 1.644853. Notice π n (Δ) can be estimated by π n (Δ), the frequency H 0 is rejected (rf). Hence, π n (Δ) is the frequency of times that U n (T)/v 0 (T) > 1.644853, under H Δ : R 0 = 1 + Δ/ √ n. By the Law of Large Number, a strong consistent estimator of π n (Δ) is given by where U n,1 (T), . . . , U n,M (T) are independent and identically distributed random variables, which have the same distribution as U n (T) under H Δ . Even, it can prove that the convergence of π n to π n , as M goes to ∞, is uniform on compact subsets of R + , I.e., for each T > 0, lim M→∞ sup 0≤Δ≤T | π n (Δ) − π n (Δ)| = 0, a.s.
In Table 4 below, M = 1, 000 simulations of π n (Δ) are carried out for a population size equals 300 and for 150 values of Δ between 0 and 14.9, with step sizes of equal length Δ = 0.  for Δ = 0 the power of the test should be close to 0.05, the random function π n cannot be considered a good estimator for π n , at least for n = 300. However, the asymptotic power function π seems to give a better approximation for π n . The values of π(Δ) are given in Table  5 below for 150 values of Δ between 0 and 14.9, with step sizes of equal length Δ = 0.1. In  Table 5. Values of π(Δ) for different values of Δ. Figure 6 below, a graphical comparison between π and π n is given for a population size equals n = 300.

Asymptotic and Estimated Power Function
Delta Probability of Rejection apf rf Figure 6. Grafics of π(Δ) and π n (Δ) for a population size equals n = 300.

Conclusions
A wide class of discrete-time stochastic epidemic models was introduced and analyzed in this work, the convergence of the stochastic model to the deterministic one as the population increases was proved. Moreover, it was proved the convergence in distribution of a process, depending on the fluctuations between the stochastic and deterministic models, to the solution to a stochastic differential equation. The statistical analysis was focused in proving asymptotic normality of natural martingale estimators, for the parameters of the model, and these results were applied to hypothesis tests for the General Epidemic Model. As a consequence, a hypothesis test for the reproduction number was stated and numerical simulations validated the size populations under which the results are useful.
The stochastic models considered here represent an alternative modeling to those using counting processes and having transitions occurring at random times. However, both are asymptotically consistent with their deterministic counterparts. The advantage of considering stochastic models at discrete times is that it is not necessary to observe the epidemic over a long period of time; however, in order to attain the asymptotical consistency mentioned, our model requires frequent observation when dealing with a large population. Another important conclusion is obtained from Subsection 4.2, where numerical simulations was carried out for the SIS epidemic model. Indeed, the simulation showed large fluctuations of the SIS stochastic epidemic model, for n = 10, regarding the deterministic one. However, by simulating the process for n = 1, 000 it could be appreciated the trajectories of the deterministic and stochastic models were quite closed. Moreover, Theorem 2 provides confidence bounds which give an insight of the fluctuations of the stochastic model regarding the deterministic one. The simulation in Subsection 4.2 for the SIS epidemic model shows the coherence between these bounds and the simulated process ( Figure 3). As a conclusion, the deterministic and stochastic model are quite different for small populations, while for large populations both models perform similarly. However, it is worth noting, although the stochastic and deterministic trajectories are similar for large size populations, an attractor state for the deterministic model is not necessarily an attractor for the stochastic model. As shown in Subsection 4.2, this is the case for the SIS epidemic model.
In order to develop statistical inference on the parameters of the model, martingale estimators are proposed. This fact allows to obtain closed form for the estimators. Even, such estimators can be obtained when the distribution of the process governing the epidemic is not completely known.
All the models belong to the class that we are presenting in this chapter assume a stochastic latent period, and consequently, the Reed-Frost model are not included here. As a matter of fact, none of these models could be considered an extension or a particular case of the Reed-Frost model. Moreover, the modeling presented in this chapter need not be Markovian, even though some Markovian epidemic models are included in this setting.