Mathematical models of propagation of a disease in given populations play a central role for understanding this propagation, for predicting the future extension of the outbreak, its extinction time, and for evaluating the efficiency of control measures. Of course the validity and the richness of results of a model strongly depend on the reliability and the accuracy of the model. A fine predictive model should be built as far as possible in a rigorous mechanistic way starting from the mechanism of exposure/infection of each individual and taking into account their variability. The population dynamic described by births, deaths, migrations should also be taken into account, especially when the incubation time is relatively long in comparison with this dynamic. Of course, the time unit of the model should be chosen in keeping with the respective durations of each health state and the data.

Nevertheless, since the asymptotic behavior, as time tends to infinity, of deterministic models in continuous time are more easily studied than that of discrete time models or stochastic models, a large literature in applied mathematical journals is devoted to such theoretical studies [19], while for statistical purposes, either descriptive non dynamical models or very simple stochastic models are used, such as a chain binomial model for the propagation of SIS or SIR diseases in closed populations [5, 6], or BGW (Bienaymé-Galton-Watson) branching processes on the clinical (or diagnosed) cases as an approximate model of the beginning of an outbreak [2, 3, 4, 6, 10, 11, 20]. These two approaches are both individual-based (each individual transition is given) and stochastic (individual variability is taken into account) but the first one takes into account the individual health state evolution S → I → S or S → I → R, where S means susceptible, I means infective or clinical cases (according to authors and purposes) and R means removed from the susceptible population, either by immunization or death; while the second approach directly models the process of I individuals, without analyzing this mechanism of exposure/infection. The behavior of these two types of processes are well-known. The chain binomial models are just Markov chains in a finite state space, and since there is no immigration and no incubation period, the state “0 infected individual” is an absorbing state. Concerning BGW processes, a huge literature exists for describing their properties. In fact these two different types of models may be written as particular cases of a large class of branching processes with individual transitions that may be population and age dependent, and which can take into account both the population dynamics and the disease dynamic.

Branching processes were initiated in the nineteen century by Bienaymé and then by Galton and Watson, for studying the extinction of some family names. Since this time, the complexity of these processes continues to increase allowing to describe more and more realistic population dynamics. These processes are based on the simple property that the population size of each considered type (such as clinical cases here) at each time is calculated as the sum of all the new individuals (“offspring”) of this type who are generated by the individuals of the population at the previous times. Since the modelled variables are integers, then the extinction time of the population of each type is finite on the set of trajectories which extinct. This is a finer and more realistic property than the asymptotic extinction time given by a deterministic model. Since the population dynamic may influence the disease dynamic, and conversely, these two dynamics should be explicitly taken into account in the model, thus leading to rigorous, but not simple, multitype models where each type should represent an health state crossed with influence factors levels. Typical examples of such influence factors are age, geographical locations. However, some authors model directly the time evolution of the incidence of clinical cases, without explicating all the intermediary steps.

In the following subsections, we present such direct models starting from the simplest one, the single-type Bienaymé-Galton-Watson process, and finishing by a general and rigorous approach that takes into account the intermediary steps and the population dynamic, and is based on age-dependent and population-dependent individual transitions. We focus on models in discrete time since they have the double advantage to be easily written as recursive models and to easily correspond to the time unit of observation, which offers a pedagogic framework and moreover facilitates the model validation and the estimation of unknown parameters. We study here the behavior of these models. From now on, all results are given conditionally to the initial value F₀ of the process and the notation “|F₀” will be therefore omitted for the sake of simplicity of formulas. The proofs are given in detail in [23] and will be omitted here.

Let I_n be the incidence of clinical (or diagnosed) cases at time n and Y_n,i represent the numbers of secondary cases generated at time n by the previous case i. Then I_n is modelled from the past incidences by (1) I n = ∑ i = 1 I n − 1 Y n , i ,where the variables {Y_n,i}_i are assumed i.i.d. (independently and identically distributed) according to a distribution L independent of the time and of the past process given F_n−1 := {I_k}_k≤n−1. We denote m, σ² the mean and variance of Y_n,1 given F_n−1. Since these first two moments are the moments influencing the behavior of the process on the non-extinction set, we will also write L ( m , σ 2 ) rather than L. This comes from the following writing of (1): I n = m I n − 1 + I n − 1 η n, where η n   : = [ ∑ i = 1 I n − 1 ( Y n , i − m ) ] [ I n − 1 ] − 1 is asymptotically distributed according to N ( 0 ,   σ 2 ) on {lim_n I_n = ∞}, which is the non-extinction set, as n → ∞.

The behavior of this process has been deeply analysed for a long time (see for example [1, 13, 16, 24]). We recall here the main properties. The trajectories of this process either die out or explode in an exponential way: P({lim_n I_n = ∞} ∪ {lim_n I_n = 0}) = 1 and there exists an integrable random variable W such that lim_n I_nm⁻ⁿ = W, a.s. (almost surely), where P(W > 0) > 0 if and only if m > 1 (supercritical case). The random variable W depends on I₀ and of L ( m , σ 2 ). This limit behavior comes from the martingale property of I_nm⁻ⁿ, that is E(I_nm⁻ⁿ|F_n−1) = I_n−1m⁻⁽ⁿ⁻¹⁾ which implies E(I_nm⁻ⁿ) = I₀. So this process reproduces the initial phase of exponential growth of an epidemic and can be used for describing this phase when the incubation period is negligible compared to the time unit. The quantity m := E(Y_n,1|F_n−1) is the current reproductive number of the process (mean number of secondary cases produced by one case during a time unit) and is known to be the bifurcation parameter of the process, that is, m ≤ 1 implies the a.s. extinction of the process. The deterministic model X_n = mX_n−1 with X₀ = I₀, derived from E(I_n|F_n−1) = I_n−1m, has the same bifurcation parameter, but when m = 1, the deterministic model persists with X_n = I₀ while the stochastic one dies out a.s.. Moreover before dying out, the stochastic process I_n|I_n ≠ 0 presents a linear increasing behavior: lim_n P(I_n[0.5σ²n]⁻¹ ≤ x|I_n ≠ 0) = 1 − exp(−x).

In addition, the probability of extinction, q, is the solution of q = f(q) := E(q^Y_1,1) and in the subcritical/critical cases m ≤ 1, the “epidemic size” N   : = ∑ k = 0 Text . I k (total number of cases generated until the extinction time T_ext.), follows a power series distribution when Y_n,1 follows itself a power series distribution, that is, P(Y_n,i = k) ∝ a_kλ^k [7, 11, 12]. In particular, when P(Y_n,i = k) = exp(−m)[k!]⁻¹m^k (Poisson(m) distribution), then N follows the Borel − Tanner(m, I₀) distribution, for m < 1, (2) P ( N = k ) = exp ( − m k ) I 0 k ( m k ) k − I 0 ( k − I 0 ) ! ,       k ≥ I 0 ,and E(N) = I₀(1 − m)⁻¹, Var(N) = I₀m(1 − m)⁻³ are increasing functions of m, I₀.

When the population is small or when the individual migrations are slow compared to the infection process, the depletion of susceptible individuals due to the infection should be taken into account, which is not the case in the BGW process. The typical bell curve form of outbreaks is due to this depletion. The simplest such model is just an extension of the BGW process, that is, I n = ∑ i = 1 I n − 1 Y n , i, where the {Y_n,i}_i are assumed i.i.d. according to a distribution L ( F n − 1 ) = :   L ( m ( F n − 1 ) ,   σ 2 ( F n − 1 ) ) given F_n−1 : {I_k}_k≤n−1. The depletion effect of the S population at time n is replaced by the fact that m(F_n−1) (resp. P(Y_n,i = 0|F_n−1)) is a decreasing (resp. increasing) function of the previous incidences of cases {I_k}_k≤n−1.

A simple example is when L ( F n − 1 ) = L ( ∑ l = 1 d n μ l I n − l ), μ ∈ [0, 1], where m ( ∑ l = 1 d n μ l I n − l )   ( resp .   P ( Y n , 1 = 0 | F n − 1 ) = p ( ∑ l = 1 d n μ l I n − l ) ) is a decreasing (resp. increasing) function of ∑ l = 1 d n μ l I n − l. For example, E(Y_n,1|Y_n,1 ≠ 0) = α and P ( Y n , 1 = 0 | F n − 1 ) = 1 − K [ K + ∑ l = 1 d n μ l I n − l ] − 1, 0 < μ ≤ 1, K > 0. We will see in this section that this kind of models may exhibit a random cycle and therefore may be used for modeling recurrent epidemics. The extremal case μ = 0 means no depletion in S, which is got by an immediate healing with no immunization, after one time unit in the state I (SIS disease) and this model is reduced to the previous BGW process.

The other extremal case μ = 1 with d_n = n corresponds either to a not necessarily immediate healing with persistent immunization or to a fatal issue (SIR disease). This process is a branching process with a long memory and has not be studied in the literature excepted by simulation [27].

The intermediate case 0 < μ ≤ 1 with d_n = d represent the possibility of recovering but with a transient immunization only, and leads to a Markovian chain of order d. The behavior of this process has been studied in [27] and is described in the following paragraphs.

Let A1: there exists a positive function f(.) such that P(Y_n,i = 0|I_n−1 = N, F_n−1) ≥ f(N) > 0, for all N ∈ ℕ₊ and any value of F_n−1 consistent with I_n−1 = N.

This assumption is checked as soon as P(Y_n,i = 0|F_n−1) is a non-decreasing function of I_n−1, I_n−2,. . ., which is strictly increasing in I_n−1.

Proposition 1 Let us assume A1. Then P(lim_n I_n = 0 ∪ lim_n I_n = ∞) = 1.

Let A2: there exists m_* < ∞ such that m(F) ≤ m_*, for all F, and lim_|F|→∞m(F) < 1, where |F| is the L₁-norm of F.

Let us define the bifurcation parameter R_∞ by R ∞ = sup { R : 0 < R < 1 ⇒ P ( lim n   I n = 0 ) = 1 } ,where R is any real quantity depending on the parameters of the distribution of {I_n}.

Proposition 2 The bifurcation parameter R_∞ is equal to lim_|F|→∞m(F). Let us assume A1 and A2. Then R_∞ < 1, i.e., the process dies out a.s.. Moreover for the deterministic trajectory {X_n} defined by X n   : = m ( F n − 1 X ) X n − 1, where F n − 1 X = { X n − 1 ,   X n − 2 ,   … }, then the bifurcation parameter is the reproductive number R₀ = lim_|F^X|→0m(F^X), that is, if R₀ < 1, then lim_n X_n = 0, while if R₀ > 1, then lim_nX_n ≠ 0.

Remark 1 When d_n = d, for all n, {I_n} may be represented as a multitype branching process Ĩ_n = (I_n, I_n−1, ..., I_n−(d−1)) =: (I_n,1, I_n,2, ..., I_n,d), which is asymptotically BGW if L ( F n − 1 ) tends to a distribution L independent of the process as time tends to ∞ on {lim_n I_n = ∞}. But the asymptotic BGW process is not positively regular at the opposite of classical BGW branching processes, since M͂(F_n−1) being defined by E(Ĩ_n|F_n−1) =: Ĩ_n−1M͂(F_n−1), then M ˜ ( F n − 1 ) = ( m ( F n − 1 ) 1 0 … 0 0 0 1 … 0 … … 0 0 0 … 1 0 0 0 … 0 )which implies that, for all n ≥ d − 1, M͂ⁿ[1, j] = m^n−(j−1), and M͂ⁿ[i, j] = 0, i > 1, where m := lim_|F|→∞ m(F) and M͂ := lim_|F|→∞ M͂(F).

Remark 2 If m((1, 0, ..., 0)) > 1 (supercritical assumption at the beginning of the disease spread), since m(F_n−1) < 1 under A2, for |F_n−1| and n sufficiently large, then we may observe oscillations until the a.s. extinction, the process increasing when |Ĩ_n−1|is small enough, and decreasing when it is too large. It is the case of the logistic model m ( F n − 1 ) = α K [ K + ∑ l = 1 d n μ l I n − l ] − 1, when d_n = d with d > 1 and m((1, 0, ..., 0)) = α(1 + μK⁻¹)⁻¹ > 1. Moreover since R₀ = α > 1, then {X_n} tends to a stable limit cycle, as n → ∞ [9, 27].

We generalize here the model of the previous section to a set of J similar diseases in competition: I n ( j ) = ∑ i = 1 I n − 1 ( j ) Y n , i ( j ) ,     j = 1 ,   … ,   J ,where, for each j, the { Y n , i ( j ) } i are i.i.d. according to L ( j ) ( F n − 1 ) = : L ( m ( j ) ( F n − 1 ) , σ 2 ( j ) ( F n − 1 ) ) given F n − 1   :   { { I k ( j ) } j = 1 , … J } k ≤ n − 1. We assume that m(F_n−1) is non increasing in I k ( j ), for each j, each k ≤ n − 1.

A simple example is when as previously L ( j ) ( F n − 1 ) = L ( j ) ( ∑ l = 1 d n μ l ∑ j ′ I n − l ( j ′ ) ), μ ∈ [0, 1], and m ( j ) ( ∑ l = 1 d n μ l ∑ j ′ I n − l ( j ′ ) )   ( resp .   P ( Y n , 1 ( j ) = 0 | F n − 1 ) = p ( j ) ( ∑ l = 1 d n μ l ∑ j ′ I n − l ( j ′ ) ) ) is a decreasing (resp. increasing) function of ∑ l = 1 d n μ l ∑ j ′ I n − l ( j ′ ). A typical example is given by different influenza viruses.

As in the single-type case, assuming m^(j)(F) < m_* < ∞, for all F, with R ∞ ( j )   : = lim ¯ | F | → ∞ m ( j ) ( F )   <   1, then, for all j, P ( lim n   I n ( j ) = 0 ) = 1, that is each disease dies out. Therefore as soon as the number of cases due to any pathogenic agent decreases or dies out, then the epidemics due to the other pathogenic agents may increase (see Figure 1).

Let the deterministic associate trajectory { X n ( j ) } be defined by X n ( j )   : = m ( j ) ( F n − 1 X ) X n − 1 ( j ), n≥ 1, X 0 ( j )   : = I 0 ( j ).

Proposition 3 The reproductive number R ∞ ( j )   : = lim ¯ | F | → ∞ m ( j ) ( F ) is the bifurcation parameter for { I n ( j ) }, that is, if R ∞ ( j ) < 1, then lim n I n ( j ) = a . s . 0, and the reproductive number R 0 ( j )   : = lim _ | F X | → 0 m ( j ) ( F X ) is the bifurcation parameter of { X n ( j ) }, that is if R 0 ( j ) < 1, then lim n X n ( j ) = 0. Moreover if R 0 ( j )   >   1, then lim ¯ n X n ( j )   >   0.

A possible extension of the previous models consists in adding an immigration. Then: (3) I n = ∑ i = 1 I n − 1 Y n , i + J n δ n ,where J_n is an immigration at time n when δ_n = 1, and δ_n is a Bernoulli variable allowing or not an immigration at time n. When {δ_n} follows some seasonality, then the model is suitable for recurrent seasonal epidemics with seasonal immigrant, such as influenza.

This type of processes has been deeply studied in the subcritical case m < 1, when ({Y_n,i}_i, J_n) has a distribution independent of n and F_n−1, with either an immigration allowed only in the periods of extinction of the process {I_n} (regenerative branching process), or when the {δ_n} are i.i.d. independent of F_n−1 (see the review article [39] and the estimation point of view in [25]).

We may also write (3) in the following way: I n = 1 { I n − 1 ≠ 0 } ∑ i = 1 I n − 1 Y ˜ n , i + 1 { I n − 1 = 0 } J n δ n ,       Y ˜ n , i = Y n , i + J n I n − 1 δ n .Consequently, when ({Y_n,i}_i, J_n, δ_n) is not time-dependent and when 1_{In−1=0}J_nδ_n = 0, for all n, then (3) may be written as a model of Section 3..

Let N n = ( N n 1 ,   … N n D ) satisfying (4) N n k = ∑ h = 1 D ∑ l = 1 d ∑ i = 1 N n − l h Y n − l , n , i ( h ) , k ,     k = 1 ,   … ,   D ,where h, k represent types and l is the maturation time for getting the “offsprings” Y n − l , n , i ( h ) , k of the individual i (number of k individuals at time n generated by i belonging to the type h at time n − l). We assume that the { Y n − l , n , i ( h ) , k } i , l are mutually independent given F_n−1 = {N_n−1, ..., N₀} and that the { Y n − l , n , i ( h ) , k } i are i.i.d. given F_n−1.

The models of the previous sections (Sections 2. and 4.) belong to this class when d_n = d.

When rigorously modeling the propagation of a disease in a given population, taking explicitly into account the population dynamic and the disease dynamic, then a type should be a health state (S, E (incubation) or I, for example) crossed with an influence factors level, and the “offsprings” { Y n − l , n , i ( h ) , k } should be expressed according to the number of newborn individuals of i, who are of the k type, and to the proper transition of i from the state h at n − l to the state k at n. For example assuming that there is no influence factors and that the set of individual health states is {S, E, I} and assuming that the life duration in the state I is at most one time unit, then, for k = I, Y n − l , n , i ( h ) , I = 1 { h = S } δ 2 , n − l + 1 , n , i E , I | h + ∑ j = 1 Y n − l + 1 , i ( h ) δ 1 , n − l + 1 , n , i , j E , I | ( h ) ,where the { δ 2 , n − l + 1 , n , i E , I | S } i are i.i.d. Bernoulli variables given F_n−1, with δ 2 , n − l + 1 , n , i E , I | S equal to 1 if the individual i undergoes transitions S → E, at time n − l + 1, and E → I, at time n, which means that his incubation period is l, the { δ 1 , n − l + 1 , n , i , j E , I | ( h ) } i , j are similar i.i.d. Bernoulli variables given F_n−1, relative to the newborn j of i, and the { Y n − l + 1 , i ( h ) } i are i.i.d variables given F_n−1, Y n − l + 1 , i ( h ) being the number of newborns of i at time n − l + 1. So Y n − l , n , i ( h ) , I is the number of I individuals generated at time n with an incubation time l from i who is in the h type at n − l.

Let us write ℕ_n := (N_n, N_n−1, ..., N_n−(d−1)). Then E(ℕ_n|F_n−1) = ℕ_n−1 M(F_n−1), where M ( F n − 1 ) = ( M n − 1 , n ( F n − 1 ) I 0 … 0 M n − 2 , n ( F n − 1 ) 0 I … 0 … … M n − ( d − 1 ) , n ( F n − 1 ) 0 0 … I M n − d , n ( F n − 1 ) 0 0 … 0 )I is the D × D identity matrix, and M n − l , n ( F n − 1 ) [ h ,   k ]   : = E ( Y n − l , n , i ( h ) , k | F n − 1 ), for h = 1, ..., D, k = 1, ..., D. Then, assuming that M(F) is invertible, for any value F of F_n−1, ℕ_n[ M(F₀)... M(F_n−1)]⁻¹ is a martingale with E(ℕ_n [ M(F₀)... M(F_n−1)]⁻¹|F₀) = ℕ₀, implying by the convergence theorem for martingale [15] that lim_n ℕ_n [ M(F₀)... M(F_n−1)]⁻¹ U^t = W_U, for any vector U, where W _U is an integrable random variable. But, except in the simple cases M(F_n−1) = M (BGW process) or M ( F n − 1 ) = M ( ∑ k N n − 1 k ) with lim_N M(N) = M (asymptotically BGW process) [30, 31, 32, 33], we cannot deduce from this martingale convergence any information on the asymptotic behavior of {N_n} itself.

In Section 5.1., we give some approaches for studying the behavior of model (4) when the total population size remains bounded. In Sections 5.2. and 5.3., we study the asymptotic behavior of limit models derived from (4) as the initial population size increases to infinity.