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Branching processes are stochastic individual-based processes leading consequently to a bottom-up approach. In addition, since the state variables are random integer variables (representing population sizes), the extinction occurs at random finite time on the extinction set, thus leading to fine and realistic predictions. Starting from the simplest and well-known single-type Bienaymé-Galton-Watson branching process that was used by several authors for approximating the beginning of an epidemic, we then present a general branching model with age and population dependent individual transitions. However contrary to the classical Bienaymé-Galton-Watson or asymptotically Bienaymé-Galton-Watson setting, where the asymptotic behavior of the process, as time tends to infinity, is well understood, the asymptotic behavior of this general process is a new question. Here we give some solutions for dealing with this problem depending on whether the initial population size is large or small, and whether the disease is rare or non-rare when the initial population size is large.

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 [

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 _{0} of the process and the notation “|_{0}” will be therefore omitted for the sake of simplicity of formulas. The proofs are given in detail in [

Let _{n}_{n,i}_{n}_{n,i}_{i}_{n−}_{1} := {_{k}_{k≤n−}_{1}. We denote ^{2} the mean and variance of _{n,}_{1} given _{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
_{n}_{n}

The behavior of this process has been deeply analysed for a long time (see for example [_{n}_{n}_{n}_{n}_{n}_{n}m^{−n} = _{0} and of
_{n}m^{−n}, that is _{n}m^{−n}_{n−1}) = _{n−1}^{−(n−1)} which implies _{n}m^{−n}) = _{0}. 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 _{n,1}_{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, _{n}_{n−1} with _{0} = _{0}, derived from _{n}|F_{n−1}) = _{n−1}_{n}_{0} while the stochastic one dies out a.s.. Moreover before dying out, the stochastic process _{n}|I_{n}_{n}_{n}^{2}^{−1} ≤ _{n}

In addition, the probability of extinction, ^{Y1,1}) and in the subcritical/critical cases _{ext.}_{n,1} follows itself a power series distribution, that is, _{n,i}_{k}λ^{k}_{n,i}^{−1}^{k}_{0}) distribution, for _{0}(1 − ^{−1}, _{0}^{−3} are increasing functions of _{0}.

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,
_{n,i}_{i}_{n−1} : {_{k}_{k≤n−1}. The depletion effect of the _{n−1}) (resp. _{n,i}_{n−1})) is a decreasing (resp. increasing) function of the previous incidences of cases {_{k}_{k≤n−1}.

A simple example is when
_{n,1}|_{n,1} ≠ 0) =

The other extremal case _{n}

The intermediate case 0 < _{n}

Let _{n,i}_{n−1} = _{n−1}) ≥ _{+} and any value of _{n−1} consistent with _{n−1} =

This assumption is checked as soon as _{n,i}_{n−1}) is a non-decreasing function of _{n−1}, _{n−2},. . ., which is strictly increasing in _{n−1}.

_{n}_{n}_{n}_{n}

Let _{*}_{*}_{|F|→∞}_{1}-norm of

Let us define the bifurcation parameter _{∞}_{n}

_{∞} _{|F|→∞}_{∞} < 1_{n}_{0} = _{|FX|→0}^{X}_{0} < 1_{n}_{n}_{0} > 1_{n}X_{n}

_{n}_{n}_{n}_{n}, I_{n−}_{1}_{n−}_{(}_{d−}_{1)}) =: (_{n,}_{1}_{n,}_{2}_{n,d}_{n}_{n}_{n−1}) _{n}|F_{n−1}) =: _{n−1}_{n−1})^{n}^{n−(j−1)}^{n}_{|F|→∞} _{|F|→∞}

_{n−1}) < 1 _{n−1}_{n−1}|_{n}^{−1})^{−1} > 1_{0} = _{n}

We generalize here the model of the previous section to a set of _{n−1}) is non increasing in

A simple example is when as previously

As in the single-type case, assuming ^{(j)}(_{*}

Let the deterministic associate trajectory

A possible extension of the previous models consists in adding an immigration. Then:
_{n}_{n}_{n}_{n}

This type of processes has been deeply studied in the subcritical case _{n,i}_{i}, J_{n}_{n−1}, with either an immigration allowed only in the periods of extinction of the process {_{n}_{n}_{n−1} (see the review article [

We may also write (_{n,i}_{i}, J_{n}, δ_{n}_{{In−1=0}}_{n}δ_{n}

Let
_{n−1}_{n−1}_{0}} and that the
_{n−1}

The models of the previous sections (Sections 2. and 4.) belong to this class when _{n}

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 (_{n−1}_{n−1}_{n}

Let us write ℕ_{n}_{n}_{n−1}_{n−(d−1)}). Then _{n}|F_{n−1}_{n−1}_{n−1}_{n−1}_{n}_{0})...
_{n−1}^{−1} is a martingale with _{n}_{0})...
_{n−}_{1})]^{−1}|_{0}) = ℕ_{0}, implying by the convergence theorem for martingale [_{n}_{n}_{0})...
_{n−}_{1})]^{−1}
^{t}_{U}, for any vector
_{
U} is an integrable random variable. But, except in the simple cases
_{n−}_{1}) =
_{N}_{n}

In Section 5.1., we give some approaches for studying the behavior of model (

Let
_{n,i}_{n−}_{1} > _{M}_{M}_{n,i}_{n−}_{1} < _{m}_{m}_{M}_{n}_{2}_{n−1}_{1}) =: _{1}_{2}) is independent of _{n}_{1}_{2})} may be difficult to compute. A solution is then to work in continuous time: in [

We assume here that the initial population size _{0} is large which allows to study the limit, as _{0} → ∞, of the following quantities: _{n}/N_{0} =: _{n}_{n}/N_{n}_{n}_{n}_{0}^{n}^{−1} =: _{n}_{N0} _{0} = _{0} or lim_{N0} _{0} = _{0}.

A simple theoretical example that we (artificially) apply here on epidemics is the following model on densities with _{n,i}_{i}_{n−1}_{n−1}_{0}} and depend on the previous cases incidences only through the condition: _{n,i}_{n−1}_{0}, that is, a massive vaccination is done as soon as the density
_{n}_{n}_{0} and proved that lim_{I0} lim_{n}_{n}_{n}_{n}_{n}_{n}_{n−}_{1}_{n−}_{1}), _{0} = 1, with _{n−}_{1}) := _{n,}_{1}_{n−}_{1} = _{n−}_{1}). Thus, for _{0} very large (which occurs when the initial time is chosen when the epidemic is large enough), the random empirical densities has the same asymptotic behaviour, as

Let us study now the empirical probabilities in the general case _{n}_{n}_{n}_{n}_{n}

Let us first study the behavior of the total population size, under the assumption that the number of newborns is independent of the mother health state.

_{n}_{n,}_{1}_{n,d}_{n}, N_{n−}_{1}_{n−}_{(}_{d−}_{1)})_{n−1}

_{l} < ∞, _{n−l,n,i}_{n−}_{1}) > 0 _{n}_{n}

{_{n}

_{n}_{n}_{n}_{n}

_{n}_{n}_{−1}) =: _{n}_{−1}_{n}^{t}_{0}^{n}^{t}^{t}^{t}. Moreover ρ_{n}_{n}_{n}_{n}_{0}^{n}^{−1}_{n}_{n}_{−}_{l}^{−l}_{0},

_{∞} ≤ 1

_{n}

Let us write _{l}_{l}ρ^{−l}_{n}_{d}

For _{l}

_{n−l,n}_{n−1}_{n−l,n}_{n−}_{1}) _{N0→∞}_{0} = _{0}_{−l} = _{l}N_{0}_{l}_{0}_{N0} _{n}_{n}|_{n}_{n}

Proposition 5 allows to use the attractors of the dynamical model {_{n}_{0} with _{0} tending to infinity, and the loss of the possibility for the extinction time of any type to be finite. Moreover the global asymptotic stability of the healthy state for {_{n}

Another approach for studying the asymptotic behavior of the process {_{n}

Let us recall model (_{n−}_{1} = {_{n−}_{1}_{0}} and the
_{n−}_{1}. We assume here that _{n−1}_{n−1}_{n−}_{1}, and the
_{n−1}

Moreover, if a ^{tot}^{tot}

Let _{n−l,h}

^{tot}, E^{tot}_{1,a,l|n−l} _{2,a,l|n−l} _{n}_{a}_{a,n}. Moreover we may write_{n−l,n,i}_{i}_{n−1} = {_{n−}_{1}_{n−}_{2}_{n−l,n,}_{1}_{n−}_{1} ∼ _{l|n−l}_{l|n−l}_{a}_{a,l|n−l}, and the_{n−l,n,i}_{i,l}

As an example of such an approach, we studied the propagation of the BSE in Great-Britain by a general model of type (

The _{a}_{a}

There is no over-contamination during the incubation period or the clinical state;

The number of newborn animals

The population is roughly stable:

The disease is rare at the initial time:

The probability for a given

Under these assumptions, then
_{n−l}_{k}_{k}_{k}_{mat.}_{inc.}_{l|n−l}

Let us assume that Φ_{l|n−l}_{n}_{l}_{l}_{l|n−l}_{n}_{n,}_{1}_{n,}_{2}_{n,d}_{n}, I_{n−}_{1}_{n−}_{(}_{d−}_{1)}) and _{n−}_{1} = {_{n−}_{1}_{n−}_{2}_{0}}.

Let _{n}|F_{n−}_{1}) =: _{n−}_{1}

_{n}_{}} _{n}_{–1}

_{n}^{t}_{0}^{n}^{t}, where ρ and_{1} = 1^{t}^{t}, implying that

Let s := (_{1}, .., _{d}_{d}_{+1} := 1,
^{(1)}(^{(}^{d}^{)}(

_{n}},
_{n}_{n−}_{1}(^{(}^{h}^{)}(_{h}_{+1} exp(−Ψ_{h}_{1})).

As in Proposition 4, the bifurcation parameter of the process is given by the first eigenvalue

_{∞}

Let us assume from now on that Ψ_{1} > 0,..., Ψ_{d}

_{n}_{n}_{n}_{n}

_{∞} > 1 _{n}_{n}_{−}_{l}_{l}I_{0}_{l}_{0}. Then

_{∞} ≤ 1 _{n}_{n}

Let us point out that the deterministic model derived from _{n}|_{n−}_{1)} = _{n−}_{1} _{n}_{n−}_{1}_{0} := _{0.} Therefore _{n}_{0}^{n}_{n}_{n}^{t}_{0}^{t}

Let _{ext.}_{n}_{Ĩ0}(lim_{n}_{n}_{Ĩ0}(_{ext.}

_{0} = (0

As a consequence, in the particular case _{0} = (_{0}_{1} is solution of the equation: 1 _{1} = 1 − exp(_{∞}_{1})). Thus, there exists a solution _{1} ≠ 1 if _{∞}_{1} decreases as _{∞}_{∞}_{1} = 1, implying _{h}

Let _{n}_{ext.}_{n}_{ext}_{∞}_{n}^{−}^{n}_{n}_{0}.^{t}_{n}^{−}^{n}_{n}^{t}^{t}^{t}^{t}^{t}_{h}^{−(}^{h−}^{1)},
_{ext.}^{n}_{n}

_{∞}, the extinction time distribution satisfies, for n_{n}_{∞}_{l}

In addition, using Proposition 8, we get the following result, allowing an exact iterative computation of {_{n}

_{∞}, the distribution_{n}_{ext.}_{0} = (0_{n}_{n,}_{1}(_{n,d}

Let
_{0} (epidemic size). Let us recall that the Borel-Tanner distribution with parameter (_{λ}, s^{l}_{λ} (^{−}^{λ}^{(1}^{−s}^{)} is the generating function of the Poisson distribution with parameter

Let ^{N}

_{0} := (_{0}, 0, ..., 0)

_{0} := (_{0}, _{−1}, , ..., _{−(}_{d}_{−1)})_{i,j}

We presented a general class of branching processes in discrete time for modeling in a stochastic way some diseases propagation when the infected period is long respectively to the time frequency of births. However when the transitions are population dependent, the long-term prediction of these processes is an open problem in the general case. We indirectly solved this problem by studying the behavior of the limit models, as the total initial population size increases to infinity, assuming at the initial time, either a non-rare disease with a density-dependence assumption, or a rare disease.

Under the first assumption, since
_{0}, that the proportion of infected individuals in the whole population,
_{n}

This result would validate and generalize the current use in epidemiology of the reproductive number as a bifurcation parameter [_{0}, the stochastic limit model is a good approximation of the epidemic growth (in the supercritical case), or of the epidemic decay (in the subcritical case). In any case, if _{0} is too small, the limit in _{0} cannot be used and therefore the limit models developed here cannot be used.

Let us finally notice that we proved that in a size-dependent model on the clinical cases, then the quantity determining the extinction of the process was the total mean number of secondary cases that will be produced in the future by a case as the whole current population is infected (_{∞}_{0}), which easily leads to the extinction of the process on one hand and the persistence of the deterministic trajectory on the other hand. However, simulations of the single-type process with population-dependent offsprings described in Section 3., showed that until its extinction, the process roughly behaved as its deterministic counterpart and the extinction time strongly depends on the parameters of _{0}. The extinction time roughly increases as _{0} increases. So the greatest difference between the behavior of this population-dependent process and its deterministic counterpart is obtained when _{0} > 1 with _{0} ≃ 1. This is the generalization of the difference observed between the BGW process and its deterministic counterpart, when _{0}(=

Two populations of infectives from similar diseases in competition following the same logistic Poisson model
^{5},