A comparison between random and stochastic modeling for a SIR model

In this article, a random and a stochastic version of a SIR nonautonomous model previously introduced in [ 19 ] is considered. In particular, the existence of a random attractor is proved for the random model and the persistence of the disease is analyzed as well. In the stochastic case, we consider some environmental effect on the model, in fact, we assume that one of the coefficients of the system is affected by some stochastic perturbation, and analyze the asymptotic behavior of the solutions. The paper is concluded with a comparison between the two different modeling strategies.

1. Introduction. The study of biological models in nonautonomous and nondeterministic frameworks have attracted the attention of many researcher over the last decades (see [8,9,10] and the references therein). In the nondeterministic case, the most used strategies consist in considering stochastic or random perturbation. These two different approaches lead to different modeling, for which different tools are available. In order to describe that, in this article we consider random and stochastic perturbation of the following deterministic model (see [3] for a detailed discussion): (t, ω) = q − aS(t) + bI(t) − γ S(t) I(t) N (t) , I(t, ω) = −(a + b + c)I(t) + γ S(t) I(t) N (t) , R(t, ω) = cI(t) − aR(t). (1) The above model has been considered in [19] in a nonautonomous framework (see also [20] for a bifurcation scenario of a similar model with two time-dependent parameters). In particular it deals with the case in which the per capita/capita infection rate varies in time (see Thieme [25] for a more detailed discussion). This can be modeled by introducing a forcing term which can be either time dependent (see [19]) or random.
Here we consider a case in which the forcing term is nondeterministic and can be modeled in two different ways: in a first model (Section 3) we consider a random coefficient and study the problem in the framework of Random Dynamical Systems (RDS for short). This allows not only to introduce a random counterpart of the concept of deterministic global attractor, but also the useful definition of random equilibria (also called random fixed points in [24]). In particular, we will use these tools to study the asymptotic and qualitative behavior of the solutions of the system under investigation. The main definitions and results concerning RDS are briefly recalled in Section 2. In the second case (which is analyzed in Section 4), we consider the system in a stochastic context. First, we consider the situation in which the system is subjected to some environmental effect, in the sense that one of the coefficients of the model is perturbed by an additive noise. This yields a stochastic system with multiplicative noise of the same intensity in all the equations of the system. In the last section of conclusions, we emphasize that the stochastic case may exhibit more modeling problems than the random one because, depending on which coefficient is perturbed by noise, the modeling technique can provide us with a more or less appropriate model to describe the real system. In this stochastic case, we use the technique based on the construction of conjugated random dynamical systems thanks to an appropriate change of variables involving the Ornstein-Uhlenbeck process. A final comparison between these two strategies is discussed in the last section.
2. Some preliminaries definitions. In this section we review on some basic concepts from the theory of random dynamical systems (for more details see [1,8,13] amongst others).
Let (X, · X ) be a separable Banach space and let (Ω, F, P) be a probability space where F is the σ−algebra of measurable subsets of Ω and P is the probability measure. We define a flow θ = {θ t } t∈R on the probability space Ω with each θ t being a mapping θ t : Ω → Ω satisfying (1) θ 0 = Id Ω , (2) θ s • θ t = θ s+t for all s, t ∈ R, (3) the mapping (t, ω) → θ t ω is measurable, (4) the probability measure P is preserved by θ t , i.e., P(θ −1 t A) = P(A) for all A ∈ F. Finally, (Ω, F, P, θ) is called a metric dynamical system [1].
Definition 2.2. (i) A random set K is a measurable subset of X ×Ω with respect to the product σ−algebra B(X) × F. (ii) The ω−section of a random set K is defined by In the case that a set K ⊂ X × Ω has closed or compact ω−sections it is a random set as soon as the mapping ω → d(x, K(ω)) is measurable (from Ω to [0, ∞)) for every x ∈ X, see [13]. Then K will be said to be a closed or a compact, respectively, random set. It will be assumed that closed random sets satisfy K(ω) = ∅ for all or at least for P−almost all ω ∈ Ω. (iii) A bounded random set K(ω) ⊂ X is said to be tempered with respect to (θ t ) t∈R if for a.e. ω ∈ Ω, lim t→∞ e −βt sup x∈K(θ−tω) x X = 0, for all β > 0; a random variable ω → r(ω) ∈ R is said to be tempered with respect to (θ t ) t∈R if for a.e. ω ∈ Ω, lim t→∞ e −βt sup t∈R |r(θ −t ω)| = 0, for all β > 0.
We regard D(X) as the set of all tempered random sets of X.
if for any K ∈ D(X) and a.e. ω ∈ Ω, there exists T K (ω) > 0 such that (iii) (attracting property) for any K ∈ D(X) and a.e. ω ∈ Ω, is the Hausdorff semi-metric for G, H ⊆ X.
Proposition 1 ( [12,15]). Let Γ ∈ D(X) be an absorbing set for the continuous random dynamical system {ϕ(t, ω)} t≥0,ω∈Ω which is closed and satisfies the asymptotic compactness condition for a.e. ω ∈ Ω, i.e., each sequence x n ∈ ϕ(t n , θ −tn ω)Γ(θ −tn ω) has a convergent subsequence in X when t n → ∞. Then the cocycle ϕ has a unique global random attractor with component subsets If the pullback absorbing set is positively invariant, i.e., Remark 1. When the state space X = R d , the asymptotic compactness follows trivially. Note that the random attractor is path-wise attracting in the pullback sense, but does not need to be path-wise attracting in the forward sense, although it is forward attracting in probability, due to some possible large deviations, see e.g., Arnold [1].

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T. CARABALLO AND R. COLUCCI 3. Random system. The random system we are concerned with is the following: where a, b, c, γ are positive constants, Examples of this kind of noise can be found, for instance, in [2]. Given an Ornstein-Uhlenbeck process Z t (see [8,11]), the following processes take values in q 0 [1 − ε, 1 + ε], the first one peaking around q 0 [1 ± ε] and the other centering at q 0 . Although it is possible to consider a more general model in which some other coefficients can be also random, for simplicity in our analysis, we have preferred to consider just one of them random because this is enough to show how the technique works.
First of all we prove that solutions corresponding to nonnegative initial conditions remain nonnegative, that is: , is positively invariant for the system (2), for each fixed ω ∈ Ω.
Proof. We quickly verify that the vector field, at the boundary of R 3 + , points inwards.
On the plane S = 0 we have thatṠ > 0, the plane I = 0 is invariant since on it we haveİ = 0 while on R = 0 we haveṘ ≥ 0.
The positive S-semi axes is invariant, in fact we have: that is The last term is bounded, in fact that is (t−t0) ).
If we start on the positive R−semi axes we have that the solution enters the plane I = 0, while on the positive I−semi axes we haveṠ,Ṙ > 0.
Using the above results we deduce: Theorem 3.3. The random dynamical system generated by system (2) possesses a global random attractor.
If we set t 0 = 0 and replace ω by θ −t ω in (8) we have It is easy to see that the solution (8) both forward and pullback converges to In details: the pullback limit reads as while the forward limit is defined as We observe that the expression of (13) coincides with that of (7) on the S−axes.
Since the computation is done with respect to the total population N = S + I + R, it is natural to wonder what happens to the three populations individually. In order to qualitatively describe the asymptotic behavior of (S, I, R), we replace, in system (2), N (t) by its forward limit N * (θ t ω). To make the computations clearer to the reader, we proceed in this informal way, however the rigorous proof by taking approximations of N * (θ t ω) for large values of t can also be carried out. Then, we obtain the following SI random model: where .
We observe that the random parameter α(θ t ω) is bounded: , for any ω ∈ Ω, (15) and that R 2 + is positively invariant. We set V = S + I, theṅ from whichV ≤ q(θ t ω) − aV, and then the forward and pullback limit v * (ω) of V (t) satisfies . By a similar argument to that used for the SIR model we conclude that the sets B η = (S, I) ∈ R 2 + : S + I ≤ q 0 (1 + ε) + η are pullback absorbing for η > 0 and positively invariant for η ≥ 0. In order to see what happens to the population I in the asymptotic behavior, we can obtain a sufficient condition for the disappearance of the disease. We consider solutions starting in B 0 . We first observe that the S-axes is invariant and that solutions both forward and pullback converges to S * on it. Then, by the second equation of the system we havė we have that lim t→∞ I(t) = 0.
Then the systems tend to a disease free configuration if (16) is satisfied. By the first equation of the SI system we havė S .
Then the setB is positively invariant. Then if we restrict onB we havė We conclude that, if γ > ab, and the disease persists.
Remark 2. We observe that the conditions for persistence/extinction of the desease are not complementary like in [19] since in the SI model under consideration we have two (randomly) perturbed coefficients (q and α) instead of one (q) as in [19]. In this case the persistence/extinction conditions are based on the upper and lower bound of α(·) (see (15)). 4. Stochastic system. The stochastic model we will consider now is in complete agreement with a largely used approach in the published literature on this topic (see, for instance, [18,4,6] and references cited therein) which consists in considering white noise that is directly proportional to the quantities S(t), I(t), R(t) in each equation respectively. Moreover, it can be considered as the result of the environmental noise effect on some of the parameters in the model. For example, if we assume that one of the parameters, say a, is affected by a noisy perturbation of the type σẆ (t), in other words, if we replace in the deterministic model the parameter a by a − σẆ (t), then the model becomes: We remark that this kind of perturbations describing environmental noise has been used is several applied situations as can be seen, for example, in [16,17,26]. In fact, if we sum the three equations of the system and set N = S + I + R, then we obtain the following perturbed equation for the total population of the system: that has been obtained, as we mentioned above, by perturbing coefficient a in each equation by the same noise (see for example [22] or [27]).
Notice that the noise considered for three populations is correlated (following the approach of [22]). Moreover, for simplicity, we have considered that the noise intensity is the same in all the equations but it does not make a substantial difference if we consider different ones in each equation, as well as different mutually independent Wiener processes in each equation. Only the computations will be more complicated, but the technique is the same.
It is easy to see that solutions, corresponding to nonnegative initial data, remain nonnegative and, as a consequence, the model is well defined.
The previous equation has a nontrivial random solution that is both forward and pullback attracting. In fact, for any initial datum N 0 we have: