Analysing Social Epidemics by Delayed Stochastic Models

Abstract

We investigate the dynamics of a delayed stochastic mathematical model to understand the evolution of the alcohol consumption in Spain. Sufficient condition for stability in probability of the equilibrium point of the dynamic model with aftereffect and stochastic perturbations is obtained via Kolmanovskii and Shaikhet general method of Lyapunov functionals construction. We conclude that alcohol consumption in Spain will be constant (with stability) in time with around 36.47% of nonconsumers, 62.94% of nonrisk consumers, and 0.59% of risk consumers. This approach allows us to emphasize the possibilities of the dynamical models in order to study human behaviour.

1. Introduction

In this paper, we propose a mathematical framework to model social epidemics. To be precise, we propose delayed and stochastic consideration on mathematical models to analyze human behaviors related to addictions.

Hereditary systems or systems with delays are very popular in researches (see, e.g., [1–4] and the references therein). In this paper, a nonlinear dynamic alcohol consumption model [5] is generalized via adding distributed delays. Sufficient conditions for existence of the positive equilibrium points of this system are obtained. It is also supposed that this nonlinear system is exposed to additive stochastic perturbations of white noise that are directly proportional to the deviation of the system current state from the equilibrium point. Such type of stochastic perturbations first was proposed in [6, 7] and successfully used later in [8–11]. One of the important point of this assumption is that the equilibrium point is the solution of the stochastic system too. In this case, the influence of the stochastic perturbations on the considered system is small enough in the neighbourhood of the equilibrium point and big enough if the system state is far enough from the equilibrium point.

The considered nonlinear system is linearized in the neighborhood of the positive point of equilibrium, and sufficient condition for asymptotic mean square stability of the zero solution of the constructed linear system is obtained via Kolmanovskii and Shaikhet general method of Lyapunov functionals construction (GMLFC) that is used for stability investigation of stochastic functional-differential and difference equations [12–22]. Since the order of nonlinearity more than 1, this condition is also sufficient one [23, 24] for stability in probability of the initial nonlinear system by stochastic perturbations.

This way of stability investigation was successfully used for investigation of different mathematical models of systems with delays: SIR epidemic model [6], predator-prey model [7, 11], Nicholson blowflies model [9], inverted pendulum [25–27].

The present paper is organized as follows. Section 2 presents the delayed model including the study of its equilibrium points. Stochastics characteristics of the delayed model are shown in Section 3. Section 4 is related to the study of stability of the equilibrium point of the delayed stochastics model proposed. The main conclusions derived from the study are presented in the last section.

2. Delayed Mathematical Model

Taking into account the proposal presented by Rosenquist et al. in [28], we consider alcohol consumption habit as susceptible to be transmitted by peer pressure or social contact. This fact lead us to propose an epidemiological type mathematical model to study this social epidemic.

Let 𝐴(𝑡) be nonconsumers, individuals that have never consumed alcohol or they infrequently have alcohol consumption, and 𝑀(𝑡) nonrisk consumers, individuals with regular low consumption; to be precise, men who consume less than 50 cc of alcohol every day and women who consume less than 30 cc of alcohol every day. Let 𝑅(𝑡) be risk consumers, individuals with regular high consumption, that is, men who consume more than 50 cc of alcohol every day and women who consume more than 30 cc of alcohol every day.

Considering homogeneous mixing [29], that is, each individual can contact with any other individual (peer pressure), dynamic alcohol consumption model is given by the following nonlinear system of ordinary differential equations with distributed delay: 𝐴̇(𝑡)=𝜇𝑃(𝑡)+𝛾𝑅(𝑡)−𝑑_𝐴𝐴(𝑡)−𝛽𝐴(𝑡)^∞₀𝑓(𝑠)𝑀(𝑡−𝑠)+𝑅(𝑡−𝑠)𝑃()𝑀̇(𝑡)(𝑡)𝑡−𝑠𝑑𝑠,=𝛽𝐴^∞₀𝑓(𝑠𝑀()())𝑡−𝑠+𝑅𝑡−𝑠𝑃()(𝑡)(𝑡),𝑅̇(𝑡)(𝑡)(𝑡)(𝑡),𝑃(𝑡)(𝑡)(𝑡)(𝑡),𝑡−𝑠𝑑𝑠−𝑑𝑀−𝛼𝑀=𝛼𝑀−𝛾𝑅−𝑑𝑅=𝐴+𝑀+𝑅 (2.1)(2.2)(2.3)(2.4) where(i)μ: birth rate in Spain;(ii)γ: rate at which a risk consumer becomes a nonconsumer; (iii)𝑑_𝐴: death rate in Spain;(iv)β: transmission rate due to social pressure to increase alcohol consumption (family, friends, marketing, TV, etc.);(v)d: augmented death rate due to alcohol consumption. Accidents at work, traffic accidents and diseases derived by alcohol consumption are considered. The information available to calculate the augmented death rate due to alcohol consumption is aggregate. This fact does not allow us to consider the difference between the augmented death rate in nonrisk and risk consumers;(vi)α: rate at which a nonrisk consumer moves to the risk consumption subpopulation.

It is supposed that the parameters 𝛼,𝛽,𝛾,𝜇,𝑑_𝐴,𝑑 and the function 𝑓(𝑠) are nonnegative and the following condition holds: ^∞₀𝑓(𝑠)𝑑𝑠=1. (2.5) In the particular case 𝑓(𝑠)=𝛿(𝑠−ℎ), where ℎ>0, 𝛿(𝑠) is Dirac's function, system (2.1) is a system with discrete delay ℎ. The case of a system without delay (ℎ=0) is considered in [5].

It is assumed that when a nonconsumer individual is infected by alcohol consumers, there is a time 𝑠 during which the alcohol consumption habit develops in nonconsumer and it is only after that time that nonconsumer individual becomes nonrisk consumer. It is also assumed that at any time 𝑡 alcohol consumers population is simply proportional to alcohol consumers population at time 𝑡−𝑠. Let 𝑓(𝑠) be the fraction of nonconsumer population that takes time 𝑠 to become alcohol consumers (nonrisk consumer).

2.1. Normalization of the Delayed Model

Put 𝑎(𝑡)=𝐴(𝑡)𝑃(𝑡)(𝑡)=𝑀(𝑡),𝑚𝑃(𝑡)(𝑡)=𝑅(𝑡),𝑟𝑃(𝑡). (2.6) From (2.1) and (2.6), it follows that 𝑎(𝑡)+𝑚(𝑡)+𝑟(𝑡)=1. (2.7) Adding the first three equations (2.1) by virtue of (2.7), we obtain 𝑃̇(𝑡)𝑃(𝑡)=𝜇−𝑑+𝑑−𝑑_𝐴𝑎(𝑡). (2.8)

It is easy to see that (𝑡)=𝐴̇(𝑡)𝑃(𝑡)(𝑡)𝑃̇(𝑡)𝑎̇−𝐴𝑃²(𝑡)=𝐴̇(𝑡)𝑃(𝑡)−𝐴(𝑡)𝑃(𝑡)×𝑃̇(𝑡)𝑃(𝑡)=𝐴̇(𝑡)𝑃(𝑡)−𝑎(𝑡)𝜇−𝑑+𝑑−𝑑_𝐴𝑎(𝑡), (2.9) and similarly (𝑡)=𝑀̇(𝑡)𝑚̇𝑃(𝑡)−𝑚(𝑡)𝜇−𝑑+𝑑−𝑑_𝐴𝑎(𝑡,(𝑡)=𝑅̇(𝑡))𝑟̇𝑃(𝑡)−𝑟(𝑡)𝜇−𝑑+𝑑−𝑑_𝐴𝑎(𝑡). (2.10) Thus, putting 𝐼𝑎_𝑡=^∞₀𝑓(𝑠)𝑎(𝑡−𝑠)𝑑𝑠, (2.11) from (2.1), (2.5), (2.7), (2.9), (2.10), we have 𝑎̇(𝑡)=𝜇+𝛾𝑟(𝑡)+𝛽𝑎(𝑡)𝐼𝑎_𝑡−𝑎(𝑡)𝛽+𝜇−𝑑−𝑑_𝐴((𝑡,(𝑡)(𝑡)(𝑡)𝐼𝑎1−𝑎))𝑚̇=𝛽𝑎−𝛽𝑎_𝑡−𝑚(𝑡)𝛼+𝜇+𝑑−𝑑_𝐴𝑎(𝑡,(𝑡)(𝑡)(𝑡)𝑟̇=𝛼𝑚−𝑟)𝛾+𝜇+𝑑−𝑑_𝐴𝑎(𝑡). (2.12)

Via (2.7), the last equation can be rejected, and, as a result, we obtain the system of two differential equations 𝑎̇(𝑡)=𝜇+𝛾−𝛾𝑚(𝑡)+𝛽𝑎(𝑡)𝐼𝑎_𝑡−𝑎(𝑡)𝛽+𝜇+𝛾−𝑑−𝑑_𝐴((𝑡,(𝑡)(𝑡)(𝑡)𝐼𝑎1−𝑎))𝑚̇=𝛽𝑎−𝛽𝑎_𝑡−𝑚(𝑡)𝛼+𝜇+𝑑−𝑑_𝐴𝑎(𝑡). (2.13)

2.2. Existence of the Equilibrium Point

Via (2.5), (2.7), (2.13), the point of equilibrium (𝑎^∗,𝑚^∗,𝑟^∗) is defined by the system of the algebraic equations: 𝜇+𝛾1−𝑎^∗=𝑎^∗𝛽−𝑑+𝑑_𝐴1−𝑎^∗+𝛾𝑚^∗,𝛽𝑎^∗1−𝑎^∗=𝑚^∗𝛼+𝜇+𝑑−𝑑_𝐴𝑎^∗,1=𝑎^∗+𝑚^∗+𝑟^∗. (2.14)(2.15)(2.16)

Lemma 2.1. If 𝑑∈[𝑑_𝐴,𝛽+𝑑_𝐴), then system (2.14) has the unique positive solution (𝑎^∗,𝑚^∗,𝑟^∗) if and only if 𝛽>𝑑−𝑑_𝐴+𝜇+𝛼𝛾𝛼+𝛾+𝜇+𝑑−𝑑_𝐴. (2.17) If 𝑑≥𝛽+𝑑_𝐴, then system (2.14) does not have positive solutions.

Proof. From two first equations of system (2.14), we have 𝜇+𝛾−𝑎^∗𝛽−𝑑+𝑑_𝐴=𝛽𝛾𝑎^∗𝛼+𝜇+𝑑−𝑑_𝐴𝑎^∗, (2.18) or 𝑄𝑎^∗2+𝐵𝑎^∗−𝐶=0,𝐵=𝛽−𝑑+𝑑_𝐴𝛼+𝛾+𝜇−𝜇𝑑−𝑑_𝐴,𝑄=𝛽−𝑑+𝑑_𝐴𝑑−𝑑_𝐴,𝐶=𝜇+𝛼𝜇+𝛾. (2.19)
Thus, via (2.14), the equilibrium point (𝑎^∗,𝑚^∗,𝑟^∗) is defined by the system of the algebraic equations (2.19) and 𝑚^∗=𝛽𝑎^∗1−𝑎^∗𝛼+𝜇+𝑑−𝑑_𝐴𝑎^∗,𝑟^∗=1−𝑎^∗−𝑚^∗. (2.20) It is easy to check that, by condition 𝑑∈[𝑑_𝐴,𝛽+𝑑_𝐴) (or 𝑄≥0), the existence of the solution 𝑎^∗ of (2.19) in the interval (0,1) is equivalent to the condition 𝐶<𝑄+𝐵 that is equivalent to (2.17). If 𝑑≥𝛽+𝑑_𝐴, then 𝑄≤0 and 𝐵<0. So, (2.19) cannot have positive roots. The proof is completed.

Example 2.2. Consider the values of the parameters 𝛼,𝛽,𝛾,𝜇,𝑑,𝑑_𝐴 from [5]: 𝛼=0.000110247,𝛽=0.0284534,𝛾=0.00144,𝜇=0.01,𝑑=0.009,𝑑_𝐴=0.008. (2.21)
Then, condition (2.17) for these values of the parameters holds, and the solution of systems (2.19) and (2.20) is 𝑎^∗=0.3647389407,𝑚^∗=0.6293831151,𝑟^∗=0.005877944497 (2.22) or in the percents 𝑎^∗=36.47%, 𝑚^∗=62.94%, 𝑟^∗=0.59%.

3. Stochastic Perturbations, Centralization, and Linearization

Let us suppose that system (2.13) is exposed to stochastic perturbations type of white noise (𝑤̇₁(𝑡),𝑤̇₂(𝑡)), which are directly proportional to the deviation of system (2.13) state (𝑎(𝑡),𝑚(𝑡)) from the point (𝑎^∗,𝑚^∗), that is, 𝑎̇(𝑡)=𝜇+𝛾−𝛾𝑚(𝑡)+𝛽𝑎(𝑡)𝐼𝑎_𝑡−𝑎(𝑡)𝛽+𝜇+𝛾−𝑑−𝑑_𝐴(1−𝑎(𝑡))+𝜎₁𝑎(𝑡)−𝑎^∗𝑤̇₁(𝑡),𝑚̇(𝑡)=𝛽𝑎(𝑡)−𝛽𝑎(𝑡)𝐼𝑎_𝑡−𝑚(𝑡)𝛼+𝜇+𝑑−𝑑_𝐴𝑎(𝑡)+𝜎₂𝑚(𝑡)−𝑚^∗𝑤̇₂(𝑡). (3.1) Here, 𝑤₁(𝑡),𝑤₂(𝑡) are the mutually independent standard Wiener processes, the stochastic differential equations of system (3.1) are understanding in Ito sense [30].

To centralize system (3.1) in the equilibrium point, put now 𝑥₁(𝑡)=𝑎(𝑡)−𝑎^∗,𝑥₂(𝑡)=𝑚(𝑡)−𝑚^∗. (3.2) Then, from (3.1), it follows that 𝑥̇₁(𝑡)=𝜇+𝛾−𝛾𝑚^∗+𝑥₂(𝑡)+𝛽𝑎^∗+𝑥₁(𝑡)𝑎^∗+𝐼𝑥_1𝑡−𝑎^∗+𝑥₁(𝑡)𝛽+𝜇+𝛾−𝑑−𝑑_𝐴1−𝑎^∗−𝑥₁(𝑡)+𝜎₁𝑥₁(𝑡)𝑤̇₁(𝑡),𝑥̇₂(𝑡)=𝛽𝑎^∗+𝑥₁(𝑡)−𝛽𝑎^∗+𝑥₁(𝑡)𝑎^∗+𝐼𝑥_1𝑡−𝑚^∗+𝑥₂(𝑡)𝛼+𝜇+𝑑−𝑑_𝐴𝑎^∗+𝑥₁(𝑡)+𝜎₂𝑥₂(𝑡)𝑤̇₂(𝑡), (3.3) or 𝑥̇₁(𝑡)=𝜇1−𝑎^∗+𝛾1−𝑎^∗−𝑚^∗−𝑎^∗1−𝑎^∗𝛽−𝑑+𝑑_𝐴−𝜇𝑥₁(𝑡)+𝛾−𝑥₁(𝑡)−𝑥₂(𝑡)+𝑥₁(𝑡)1−2𝑎^∗𝑑−𝑑_𝐴−𝛽𝑥₁(𝑡)1−𝑎^∗+𝛽𝑎^∗𝐼𝑥_1𝑡−𝑥²₁(𝑡)𝑑−𝑑_𝐴+𝛽𝑥₁(𝑡)𝐼𝑥_1𝑡+𝜎₁𝑥₁(𝑡)𝑤̇₁(𝑡),𝑥̇₂(𝑡)=𝛽𝑎^∗1−𝑎^∗−𝑚^∗𝛼+𝜇+𝑑−𝑑_𝐴𝑎^∗+𝛽𝑥₁(𝑡)1−𝑎^∗−𝑚^∗𝑥₁(𝑡)𝑑−𝑑_𝐴−𝛽𝑎^∗𝐼𝑥_1𝑡−𝑥₂(𝑡)𝛼+𝜇+𝑑−𝑑_𝐴𝑎^∗−𝛽𝑥₁(𝑡)𝐼₁𝑥_1𝑡−𝑥₂(𝑡)𝑥₁(𝑡)𝑑−𝑑_𝐴+𝜎₂𝑥₂(𝑡)𝑤̇₂(𝑡). (3.4) Via (2.14) from (2.16), it follows that 𝑥̇₁(𝑡)=−𝑎₁₁𝑥₁(𝑡)−𝛾𝑥₂(𝑡)+𝛽𝑎^∗𝐼𝑥_1𝑡+𝛽𝑥₁(𝑡)𝐼𝑥_1𝑡−𝑑−𝑑_𝐴𝑥²₁(𝑡)+𝜎₁𝑥₁(𝑡)𝑤̇₁(𝑡),𝑥̇₂(𝑡)=𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡)−𝛽𝑎^∗𝐼𝑥_1𝑡−𝛽𝑥₁(𝑡)𝐼𝑥_1𝑡−𝑥₁(𝑡)𝑥₂(𝑡)𝑑−𝑑_𝐴+𝜎₂𝑥₂(𝑡)𝑤̇₂(𝑡), (3.5) where 𝑎₁₁=𝜇+𝛾+𝛽−𝑑+𝑑_𝐴1−𝑎^∗+𝑑−𝑑_𝐴𝑎^∗,𝑎₂₁=𝛽𝛼+𝜇1−𝑎^∗𝛼+𝜇+𝑑−𝑑_𝐴𝑎^∗,𝑎₂₂=𝛼+𝜇+𝑑−𝑑_𝐴𝑎^∗. (3.6) Note that, for 𝑑∈[𝑑_𝐴,𝛽+𝑑_𝐴), the numbers 𝑎₁₁, 𝑎₂₁, 𝑎₂₂ are positive.

Rejecting the nonlinear terms in (3.5), we obtain the linear part of system (3.5) 𝑥̇₁(𝑡)=−𝑎₁₁𝑥₁(𝑡)−𝛾𝑥₂(𝑡)+𝛽𝑎^∗𝐼𝑥_1𝑡+𝜎₁𝑥₁(𝑡)𝑤̇₁(𝑡),𝑥̇₂(𝑡)=𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡)−𝛽𝑎^∗𝐼𝑥_1𝑡+𝜎₂𝑥₂(𝑡)𝑤̇₂(𝑡). (3.7)

4. Stability of the Equilibrium Point

Note that nonlinear system (3.5) has the order of nonlinearity more than 1. Thus, as it follows from [23, 24], sufficient conditions for asymptotic mean square stability of the zero solution of linear part (3.7) of nonlinear system (3.5) at the same time are sufficient conditions for stability in probability of the zero solution of nonlinear system (3.5) and therefore are sufficient conditions for stability in probability of the solution (𝑎^∗,𝑚^∗) of system (3.1).

To get sufficient conditions for asymptotic mean square stability of the zero solution of system (3.7), rewrite this system in the form 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥_𝑡+𝜎(𝑥(𝑡))𝑤̇(𝑡), (4.1) where 𝑥(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡)^′,𝑤(𝑡)=𝑤₁(𝑡),𝑤₂(𝑡)^′,𝐵𝑥_𝑡=𝛽𝑎^∗𝐼𝑥_𝑡,−𝛽𝑎^∗𝐼𝑥_𝑡^′,𝐴=−𝑎₁₁𝑎−𝛾₂₁−𝑎₂₂(𝑥(𝑡=𝜎,𝜎))¹𝑥₁(𝑡)00𝜎₂𝑥₂(𝑡). (4.2)

Definition 4.1. The trivial solution of (2.20) is called as follows.(i)Mean square stable if for any 𝜀>0 there exists a 𝛿>0 such that 𝐄|𝑥(𝑡,𝜙)|²<𝜀 for any 𝑡≥0 provided that the initial function 𝑥(𝑠)=𝜙(𝑠), 𝑠≤0, satisfies the condition sup_𝑠≤0𝐄|𝜙(𝑠)|²<𝛿.(ii)Asymptotically mean square stable if it is mean square stable and satisfies the condition lim_𝑡→∞𝐄|𝑥(𝑡,𝜙)|²=0 for each initial function 𝜙.(iii)Stable in probability if for any 𝜀₁>0 and 𝜀₂>0 there exists 𝛿>0 such that the solution 𝑥(𝑡,𝜙) of (2.20) satisfies the condition 𝐏{sup_𝑡≥0|𝑥(𝑡,𝜙)|>𝜀₁/𝔉₀}<𝜀₂ for any initial function 𝜙 such that 𝐏{sup_𝑠≤0|𝜙(𝑠)|≤𝛿}=1.

Following the GMLFC [12–14] for stability investigation of (4.1), consider the auxiliary equation without memory 𝑦̇(𝑡)=𝐴𝑦(𝑡)+𝜎𝑦(𝑡)𝑤̇(𝑡), (4.3) and the matrix equation 𝐴′𝑃+𝑃𝐴+𝑃_𝜎=−𝐶, (4.4) where 𝑝𝑃=¹¹𝑝₁₂𝑝₁₂𝑝₂₂,𝑃_𝜎=𝑝₁₁𝜎²₁00𝑝₂₂𝜎²₂,𝐶=𝑐001, (4.5)𝑐>0, the matrix 𝐴 is defined in (3.6), (4.2).

If matrix equation (4.4) has a positive definite solution 𝑃, then the function 𝑣(𝑦)=𝑦′𝑃𝑦 is a Lyapunov function for (4.3).

Note that matrix equation (4.4) can be represented as the system of the equations 2−𝑝₁₁𝑎₁₁+𝑝₁₂𝑎₂₁+𝑝₁₁𝜎²₁2=−𝑐,−𝑝₁₂𝛾−𝑝₂₂𝑎₂₂+𝑝₂₂𝜎²₂=−1,−𝑝₁₁𝛾−𝑝₁₂𝑎₁₁+𝑎₂₂+𝑝₂₂𝑎₂₁=0, (4.6) with the solution 𝑝₁₁=𝑐+2𝑎₂₁𝑝₁₂2𝐴₁₁,𝑝₁₂=𝑎₂₁𝐴₁₁−𝑐𝛾𝐴₂₂2𝐴,𝑝₂₂=1−2𝛾𝑝₁₂2𝐴₂₂, (4.7) where 𝐴_𝑖𝑖=𝑎_𝑖𝑖−𝛿_𝑖,𝛿_𝑖=12𝜎²_𝑖,𝑖=1,2,𝐴=𝐴₁₁𝐴₁₂𝐴₂₂+𝛾𝑎₂₁𝐴₁₁+𝐴₂₂,𝐴₁₂=𝑎₁₁+𝑎₂₂. (4.8)

Lemma 4.2. If 𝛿₁<𝑎₁₁ and 𝛿₂<𝑎₂₂, that is, 𝐴₁₁>0,𝐴₂₂>0, (4.9) then the matrix 𝑃=‖𝑝_𝑖𝑗‖ with entries (4.7) is a positive definite one.

Proof. Note that via (4.9) 𝛾𝑎₂₁𝐴_𝑖𝑖<𝐴, 𝑖=1,2. Thus, from (4.7), it follows that for arbitrary 𝑐>0𝑝₁₂<𝑎₂₁𝐴₁₁<12𝐴2𝛾,𝑐+2𝑎₂₁𝑝₁₂=𝑐1−𝛾𝑎₂₁𝐴₂₂𝐴+𝑎²₂₁𝐴₁₁𝐴>0. (4.10) Therefore, via (4.9), 𝑝₁₁>0, 𝑝₂₂>0. Let us show that 𝑝₁₁𝑝₂₂>𝑝²₁₂. Really, from 𝑐+2𝑎₂₁𝑝₁₂1−2𝛾𝑝₁₂4𝐴₁₁𝐴₂₂>𝑝²₁₂, (4.11) it follows that 4𝐵𝑝²₁₂−2(𝑎₂₁−𝑐𝛾)𝑝₁₂<𝑐 by 𝐵=𝐴₁₁𝐴₂₂+𝛾𝑎₂₁. Substituting into this inequality 𝑝₁₂ from (4.7), we have 𝐵(𝑎₂₁𝐴₁₁−𝑐𝛾𝐴₂₂)²−𝐴(𝑎₂₁−𝑐𝛾)(𝑎₂₁𝐴₁₁−𝑐𝛾𝐴₂₂)<𝑐𝐴² or 𝑐²𝛾²𝐴₂₂𝐴−𝐵𝐴₂₂+𝑐𝐴₁₁𝐴₂₂𝐴𝐴₁₂+2𝛾𝑎₂₁𝐵+𝑎²₂₁𝐴₁₁𝐴−𝐵𝐴₁₁>0. (4.12) Since 𝐵𝐴_𝑖𝑖<𝐴, 𝑖=1,2, the obtained inequality holds for arbitrary 𝑐>0. Thus, for arbitrary 𝑐>0, the matrix 𝑃 with entries (4.7) is a positive definite one. The proof is completed.

Theorem 4.3. If conditions (4.9) hold and for some 𝑐>0 the matrix 𝑃 entries (4.7) satisfy the condition 𝛽𝑎^∗𝑝₁₂−𝑝₂₂²+2𝛽𝑎^∗𝑝₁₁−𝑝₁₂<𝑐 (4.13) then the solution (𝑎^∗,𝑚^∗) of system (3.1) is stable in probability.

Proof. Note that the order of nonlinearity of system (3.1) is more than one. Therefore, via [23, 24] to get for this system conditions of stability in probability, it is enough to get conditions for asymptotic mean square stability of the zero solution of linear part (3.7) of this system. Following the GMLFC [13–15], we will construct a Lyapunov functional for system (3.7) in the form 𝑉=𝑉₁+𝑉₂, where 𝑉₁=𝑥′𝑃𝑥, 𝑥=(𝑥₁,𝑥₂)′, 𝑃 is a positive definite solution of system (4.6) with entries (4.7) and 𝑉₂ will be chosen below.
Let 𝐿 be the infinitesimal operator [30] of system (3.7). Then, via (3.7), (4.6), 𝐿𝑉₁=2𝑝₁₁𝑥₁(𝑡)+𝑝₁₂𝑥₂(𝑡)−𝑎₁₁𝑥₁(𝑡)−𝛾𝑥₂(𝑡)+𝛽𝑎^∗𝐼𝑥_1𝑡+𝑝₁₁𝜎²₁𝑥²₁(𝑡)+2𝑝₁₂𝑥₁(𝑡)+𝑝₂₂𝑥₂(𝑡)𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡)−𝛽𝑎^∗𝐼𝑥_1𝑡+𝑝₂₂𝜎²₂𝑥²₂(𝑡)=−𝑐𝑥²₁(𝑡)−𝑥²₂(𝑡)+2𝛽𝑎^∗𝑝₁₁−𝑝₁₂𝑥₁(𝑡)+𝑝₁₂−𝑝₂₂𝑥₂(𝑡)𝐼𝑥_1𝑡. (4.14) Via (2.11), (2.5), we have 2𝑥₁(𝑡)𝐼(𝑥_1𝑡)≤𝑥²₁(𝑡)+𝐼(𝑥²_1𝑡) and 2𝑥₂(𝑡)𝐼(𝑥_1𝑡)≤𝛼𝑥²₂(𝑡)+𝛼⁻¹𝐼(𝑥²_1𝑡) for some 𝛼>0. Using these inequalities, we obtain 𝐿𝑉₁≤−𝑐𝑥²₁(𝑡)−𝑥²₂(𝑡)+𝛽𝑎^∗𝑝₁₁−𝑝₁₂𝑥²₁(𝑡)+𝐼𝑥²_1𝑡+𝛽𝑎^∗𝑝₁₂−𝑝₂₂𝛼𝑥²₂(𝑡)+𝛼⁻¹𝐼𝑥²_1𝑡=𝛽𝑎^∗𝑝₁₁−𝑝₁₂−𝑐𝑥²₁(𝑡)+𝛽𝑎^∗𝑝₁₂−𝑝₂₂𝛼−1𝑥²₂(𝑡)+𝑞𝐼𝑥²_1𝑡, (4.15) where 𝑞=𝛽𝑎^∗𝑝₁₁−𝑝₁₂+𝑝₁₂−𝑝₂₂𝛼⁻¹. (4.16)
Putting 𝑉₂=𝑞^∞₀𝑓(𝑠)^𝑡_𝑡−𝑠𝑥²₁(𝜃)𝑑𝜃𝑑𝑠, (4.17) via (2.5), (2.11), we have 𝐿𝑉₂=𝑞(𝑥²₁(𝑡)−𝐼(𝑥²_1𝑡)). Therefore, via (4.15), (4.16) for the functional 𝑉=𝑉₁+𝑉₂, we have 𝐿𝑉≤2𝛽𝑎^∗𝑝₁₁−𝑝₁₂+𝛽𝑎^∗𝑝₁₂−𝑝₂₂𝛼⁻¹−𝑐𝑥²₁(𝑡)+𝛽𝑎^∗𝑝₁₂−𝑝₂₂𝛼−1𝑥²₂(𝑡). (4.18) Thus, if 2𝛽𝑎^∗𝑝₁₁−𝑝₁₂+𝛽𝑎^∗𝑝₁₂−𝑝₂₂𝛼⁻¹<𝑐,𝛽𝑎^∗𝑝₁₂−𝑝₂₂𝛼<1, (4.19) then the zero solution of system (3.7) is [3] asymptotically mean square stable. From (4.19), it follows that 𝛽𝑎^∗𝑝₁₂−𝑝₂₂𝑐−2𝛽𝑎^∗𝑝₁₁−𝑝₁₂1<𝛼<𝛽𝑎^∗𝑝₁₂−𝑝₂₂. (4.20) Thus, if for some 𝑐>0 condition (4.13) holds, then there exists 𝛼>0 such that conditions (4.20) (or (4.19)) hold too, and therefore the zero solution of system (3.7) is asymptotically mean square stable. From here and [23, 24], it follows that the zero solution of system (3.5) and therefore the equilibrium point of system (3.1) is stable in probability. The proof is completed.

Example 4.4. Consider system (3.1) with the value of the parameters 𝛼, 𝛽, 𝛾, 𝜇, 𝑑, 𝑑_𝐴 and the equilibrium point (𝑎^∗,𝑚^∗,𝑟^∗) given in (2.21) and (2.22) and the levels of noises 𝜎₁=0.028969, 𝜎₂=0.142252. We consider this value for 𝜎₁ and 𝜎₂ as an example.
From (3.6), it follows that the values of system (3.7) parameters are 𝑎₁₁=0.029245, 𝑎₂₁=0.017446, 𝑎₂₂=0.010475, and conditions (4.9), hold: 𝛿₁=0.00042<𝑎₁₁, 𝛿₂=0.010118<𝑎₂₂. Put 𝑐=20. Then, via (4.7), 𝑝₁₁=477.4438, 𝑝₁₂=215.6615, 𝑝₂₂=530.4124 and condition (4.13) holds: (𝛽𝑎^∗|𝑝₁₂−𝑝₂₂|)²+2𝛽𝑎^∗|𝑝₁₁−𝑝₁₂|=16.1036<20. Thus, system (3.1) solution is stable in probability.

Example 4.5. Consider system (3.1) with the previous values of all parameters except for the levels of noises that are 𝜎₁=0.0075, 𝜎₂=0.0077. These values of 𝜎₁ and 𝜎₂ are selected taking into account sample errors of the monitoring of the epidemic (alcohol consumption) in Spain in the period 199720100332007. More details about these sample errors are shown in [31].
Then, conditions (4.9) hold: 𝛿₁=0.000028<𝑎₁₁, 𝛿₂=0.000030<𝑎₂₂. Put 𝑐=4. Then, via (4.7), 𝑝₁₁=78.6856, 𝑝₁₂=17.1347, 𝑝₂₂=45.5060, and condition (4.13) holds: (𝛽𝑎^∗|𝑝₁₂−𝑝₂₂|)²+2𝛽𝑎^∗|𝑝₁₁−𝑝₁₂|=1.36<4. Thus, system (3.1) solution is stable in probability.
Let us get now two corollaries from Theorem 4.3 which simplify verification of stability condition (4.13). Put 𝐵₀=𝐴₂₂𝐴₁₂+𝛾+𝛾𝑎₂₁2𝐴,𝐵₁=𝑎₂₁𝑎₂₁−𝐴₁₁,𝐷2𝐴₀=𝛾𝐴₂₂+𝛾2𝐴,𝐷₁=𝐴₁₁𝐴₁₂−𝑎₂₁+𝛾𝑎₂₁,𝑓(𝑐)=2𝐴𝛽𝑎^∗2𝐷₀𝑐+𝐷₁²+2𝛽𝑎^∗𝐵₀𝑐+𝐵₁−𝑐,𝑆=𝛽𝑎^∗𝐷₀²𝐷₁𝐷₀−𝐵₁𝐵₀²+𝐵₁𝐵₀,𝑅=2𝛽𝑎^∗𝐵₀1−2𝛽𝑎^∗𝐵₀2𝛽𝑎^∗𝐷₀²−𝐷₁𝐷₀+B₁𝐵₀1,𝑄=4𝛽𝑎^∗𝐷₀²−𝐷₁𝐷₀−𝐵²₀𝐷²₀. (4.21)

Remark 4.6. From (4.7) and (4.21), it follows that 𝑝₂₂−𝑝₁₂=𝐷₀𝑐+𝐷₁ and 𝑝₁₁−𝑝₁₂=𝐵₀𝑐+𝐵₁. Thus, condition (4.13) is equivalent to the condition 𝑓(𝑐)<0.

Corollary 4.7. If conditions (4.9) hold and 𝑆<0, then the solution of system (3.1) is stable in probability.

Proof. Note that from 𝑆<0 follows that 𝐵₁<0. Putting 𝑐₀=−𝐵₁/𝐵₀>0, we obtain 𝑓(𝑐₀)=𝑆<0, that is, condition (4.13) holds. The proof is completed.

Corollary 4.8. If conditions (4.9) hold and 0≤𝑅<𝑄, then the solution of system (3.1) is stable in probability.

Proof. Let us suppose that 𝐵₀𝑐+𝐵₁≥0. Then, the minimum of the function 𝑓(𝑐) is reached by 𝑐₀=1−2𝛽𝑎^∗𝐵₀2𝛽𝑎^∗𝐷₀²−𝐷₁𝐷₀𝐵≥−¹𝐵₀. (4.22) Substituting 𝑐₀ into the function 𝑓(𝑐), we obtain that the condition 𝑓(𝑐₀)<0 is equivalent to the condition 0≤𝑅<𝑄. The proof is completed.

Example 4.9. Consider system (3.1) with the values of the parameters from Example 4.4. Calculating 𝑆, 𝑅, 𝑄, we obtain, 𝑆=4.56>0, 𝑅=740.0<𝑄=1313.59. From Corollary 4.8, it follows that the solution of system (3.1) is stable in probability.

Example 4.10. Consider system (3.1) with the values of the parameters from Example 4.5. Calculating 𝑆, 𝑅, 𝑄, we obtain 𝑆=−0.38<0, 𝑅=2499.07<𝑄=4708.10. From both Corollary 4.7 and Corollary 4.8 it follows that the solution of system (3.1), is stable in probability.

5. Conclusions

In this work, a modelling approach based on delayed and stochastic differential equations is proposed to understand social behaviours and their evolutionary trends. Taking into account this approach, we can know how social habits can evolve in the future. Considering the study proposed related to the alcohol consumption habit, we note that around 63.5% of Spanish population is (and will be) alcohol consumers (risk or nonrisk consumers). The existence of a equilibrium point (stable in probability) in 𝑎^∗=36.47% (a: nonconsumers), 𝑚^∗=62.94% (m: nonrisk consumers), 𝑟^∗=0.59% (r: riskconsumers), see (2.22), allows us to confirm it.

This mathematical approach is an example as to how delayed and stochastic models can also be an useful tool to model human behaviour. We consider that this approach can be an interesting framework for public health authorities and policy makers.

Note also that the system (2.1) with delay is considered here first. The conditions for the existence of the equilibrium point of the considered system by stochastic perturbations are new result too. The main result of the paper is the conditions for stability in probability of the equilibrium point by stochastic perturbations. The here proposed research method can be used for stability investigation of other important models.