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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.

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., [

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 [

This way of stability investigation was successfully used for investigation of different mathematical models of systems with delays: SIR epidemic model [

The present paper is organized as follows. Section

Taking into account the proposal presented by Rosenquist et al. in [

Let

Considering homogeneous mixing [

It is supposed that the parameters

It is assumed that when a nonconsumer individual is

Put

It is easy to see that

Via (

Via (

If

From two first equations of system (

Thus, via (

Consider the values of the parameters

Then, condition (

Let us suppose that system (

To centralize system (

Rejecting the nonlinear terms in (

Note that nonlinear system (

To get sufficient conditions for asymptotic mean square stability of the zero solution of system (

The trivial solution of (

Following the GMLFC [

If matrix equation (

Note that matrix equation (

If

Note that via (

If conditions (

Note that the order of nonlinearity of system (

Let

Putting

Consider system (

From (

Consider system (

Then, conditions (

Let us get now two corollaries from Theorem

From (

If conditions (

Note that from

If conditions (

Let us suppose that

Consider system (

Consider system (

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

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 (