The NERA model incorporating cellular automata approach and the analysis of the resulting induced stochastic mean field

Abstract

A stochastic process depicting the spreading dynamics of illicit drug consumption in a given population forms the crux of this work. A probabilistic cellular automaton (PCA) model is developed to examine the effects of the social interactions between nonusers and drug users. The model, called NERA, comprises four classes of individuals, namely, nonuser (N), experimental user (E), recreational user (R), and addict user (A). The stochastic process evolves in time by local transition rules. By means of dynamical simple mean field approximation, a nonlinear system of differential equations illustrating the dynamics of the PCA is obtained. The existence and uniqueness of a positive solution of the model is established, and the fixed points of the system are sought to perform the stability analysis. Furthermore, a stochastic mean field (SMF) approach to the NERA system is introduced. SMF extends the latter model to integrate the stochastic behaviour of drug consumers in a given environment. The SMF system is shown to exhibit a unique global solution which is stochastically ultimately bounded. Simulations of the cellular automaton and mean field analysis are used to study the evolution of the model. Verification and validation are carried out using data available on the consumption of cannabis in the state of Washington (Darnell and Bitney in ${I}-502$ evaluation and benefit-cost analysis: second required report, Washington state institute for public policy. Technical Report, 2017). These numerical experiments confirm that the NERA model can help in the analysis and quantification of the spatial dynamics of illicit drug usage in a given society and eventually provide insight to policy-makers on different steps to be taken to curb this social epidemic.

Appendices

Appendix

Appendix A: Theorems

Theorem 2

(Haddad and Chellaboina 2011) Let $D \subset \mathbb {R}^{n}$ be open and let $f: D\mapsto \mathbb {R}^{n}$. If f is continuously differentiable on D, then f is locally Lipschitz continuous on D.

Theorem 3

(Khalil 2002) Let F(X(t)) be Lipschitz continuous. Then, the Cauchy problem related to system (2) has a unique local solution.

Appendix B: Derivation of the reproduction number, $R_0$

Let $x=(N,E,R,A)^T$, denotes the state vector for the NERA model with each component of $x \ge 0$.

Let ${\mathcal {H}}_{i}(x)$ represents the arrival of new drug users in class i and ${\mathcal {T}}_{i}(x) = {\mathcal {T}}^-_{i}(x) - {\mathcal {T}}^+_{i}(x)$, where ${\mathcal {T}}^+_{i}(x)$ represents the rate of transfer of users into class i by all other means and ${\mathcal {T}}^-_{i}(x)$ denotes the rate of transfer of users out of class i. In this sense, the NERA model can be written in the form:

$$\begin{aligned} \dfrac{\mathrm{{d}}x_{i}}{\mathrm{{d}}t}=f_{i}(x)={\mathcal H}_{i}(x)-{\mathcal T}_{i}(x) \quad i=1, \ldots , 4, \end{aligned}$$

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where

$$\begin{aligned} {\mathcal {H}}=\left( \begin{array}{l} {\mathcal {H}}_{2}\\ {\mathcal {H}}_{3}\\ {\mathcal {H}}_{4}\end{array}\right) =\left( \begin{matrix}r_{1}NE + r_{1}NR \\ 0\\ 0\end{matrix}\right) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {T}}=T^-_{i}(x) - T^+_{i}(x)=\begin{pmatrix} r_{2}ER + r_{3}EN \\ -r_{2}ER + r_{4})R +r_5RN\\ -r_{4}R +r_{6}A\end{pmatrix}. \end{aligned}$$

Evaluating matrices H and T at the DFE, $P_{0}=(1,0,0,0)$, we get:

$$\begin{aligned} H= \left( \begin{array}{c@{\quad }c@{\quad }c} r_{1}N &{} r_{1}R &{}0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 \end{array}\right) \quad \text{ and } \quad T= \left( \begin{array}{ccc} r_{2}R + r_{3}N &{} r_{2}E &{}0\\ -r_{2}R &{} -r_{2}E+r_4+r_5N &{}0\\ 0&{} -r_{4} &{} r_{6} \end{array}\right) . \end{aligned}$$

Therefore:

$$\begin{aligned} HT^{-1}= \left( \begin{array}{c@{\quad }c@{\quad }c} \dfrac{r_{1}}{r_{3}} &{} 0 &{}0\\ 0 &{} 0 &{}0\\ 0 &{} 0 &{}0\end{array}\right) , \end{aligned}$$

where $HT^{-1}$ represents the next-generation matrix of model (2).

According to Theorem 2 in Van den Driessche and Watmough (2002), it follows that the spectral radius of $HT^{-1}$ gives $R_{0}$. As a result, the reproduction number of model (2) is given by:

$$\begin{aligned} R_{0} = \frac{r_{1}}{ r_{3}}. \end{aligned}$$

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Appendix C: Stability analysis of the drug-free equilibrium, DFE

The stability analysis of the drug-free state is studied by analysing the Jacobian matrix, J, of the system. The matrix, J, is given as follows:

$$\begin{aligned} J= \begin{bmatrix} -r_{1}E - r_{1}R+r_{3}E+r_{5}R &{} -r_{1}N+r_{3}N &{} -r_{1}N+r_{5}N &{} r_{6}\\ r_{1}E + r_{1}R-r_{3}E &{} -(r_{3}N+r_{2}R-r_{1}N) &{} r_{1}N-r_{2}E &{} 0\\ -r_{5}R &{} r_{2}R &{} -(r_{4}+r_{5}N-r_{2}E) &{} 0\\ 0 &{} 0 &{} r_{4} &{} -r_{6} \end{bmatrix}. \end{aligned}$$

Evaluating the Jacobian matrix, J at the DFE, $P_{0} = (1, 0, 0, 0)$ gives:

$$\begin{aligned} J(P_{0})= \begin{bmatrix} 0 &{} -r_{1}+r_{3} &{} -r_{1}+r_{5} &{} r_{6}\\ 0 &{}r_{1}-r_{3} &{} r_{1} &{} 0\\ 0 &{} 0 &{} -r_{4}-r_{5} &{} 0\\ 0 &{} 0 &{} r_{4} &{} -r_{6} \end{bmatrix}. \end{aligned}$$

Solving $J(P_{0})$, the following eigenvalues are obtained:

$$\begin{aligned} \lambda _{1}= & {} 0 ,\nonumber \\ \lambda _{2}= & {} r_{1}-r_{3}=r_3(R_0-1),\nonumber \\ \lambda _{3}= & {} -r_{4}-r_{5},\\ \lambda _{4}= & {} -r_{6}. \nonumber \end{aligned}$$

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Clearly, $\lambda _{3}$ and $\lambda _{4}$ are negative. $\lambda _{2}$ is also negative when $R_{0} <1$. Since there is the presence of a zero eigenvalue, no conclusion can be reached pertaining to the stability of the critical point (1, 0, 0, 0) as the linearisation test fails.