Poverty and the Effects of Drug Addiction in a Deterministic and Stochastic Model

Nowadays directly goes to addict the young and moderate drug used people. They are not recovered quickly from drug usage after they have recovered under the proper guidelines and counseling in particular periods. The poverty/poor people may contact drug-addicted people, who are also easily addicted to drug usage and then face so many struggles in daily life and run in regular life.

Currently, endemic research is getting more and more grounded in reality thanks to societal issues. Based on the association between poverty and drug addiction. So, we are beginning to create mathematical models.

“Drug dependence must be treated primarily as a disease and not a crime. Countries increasingly realize the benefits of drug treatment for not only individuals themselves but also for society and the economy [1]”.

Only a small number of samples of mouth fluid were required. Compared to the traditional ELISA test, this sensing approach improves the sensitivity and specificity of ketamine analysis. Additionally, when resources are limited, the dissection is carried out using a smart phone app that does not require expensive equipment. In addition to ketamine, the cP-ELISA sensing system provides a high-quality, quick-testing platform that uses amphetamines, cocaine, and opioids [2].

Drug abuse has been one of the biggest problems for governments everywhere. Risky injectables and sexual contact among drug users are grave public health issues because the danger of transmission of blood-borne infections includes HIV, HCV, and HBV, particularly among the underprivileged and most at-risk groups.

On top two misused drugs in Taiwan, according to medical institutes, continue to be historically abused substances like heroin and methamphetamine. On the other side, throughout time, the number of people who take new drugs like ketamine has progressively increased, and MDMA abuse has returned [3].

The Joint UNODC-WHO initiative closely connects with the WHO's Mental Health Gap Action Programme (mhGAP) was established in November 2008 to develop solutions for scaling up mental, neurological, and drug use disorder treatment. One of the eight priority conditions is disorders caused by illegal drug use [4].

The joint UNODC-WHO programme:

Leads a global collaborative effort to improve drug use disorder treatment and care coverage and quality in low- and middle-income nations.

Promotes the development of comprehensive and integrated treatment programmes that may link local, state, and federal agencies to offer drug addicts a continuum of care.

Maps are regional needs, governing laws, and available resources and programmes for drug addiction treatment.

Supports revisions to policy and law to achieve drug policy balance and promote compassionate and effective drug prevention, treatment, and care.

Improves exposure in rural and isolated locations by developing low-cost outreach care and treatment services.

The prevention, rehabilitation, and treatment of drug use disorders are incorporated into the conventional healthcare system as a built-in component of the integrated healthcare continuum, working with NGOs to ensure perfect cooperation with the healthcare system.

Supports local networks of top-notch service providers that are devoted to social services, HIV/AIDS prevention, and treatment [4].

Prostitution is the street link to poverty. Female prostitutes recognize as members of racial and ethnic minorities who are poor, uneducated and have a dear of marketable skills. Legislators have chastised and continue to condemn these ladies because they view them as wicked persons. However, consistency in implementing prostitution laws, including laws prohibiting patronizing prostitutes, has been and is inequitable [5].

Tobacco contains a very addictive substance called nicotine. Use of tobacco products may increase the chance of developing heart and lung ailments, more than 20 different kinds or subtypes of cancer, and a wide range of other life-altering illnesses [6].

Even non-smokers can die from tobacco use. And also linked to the poor health consequences, second-hand smoke exposure accounts for over 1.2 million yearly fatalities. Nearly half of all youngsters breathe tobacco smoke-polluted air, and 65 000 young people pass away from diseases each year linked to second-hand smoke. Smoking during pregnancy may result in long-term health issues for the unborn child [6].

It has been determined by experts from all fields, including sociologists and economists, that poverty and crime have an "intimate" connection. World Bank and the United Nations see crime as the primary barrier to a country's development. Governments coping with the effects of poverty usually have to deal with the problem of crime as they work to develop their nation's economy and society. Uncertainty and instability are made by crime, making it difficult for companies to thrive(at micro and macroeconomic levels) [7-9].

Heavy drinking links to an increased risk of cancer. Fatty liver disease and cirrhosis are two conditions that can result from it. The brain and other organs may be potentially harmed by drinking. During pregnancy, liquor might be harmful to your baby. Alcohol also raises the chances of dying in vehicle accidents, illnesses, homicides, and suicides [10, 11].

Changes such as increased rainfall, for example, will impact African dry lands and thus increase the burden of vector-borne water-related diseases in regions already vulnerable to poverty [12].

Sakib et al. [13] they have planned five compartments. but now we have simplified in our model into four compartment.

In this paper is organized as follows: Section 2 portrayed the basic model and analysis which comprises the subsections as basic reproduction number, equilibria and their stability analysis. Section 3 describes the Stochastic Model, section 4 as we display the numeric results and simulating the design of Python showing the figure of numerical simulation on the deterministic model and stochastic model. At the last, we have given our conclusion in section 5.

In this section, we extend our deterministic model to a stochastic model because the issue of drug addiction may be detected more accurately using stochastic models. Based on the method's assumptions, an SDE (Stochastic Differential Equation) model is derived [18-21]. Let X(t)=(X₁(t), X₂(t), X₃(t), X₄(t))^T be a continuous random variable for (H(t), P(t), D(t), R(t))^T and T means the transpose of a matrix.

Let $\Delta X=X(t+\Delta t)-X(t)=\left(\Delta X_1, \Delta X_2, \Delta X_3, \Delta X_4\right)^T$ signify the random vector for the change in random variables during time interval $\Delta t$. Here, we'll create the transition maps for the SDE model, which outline all relevant transitions between states. Based on our ODE model system (1), we can see that the exist 12 possibilities between the states during the small interval $\Delta t$. We are discussed the state changes and their probabilities in Table 2.

Let us consider the case the recruitment of drug addictions becomes high class / poverty and drug addiction. In this case, the state change $\Delta X$ denotes by $\Delta X=(1,0,0,0)$ its probability of the change is given by:

$\begin{gathered}\operatorname{Prob}\left(\Delta X_1, \Delta X_2, \Delta X_3, \Delta X_4\right)= \\ (1,0,0,0) \mid\left(X_1, X_2, X_3, X_4\right)=P_1=\mu N \Delta t+o(\Delta t) .\end{gathered}$

One will rapidly to solve the expectation change $E(\Delta X)$ and its covariance matrix $V(\Delta X)$ connected with $\Delta X \mathrm{b}$ ignoring the terms higher than $o(\Delta t)$. The expectation of $\Delta X$ is given by:

$E(\Delta X)=\sum_{\mathrm{i}=1}^{12} \mathrm{P}_{\mathrm{i}}(\Delta \mathrm{X})_{\mathrm{i}} \Delta \mathrm{t}=\left(\begin{array}{c}\mu N-\left(\mu+\theta_1\right) X_1-\alpha_1 X_1 \frac{X_3}{N} \\ \theta_1 X_1-\left(\mu+\theta_2\right) X_2-\alpha_2 X_2 \frac{X_3}{N}+l_2 X_4 \\ \alpha_1 X_1 \frac{X_3}{N}+\alpha_2 X_2 \frac{X_3}{N}-\mathrm{kX}_3+l_1 X_4 \frac{X_3}{N} \\ \gamma X_3-l_1 X_4 \frac{X_3}{N}-l_2 X_4-\mu X_4\end{array}\right) \Delta t=f\left(X_1, X_2, X_3, X_4\right) \Delta t$                (2)

Here, the ODE system (1) can be noted that the expectation vector and the function f are in the same form. Since,

$\begin{gathered}\mathrm{V}(\Delta \mathrm{X})=\mathrm{E}\left((\Delta \mathrm{X})(\Delta \mathrm{X})^{\mathrm{T}}\right)-\mathrm{E}(\Delta \mathrm{X})\left(\mathrm{E}(\Delta \mathrm{X})^{\mathrm{T}}\right) \\ \text { and } E\left((\Delta X)(\Delta X)^T\right)=f(X)\left(f(X)^T\right) \Delta t\end{gathered}$

Table 2. The probability of possible states and changes

It can be approximately diffusion matrix $\mathrm{V}$ times $\Delta t$ by neglecting the term of $(\Delta t)^2$ such that $V(\Delta X) \approx$ $E\left((\Delta X)(\Delta X)^T\right)$

Hence,

$E\left((\Delta X)(\Delta X)^T\right)=\sum_{i=1}^{12} P_i\left((\Delta X)_i(\Delta X)_i^T\right) \Delta t=\left(\begin{array}{cccc}V_{11} & V_{12} & V_{13} & 0 \\ V_{21} & V_{22} & V_{23} & V_{24} \\ V_{31} & V_{32} & V_{33} & V_{34} \\ 0 & V_{42} & V_{43} & V_{44}\end{array}\right) \cdot \Delta \mathrm{t}=\Omega \cdot \Delta \mathrm{t}$             (3)

where, we solved the above diffusion matrix is symmetric, positive-definite and each component of this 4 x 4 matrix can be attained as follows:

$\begin{array}{cc} V_{11}=P_1+P_2+P_3+P_4, & V_{14}=V_{41}=0, \\ V_{22}=P_3+P_5+P_6+P_7, & V_{24}=V_{42}=-P_7 . \\ V_{33}=P_4+P_6+P_8+P_9+P_{10}, & V_{12}=V_{21}=-P_3, \\ V_{44}=P_7+P_8+P_{10}+P_{11} . & V_{23}=V_{32}=-P_6, \\ V_{13}=V_{31}=-P_4, & V_{34}=V_{43}=-P_8 . \end{array}$

We are following the approach considered in the research [18] and construct a matrix M such that V=MM^T, where M is a 4 x 10 matrix.

$\mathrm{M}=\left(\begin{array}{cccccccccc}\mathrm{m}_{101} & \mathrm{~m}_{102} & \mathrm{~m}_{103} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathrm{~m}_{202} & 0 & \mathrm{~m}_{204} & \mathrm{~m}_{205} & \mathrm{~m}_{206} & 0 & 0 & 0 & 0 \\ 0 & 0 & \mathrm{~m}_{303} & 0 & \mathrm{~m}_{305} & 0 & \mathrm{~m}_{307} & \mathrm{~m}_{308} & \mathrm{~m}_{309} & 0 \\ 0 & 0 & 0 & 0 & 0 & \mathrm{~m}_{406} & \mathrm{~m}_{407} & 0 & \mathrm{~m}_{409} & \mathrm{~m}_{410}\end{array}\right)$

where,

$\begin{aligned} m_{101} & =\sqrt{\mu N-\mu X_1}, & m_{305} & =\sqrt{\alpha_2 X_2 \frac{X_3}{N},} \\ m_{102} & =-\sqrt{\theta_1 X_1}, & m_{307} & =-\sqrt{\gamma X_3}, \\ m_{103} & =-\sqrt{\alpha_1 X_1 \frac{X_3}{N},} & m_{308}= & -\sqrt{\left(\mu+\mu_1\right) X_3}, \\ m_{202} & =\sqrt{\theta_1 X_1} & m_{309} & =\sqrt{l_1 X_4 \frac{X_3}{N}}, \\ m_{204} & =-\sqrt{\left(\mu+\theta_2\right) X_2}, & m_{406} & =-\sqrt{l_2 X_4}, \\ m_{205} & =-\sqrt{\alpha_2 X_2 \frac{X_3}{N},} & m_{407} & =\sqrt{\gamma X_3}, \\ m_{206} & =\sqrt{l_2 X_4}, & m_{409} & =-\sqrt{l_1 X_4 \frac{X_3}{N},} \\ m_{303} & =\sqrt{\alpha_1 X_1 \frac{X_3}{N},} & m_{410} & =-\sqrt{\mu X_4}\end{aligned}$

Now, the form of Ito-Stochastic differential model is given:

$d(X(t))=f\left(X_1, X_2, X_3, X_4\right) d t+M \cdot d W(t)$

with initial condition:

$X(0)=\left(X_1(0), X_2(0), X_3(0), X_4(0)\right)^T$

$\begin{aligned} & \text { and } \quad \text { a } \quad \text { Wiener } \quad \text { process, } \quad W(t)=\left(W_1(t), W_2(t), W_3(t), W_4(t), W_5(t), W_6(t), W_7(t), W_8(t)\right.  \end{aligned}$, $\left.W_9(t), W_{10}(t)\right) T$ Keeping in view of the above facts, we get the SDE model as follows:

$\begin{aligned} & d H=\left(\mu N-\left(\mu+\theta_1\right) X_1-\alpha_1 X_1 \frac{X_3}{N}\right) d t-\sqrt{\mu N-\mu X_1} d W_1-\sqrt{\theta_1 X_1} d W_2-\sqrt{\alpha_1 X_1 \frac{X_3}{N}} d W_3\end{aligned}$            (4)

$\begin{gathered}d P=\left(\theta_1 X_1-\left(\mu+\theta_2\right) X_3-\alpha_2 X_2 \frac{X_3}{N}+l_2 R\right) d t +\sqrt{\theta_1 X_1} d W_2-\sqrt{\left(\mu+\theta_2\right) X_2} d W_4 -\sqrt{\alpha_2 X_2 \frac{X_3}{N}} d W_5+\sqrt{\mathrm{l}_2 X_4} \mathrm{dW}_6\end{gathered}$            (5)

$\begin{gathered}d D=\left(\alpha_1 X_1 \frac{X_3}{N}+\alpha_2 X_2 \frac{X_3}{N}-k X_3+l_1 X_4 \frac{X_3}{N}\right) d t +\sqrt{\alpha_1 X_1 \frac{X_3}{N}} d W_3+\sqrt{\alpha_2 X_2 \frac{X_3}{N}} d W_5-\sqrt{\gamma X_3} d W_7 -\sqrt{\left(\mu+\mu_1\right) X_3} d W_8+\sqrt{l_1 X_4 \frac{X_3}{N}} d W_9\end{gathered}$            (6)

$\begin{aligned} & d R=\left(\gamma X_3-l_1 X_4 \frac{X_3}{N}-l_2 X_4-\mu X_4\right) d t \quad+\sqrt{\gamma X_3} d W_7-\sqrt{l_1 X_4 \frac{X_3}{N}} d W_9-\sqrt{\mu X_4} d W_{10}\end{aligned}$            (7)