MATHEMATICAL MODELING OF DRUG ABUSE, UNEMPLOYMENT AND MENTAL STRESS ON POPULATION DYNAMICS OF MENTAL ILLNESS

,


INTRODUCTION
Patients, professionals, and members of the general public all have different actions and emotional responses based on how they think the world views their experiences with mental diseases.
People's behavioral and emotional reactions are heavily influenced by their beliefs about the events that occur [1]. Alterations in one's emotional state, level of thought, or pattern of conduct are all symptoms of a mental disorder (or a combination of the three). They are non-communicable diseases (NCDs), according to Daud & Qing [2] related to distress and social, work, or family issues. Non-communicable diseases (NCDs) are presently the main health and development threat to people worldwide. Currently, the stratified heterogeneity of NCD fatalities is seldom addressed [3].
Through several studies conducted in High-Income Countries (HIC), there has been an adoption of effective systems and approaches to mental health. Several Low and Middle-Income Countries (LMIC) have attempted to address mental health challenges. The lack of enough resources and logistics to achieve a mental illness-free community remains a big challenge [4]. Mental disorders are distressing and disturbing and pose an enormous burden in terms of cost, morbidity, and mortality, according to Thakur & Roy [5], which delay the accomplishment of sustainable developmental goals for a country. Since addressing mental health is crucial to achieving universal health coverage, as outlined by the "Big Four Agenda" and necessary to realize Vision 2030, it has been thrust to the forefront of Kenyan policy priorities.
The World Health Organization (WHO) Global Mental Health Action Plan 2013-2020 alludes that mental health remains a key determinant of any country's overall health and socio-economic development. Regarding WHO, a variety of outcomes for individuals within a given society, such as healthier lifestyles; better physical health; improved recovery from illness; fewer limitations in daily living; higher education attainment; greater productivity, employment, and earnings; better family relationships; social cohesion and engagement and improved quality of life, largely depend on the state of mental health. Mathematical models have been utilized for years to study biological sciences to understand diverse aspects of non-communicable diseases such as diabetes mellitus [6]. transmission rates from the employed class Q(t) while 12 , and 3  transmission rates from the unemployed subpopulation P(t) to join either mental stress or drug abuse classes.
The study considered drug abuse and mental stress as the main stressors (factors) leading to mental illness. Though the state of employment has been captured, it only helps this study to identify the advanced effect it can cause on the "main stressors" leading to mental illness. Y(t) is thus taken as the total population of those individuals with mental illness at any given time t. the transmission rates to mental illness class Y(t) are 1 ,  11 , and 2  which are for those individuals suffering mental stress and abusing drugs regardless of the state of employment.
Mental illness does not cause death; it triggers the causes of death, such as violent actions and suicide, and the patient can recover from mental illness. This study considered that, mental illness patients only die through natural death  and those individuals  recovering from mental diseases have gone through proper and adequate counseling, undergo sufficient therapy, have completely healed from drug abuse, and are capable of accessing good health facilities with the right support system for patients as well as family care system and thus they cannot get mental disorders once again. The total population N(t) at any given time, t, is thus given by; where t [0, ] and > 0.


The portion of people who get a job at any given time t. 1 1  − The number of people without an appointment at any given time t

Modal Equation
The system of ordinary differential equations governing the S-M-P-Q-R-X-Y-Z mental illness model is given by the system of Equation (2) as;

MODEL ANALYSIS
Mathematical analysis of the formulated model system (2) is presented to in this section. The study shows the system of ordinary differential equation (2) governing the model is well-posed.

Positivity of the SMPQRXYZ Model
The positivity of the SMPQRXYZ Model is determined to ascertain the existence of all state variables on the real domain, .

Theorem 1:
 then the solution set of the modal system of equation (2) for the initial data set ( )( ) , , , , , , , Proof: Consider the equation on mental illness from the system of model equation (2): Equation (4) represents a first-order linear differential inequality and thus can be solved using the separation of variables to obtain: In the absence of mental illness and applying initial conditions, (6) become, By considering the same approach in the remaining seven equations in the system of model equation (2), the results are; Therefore, the set solutions ( ), ( ), ( ), ( ), ( ), ( ), ( ) and lie in the positive quadrant t  > 0, which proves the theorem.

3.2: Boundedness of the solution
The consequences restricting a population's growth are vital in analogy to a dynamic population system [7]. To study the boundedness of the solution of the system around the steady states, all the state variables and parameters of the S-M-P-Q-R-X-Y-Z Mental Illness Modal are assumed to be positive t  > 0. In this regard, boundedness is determined by the following theorem: Substituting model equation (2) to equation (15) and simplifying further yields; If there is no mental illness at any given time ,0 t  = and thus (17) reduces to Using integrating factors method, we proceed as follows; , By product rule and taking μ εμ; the flow generated by the model can thus be considered for analysis since the S-M-P-Q-R-X-Y-Z Mental Illness Modal is mathematically well-posed and biologically meaningful.

Steady States
To study the stability of the proposed SMPQRXYZ model, the equilibrium points of the system need to be determined. Let 1     = + + , 5 1 2     = + + , 6 1 2 3      = + + + and 7    =+ , then substitute in system (2) to form system (25). Based on different mental illness stressors considered in this study, the possible equilibrium points considered are: Case 1: All stress factors of mental illness exist; The equilibrium point is   ; the mental stress free-equilibrium point   3  3  3  3  3  3  3 ( , , , , , )

Stability analysis
The stability analysis of the equilibrium points is determined for the proposed model. The study considered the local stability of 0 1 2 3 , , , E E E E and 4 E . The global stability of 3 E and 4 E is also studied to ascertain which stress factor(s) pose a greater impact on mental illness.

Local stability
A variation matrix is constructed, and the nature of the eigenvalues of the matrix is determined. A point is said to be locally asymptomatic stable if the eigenvalues of the variation matrix are negative, otherwise unstable.

4.5.2: Global Stability
The Lyapunov function below is proposed to determine the global stability by performing analysis near the equilibrium points of the system. Considering mental illness caused by mental stress and drug abuse, two scenarios were considered ignored to study for global stability.
The system will be globally asymptotically stable if The system will be globally asymptotically stable if

NUMERICAL SIMULATIONS
In this section, the numerical analysis is done with the help of Wolfram Mathematica to substantiate the analytical findings. Values for several parameters were calculated based on the literature study, and these estimates are cited in Table 2. The study relied heavily on previously publish ed works related to this study due to difficulty and unavailability of data on mental illness

DISCUSSION OF THE RESULTS
In this study, state of employment, substance abuse and mental stress are selected as the main factors triggering mental illnesses. Figure 2 describes the dynamic simulation of the total population with respect to time for our modal using the original parameters cited in Table 2 this study noticed similar effects. The population of those abusing drugs increased significantly as the value of ( θ 3 ) increased and vice versa. Unemployed population was found to be more susceptible to drugs(substance) abuse due to idleness, peer pressure, and criminal cases.
To study the effects of unemployment on mental stress, the study varied the value of (θ 3 ), which directly affected the number of those with a source of income, as shown in figure 5 below. There was a sharp increase in the unemployed population with mental stress M(t) as time t increased, but this trend was seen to attain a maximum point and then dropping significantly with time t.
Moreover, a slight change in (θ 3 ) showed the similar trend with numbers in M(t) class vary proportionally as (θ 3 ) increased. This indicated an increase in the mentally stressed population as more people lost a source of income.

CONCLUSION
Based on eight differential equations(ODE's) described in system (2), an SMPQRXYZ mathematical model to study mental disease was established. Eight diverse categories are established to reflect distinct subpopulations, including the vulnerable, the employed, the jobless, drug abusers, the psychologically stressed, and the mentally sick. Employment status is often believed to be a primary determinant in substance addiction and mental health problems.
Positiveness, boundedness, and local and global stability analyses of the equilibrium point were determined to establish the well-possedness of the mental disease model's equations.
The study constructed five equilibrium points of the system 0   The study suggests that parameters causing mental stress ( 1 1 1 2 3 , , , ,      ) and substance abuse ( 2 2 2 2 3 , , , ,      ) within a given population set up to be controlled for their great impact on mental illness. A clear policy on employment for those graduating from formal education and improvement in the general working condition for the employees will lead to reduced cases of drug abuse and mental stress which will help reduce mental illness within the community. Doing real data fitting, Modelling, and simulating the competition of mental stress and substance abuse on mental illness using Holing type II response would be an interesting area for future studies.

RECOMMENDATION
There are not enough records for people with mental illness for several reasons. Disconcerting follow-up visit costs, wasted time in overcrowded mental health clinics, and a rise in inaccurate diagnosis-especially from so-called "quacks" in the psychiatric and medical communities-have made patients wary of seeing their psychiatrists for treatment updates. Classifiers identify instances as normal, addiction-free, dependence-with-conditions, or disorder-free. On the other hand, there has been a recent increase in cases of personality disorders, mental stress, and drug misuse.
While the government of Kenya has made some attempts to improve mental health via the ministry of health (MOH), as seen by the 'KENYA MENTAL HEALTH ACTION PLAN 2021-2025(KMHAP)', researchers still lack access to adequate data on mental health. As one of the LMICs, Kenya's Ministry of Health (MOH) is urged by this study to make its mental health data publicly available without compromising patients' right to privacy, to fully implement the policies outlined in the Kenya Mental Health Action Plan (KMHAP) for the years 2021 through 2025, and to make available sufficient capitation to fund researchers engaged in mathematical and biological modeling. This will be a nice way to inspire more people to study mental health in order to provide a precise mathematical forecast of future trends, which can then be used to inform improved mental health policies that will help the community as a whole thrive.
Mathematical modeling and simulation of drug addiction and mental stress on population dynamics of mental disease need more study in a number of areas. In the first place, we need 25 POPULATION DYNAMICS OF MENTAL ILLNESS further study into the connection between substance abuse and the development, prevalence, and intensity of mental diseases. Stress on the mind may have an effect on the beginning, development, and severity of mental disease, hence studies examining this connection are necessary.
Second, there is a need for further study into the population dynamics of mental disease, especially with regards to the manifestation and variation of mental illnesses among various demographic groups and geographic locations. Population-level variables connected to the frequency and severity of mental illness might be determined by studying the prevalence and severity of mental illnesses in various geographical locations and demographic groups.
Lastly, greater study is required to enhance current models of mental disease and investigate new models that more accurately portray the intricacies of mental health issue etiology, clinical presentation, and treatment. To better understand the interaction between many variables affecting mental health at the individual, group, or regional levels, such models might combine elements of epidemiology, system dynamics, and population-level dynamics.