SEIHR EPIDEMIOLOGICAL MODEL: A MATHEMATICAL DYNAMICS FOR TREATING DRUG ABUSE IN SOCIETY

Abstract

This study presents a deterministic mathematical model of SEIHR (Susceptible, Exposed, Infected/Addict, Hospitalized/Treated, Recovered).  In the SEIHR model, each compartment in this system represents the following: S (Susceptible): Non-addicted individuals who have not been exposed to drugs or addicts and have not started using. They can move to compartment E through social interactions with addicts in compartment I. E (Exposed): Individuals who have been exposed and have begun experimenting, but have not yet become fully addicted or socially "infected." This compartment represents the behavioral latency stage 7 before developing into active addiction. I (Infected/Addict): Individuals who have reached a state of active addiction and are the primary source of "social contagion" to the susceptible population. Compound H (Healthy or Hospitalized): During the time an individual is in compartment H, they are outside the cycle of active addiction, giving them temporary "immunity" from social contagion or exposure to the risks of active addiction. R (Recovered): Individuals who have recovered from active addiction (either through treatment H or self-healing ). They may relapse and return to a state of addiction I (similar to SEIRS models that suggest temporary immunity). The age-structured SEIHR basic reproduction number, ℝ₀, is computed using the next-generation matrix approach and established to be ℝ₀ =0.2162.    By analyze the suggested system by finding it's equilibrium points which are two equilibrium points, SEIHR, E⁰, and a unique SEIHR -endemic equilibrium point, E, are established, and after that their stability are checked locally. E⁰ stable whenever ℝ₀ ≤ 1 The local bifurcation is also shown which happens when the equilibrium points are changed from stable to unstable or vice versa. The positivity of the functions S(t), E(t), I(t), H(t) and R(t) as solutions of the system is proved. The uniformly boundedness of the solutions of the system under consideration is also proved. The basic reproduction number is used to prove the stability of all equilibrium points as well as the method of the nature of the eigen values of the Jacobian at each equilibrium point.  Sotomayor's theorem is used to show that the system has saddle node bifurcation. As a result of this paper the susceptible population S(t) increasing rapidly at the beginning because they don't know the danger then it stabilizes acceptable number similar interpretation is given for E(t), I(t), H(t) and R(t).