Question 3: Consider the following dynamical interaction of giving-up smoking model with constant birth rate À for the potential smoker individual: = A-BVPL-(d+μ)P, dt dL BVPL-(y+d+µ)L, dt ds = L-(8+d+u)S, dt dQ = 8S - (μ+d)Q, dt where P(t), L(t), S(t), and Q(t) denote numbers of potential smokers, occasional smokers, smokers and quit smokers at time t, respectively. The parameter u is natural death rate, y is recovery rate from infection, 3 is the transmission coefficient, & is quit rate of smoking, d represents death rate for potential smokers, occasional smokers, smoker and quit smoker related to smoking disease. The total population size at time t is N(t) = P(t) + L(t) + S(t) + Q(t). dN 1. Show that -=A-(μ+d) N. dt 2. Find the exact solution of this equation given that N(0) = No. 3. Find the disease free equilibrium of this model (smoking is considered as a disease or infection).

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