Chapter 7.8, Problem 81E

In a study of the spread of illicit drug use from an enthusiastic user to a population of N users, the authors model the number of expected new users by the equation

$\gamma ={\displaystyle {\int }_0^{\infty }\frac{cN\left(1-e^{-kt}\right)}{k}}{e}^{-\lambda t}dt$

where c, k and λ are positive constants. Evaluate this integral to express γ io tenns of c, N, k, and λ.

Source: F. Hoppensteadt et al., ‘“Threshold Analysis of a Drug Use Epidemic Model,” Mathematical Biosciences 53 (1981): 79‒87.