Cigarette smoking on college campuses: an epidemical modelling approach

Abstract

Cigarette smoking on college campuses has become a significant public health issue, which in turn led to an increasing focus on establishing programs to reduce its prevalence. In this paper, a compartmental model depicting the spread and cessation of the smoking habit on college campuses, obtained using theoretical principles often employed in mathematical epidemiology, is proposed and analysed. The existence and stability of the habitual smoking-free and habitual smoking-persistent equilibria, respectively, are explored in terms of a threshold parameter, hereby called the smokers generation number and denoted by $R_c$. A sensitivity analysis indicates that $R_c$ is the most sensitive to the contact rate between habitual-smokers and occasional-smokers and to the rate of successfully quitting smoking. Numerical simulations of the proposed optimal control strategies reveal that the most effective approach to reduce the prevalence of cigarette smoking and possibly achieve a smoking-free campus should combine both control measures, namely allocating mandatory smoking rooms together with educating the public on the harmful effects of smoking and providing large scale guidance, counselling and support therapy to help students quit smoking.

Appendix

1.1 A1: Invariant region and positivity of solutions

Theorem 3

If $O_s(0)$, $S_s(0)$ and $Q_s(0)$ are non-negative, then $O_s(t)$, $S_s(t)$ and $Q_s(t)$ are also non-negative for all $t>0$. Moreover, the region ${\mathcal {D}}$ given by

$$\begin{aligned} {\mathcal {D}} = \{ (O_s(t), S_s(t), Q_s(t)) \in R^3_{+}: O_s(t) + S_s(t) + Q_s(t) \le \dfrac{\varLambda }{g} \} \end{aligned}$$

is positively invariant.

Proof

Let $(O_s(0), S_s(0), Q_s(0))$ be a set of non-negative initial conditions and denote by $\ [ 0, t_{\max })$ the maximum interval of existence of the corresponding solution. Let $t_1 = \sup \{0< t < t_{\max }: O_s, S_s \, \text {and} \, Q_s \, \text {are positive on} \ [0, t\ ] \}.$ Since $O_s(0)$, $S_s(0)$, and $Q_s(0)$ are non-negative, then $t_1 > 0$.

If $t_1 < t_{\max }$ then, by using the variation of constants formula to the first equation of the system (1), we obtain

$$\begin{aligned} O_s(t_1) = {\mathcal {U}}(t_1, 0)O_s(0) + \int _{0}^{t_1} \varLambda {\mathcal {U}}(t_1, \omega )d\omega , \end{aligned}$$

where

$$\begin{aligned} {\mathcal {U}}(t_1, \omega ) = e^{\int _{\omega }^{t_1}(\beta + g)(O_s)dO_s}. \end{aligned}$$

This implies that $O_s(t_1) > 0.$ It can be shown in the same manner that this is the case for the other variables as well. This contradicts the fact that $t_1$ is the supremum because at least one of the variables should be equal to zero at $t_1$. Therefore $t_1 = t_{\max }.$

Let $N(t) = O_s(t) + S_s(t) + Q_s(t)$. Summing up all equations of (1), we have $\dfrac{dN(t)}{dt}(t) = \varLambda - gN(t).$ This implies

$$\begin{aligned} N(t)=\dfrac{\varLambda }{g}+\left( N(0)-\dfrac{\varLambda }{g}\right) e^{-gt}. \end{aligned}$$

Then $N(t) \le \dfrac{\varLambda }{g}$ if $N(0) \le \dfrac{\varLambda }{g}$, which establishes the invariance of ${\mathcal {D}}$ as required. $\square$

1.2 A2: The smokers generation number $R_c$

The smokers generation number for the system (1) is obtained using the next generation method. For this system, ${\mathcal {F}}$ and ${\mathcal {V}}$ are given by

$$\begin{aligned} {\mathcal {F}} = \left[ \begin{array}{c} \beta O_s S_s \end{array} \right] , \qquad {\mathcal {V}} = \left[ \begin{array}{c} (\gamma + g) S_s \end{array} \right] . \end{aligned}$$

The associated Jacobian matrices of ${\mathcal {F}}$ and ${\mathcal {V}}$ at the smoking-free equilibrium are given by

$$\begin{aligned} F = \left[ \begin{array}{c} \dfrac{\beta \lambda (g\delta + \tau )}{g(g + \tau )} \end{array} \right] , \qquad V = \left[ \begin{array}{c} (\gamma + g) \end{array} \right] . \end{aligned}$$

The smokers generation number $R_c$ equals the spectral radius (dominant eigenvalue) of the matrix $FV^{-1}$, given by:

$$\begin{aligned} R_c = \dfrac{\beta \varLambda (g \delta + \tau )}{g (\gamma + g)(g + \tau )}. \end{aligned}$$

(13)

1.3 A3: The stability of the habitual smoking-persistent equilibrium

Theorem 4

The habitual smoking-persistent equilibrium $E^*$ is locally asymptotically stable if $R_c > 1$ and unstable if $R_c < 1$.

Proof

The coordinates of the habitual smoking-persistent equilibrium $E^*$, obtained by solving the equilibrium relations associated to the system (1), are

$$\begin{aligned} O^*_s&= \dfrac{g + \gamma }{\beta }, \end{aligned}$$

(14)

$$\begin{aligned} S^*_s&= \dfrac{(g + \tau )(\gamma + g)(R_c - 1)}{\beta (g + \tau + \gamma )}, \end{aligned}$$

(15)

$$\begin{aligned} Q^*_s&= \dfrac{\beta \varLambda \bigg ( g(1 - \delta ) + \gamma \bigg ) - g\gamma (g + \gamma )}{g\beta (g + \tau + \gamma )} . \end{aligned}$$

(16)

The Jacobian matrix of the system (1), evaluated at the habitual smoking-persistent equilibrium $E^{*}$ is

$$\begin{aligned} J(O^*_s, S^*_s, Q^*_s) = \left[ \begin{array}{c c c} -\dfrac{(g + \tau )(\gamma + g)(R_c - 1)}{g + \tau + \gamma } - g &{} -(\gamma + g) &{} \tau \\ \dfrac{(g + \tau )(\gamma + g)(R_c - 1)}{g + \tau + \gamma } &{} 0 &{} 0\\ 0 &{} \gamma &{} -(g + \tau ) \end{array} \right] . \end{aligned}$$

Solving for the eigenvalues of the Jacobian matrix at the habitual smoking-persistent equilibrium, we obtain the characteristic polynomial:

$$\begin{aligned} p(\lambda ) = \lambda ^3 + A_2 \lambda ^2 + A_1 \lambda + A_0 , \end{aligned}$$

(17)

in which

$$\begin{aligned} A_0&= (R_c -1)g(\gamma + g)(g + \tau ), \end{aligned}$$

(18)

$$\begin{aligned} A_1&= \dfrac{(g + \tau )(g^2 + (\tau + \gamma )g + \gamma (\tau + \gamma ))(R_c - 1)}{g + \tau + \gamma } + \dfrac{(g + \tau )(g^2 + (\tau + \gamma )g)}{g + \tau + \gamma }, \end{aligned}$$

(19)

$$\begin{aligned} A_2&= \dfrac{(g^2 + (\tau + \gamma )g + \gamma \tau )(R_c - 1)}{g + \tau + \gamma } + \dfrac{g^2 + (\tau + \gamma )g + \tau (\tau + \gamma )}{g + \tau \gamma }. \end{aligned}$$

(20)

We use the Routh - Hurwitz stability criterion to determine the stability of the characteristic polynomial equation associated to (17). From this criterion, if conditions

$$\begin{aligned} A_2> 0, \quad A_0> 0, \quad A_1 A_2 > A_0 \end{aligned}$$

hold true, then all the roots of the characteristic polynomial equation have negative real parts which means that the habitual smoking-persistent equilibrium is stable.

From equations (18) and (20) above, the first two conditions are true for $R_c > 1$, as in this case $A_2$ and $A_0$ are both positive quantities. The third condition reduces to

$$\begin{aligned}&\bigg [g^4 + (\gamma + \tau )g^3 + (\gamma + \tau )^2 g^2 + (\gamma + \tau )(\gamma \tau + \gamma ^2)g + (\gamma + \tau )\tau \gamma ^2\bigg ](g + \tau )(R_c - 1)^2 \\&+\bigg [g^4 + (\gamma + \tau )g^3 + (\gamma \tau + \tau ^2) g^2 + (\gamma + \tau )(\gamma \tau + \tau ^2)g + (\gamma + \tau )^2 \tau \gamma \bigg ](g + \tau )(R_c - 1) \\&+ \bigg [g^4 + (\gamma + \tau )g^3 + (\gamma + \tau )^2 g^2 + (\gamma + \tau )(\gamma \tau + \tau ^2)g \bigg ](g + \tau )>0 \end{aligned}$$

which holds true for all parameter values such that $R_c > 1$. Thus, the habitual smoking-persistent equilibrium is stable when $R_c > 1$ by the Routh - Hurwitz criterion. $\square$