Our objective in this section is to extend the initial model to include two intervention methods, called controls, represented as functions of time and assigned reasonable upper and lower bounds, each representing a possible method of rumor intervention.
Historically, rumor outbreaks have tended to reach the attention of authorities only after transmission has been amplified by inadequate infection control. Conversely, as with any breaking news story, information is often fluid and the authoritative media updated the story with the official explanation as soon as it was provided. If there is a rumor spread in emergency event, the official has the obligation to state whether it is true or not, it helps the person to understand the situation better. After the 9magnitude earthquake in Fukushima, incurring nuclear leakage accidents in 2011, some rumors said that taking materials containing iodine could help ward of nuclear radiation, which led to the public rushing for everything containing iodine, such as Chinese snapping up iodized salt, Americans rushing for iodine pills, Russians hoarding iodine, and Korean residents rushing for seaweed. The science knowledge is that eating iodized salt cannot prevent people from radiation, which encourages lurkers to protect themselves from the rumor and attempt to convert lurkers into stiflers. Official media as authority announcing the news, after receiving the true information, the public will not be confused by the rumors. It is imperative that announcements be enacted and executed to stop the production and transmission of the rumors in emergency event and attempt to convert spreaders into stiflers.
Generally, the lurker individual becomes a spreader when being convinced of the truth of the rumor and then decides to inform others. However, note that ‘a convinced’ lurker can possibly refuse to spread the rumor, or alternately a spreader can lose interest in the rumor and then decide not to spread the rumor any further. In these two situations, both become stiflers. A stifler is therefore either an individual who knows the rumor but who is not spreading it or a spreader who, with time, loses interest and is no longer spreading the rumor. Furthermore, one can notice that the tendency of accepting a rumor as credible information differs from one lurker to another. This can be explained by the strong background knowledge that some of the lurker individuals possess. These types of lurker individuals, once aware of the rumor, generally raise some reasonable questions and/or logical arguments in order to assess the credibility or the validity of the rumor. Science education is therefore among the factors that also contribute to the cessation of a rumor spreading and is an important aspect that has not been considered in previous studies.
We will integrate the essential components into one XWYZtype model to accommodate the dynamics of rumor outbreak determined by populationspecific parameters such as the effect of contact reduction when infectious and stifler individuals are reported in the official media.
Let \(u_{\eta}\) and \(u_{\gamma}\) be the control variables for science education and official media coverage, respectively, where \(\eta=1\theta\). Thus, model (2) now reads
$$ \begin{cases} \frac{dx}{d t}=1\beta xyx,\\ \frac{dw}{d t}=\beta xy\theta\alpha w u_{\eta}\eta \alpha ww,\\ \frac{dy}{d t}=\theta\alpha w\lambda(y+z)y(1u_{\gamma}) yy,\\ \frac{dz}{d t}=u_{\eta}\eta\alpha w+\lambda(y+z)y+(1u_{\gamma}) yz. \end{cases} $$
(28)
The balance of multiple intervention methods can differ between populations. A successful mitigation scheme is one which reduces rumor infectious with minimal cost. A control scheme is assumed to be optimal if it maximizes the objective functional
$$ W\bigl(u_{\eta}(t), u_{\gamma}(t)\bigr)=\int ^{t_{f}}_{t_{0}}\bigl[B_{0}\bigl(x(t)+z(t) \bigr)B_{1} y(t)B_{2}\bigl({u_{\eta}(t)}^{2}+{u_{\gamma}(t)}^{2} \bigr)\bigr] \,dt. $$
(29)
The first two terms represent the benefits of the ignorant and stifler populations. The parameters \(B_{0}\) represent the weight constraints for the ignorant and stifler populations, \(B_{1}\) and \(B_{2}\) represent the weight constraints for the infected population and the control, respectively. They can also represent balancing coefficients transforming the integral into dollars expended over a finite time period of \([t_{0}, t_{f} ]\). The goal is to maximize the populations of ignorant and stifler individuals, minimize the population of infectives, and maximize the benefits of official media coverage and vaccination, while minimizing the systemic costs of both rumor vaccination and official media coverage. The terms \(B_{2} {u_{\eta}(t)}^{2}\) and \(B_{2} {u_{\gamma}(t)}^{2}\) represent the maximal cost of education, implementation and campaigns on both rumor vaccination and official media coverage. \(x(t)\) and \(z(t)\) account for the fitness of the ignorant and the stifler groups. We thus seek optimal controls
$$ W\bigl(u^{\ast}_{\eta}(t),u^{\ast}_{\gamma}(t) \bigr)=\max\bigl[W\bigl(u_{\eta}(t), u_{\gamma}(t)\bigr) \bigl(u_{\eta}(t), u_{\gamma}(t)\bigr)\in U\bigr], $$
(30)
where \(U=\{u_{\eta}, u_{\gamma} u_{\eta}, u_{\gamma}\mbox{ measurable, } 0\leq a_{11}\leq u_{\eta}\leq b_{11}\leq1, 0\leq a_{22}\leq u_{\gamma}\leq b_{22}\leq1, t\in[t_{0}, t_{f}] \}\). The basic framework of this problem is to characterize the optimal control. The existence of an optimal control can be obtained by using a result by Joshi [35].
Theorem 5.1
Consider the control problem with the system of equations (9)(12). There exists an optimal control, such that
\(\max\{W(u_{\eta}(t), u_{\gamma}(t))(u_{\eta}(t), u_{\gamma}(t))\in U\}=W(u^{\ast}_{\eta}(t),u^{\ast}_{\gamma}(t))\).
Proof
To prove this theorem on the existence of an optimal control, we use a result from Fleming and Rishel [36] (Theorem 4.1 pp.6869), where the following properties must be satisfied: (I) The set of controls and corresponding state variables is nonempty; (II) the control set U is closed and convex; (III) the righthand side of the state system is bounded above by a linear function in the state and control; (IV) the integrand of the functional is concave on U and is bounded above by \(c_{2}c_{1} (u_{\eta}(t)^{2}+u_{\gamma}(t)^{2})\), where \(c_{1}>0\), \(c_{2}>0\), and \(k>1\).
An existence result in Lukes [37] (Theorem 9.2.1) for the system of equations (28) for bounded coefficients is used to give the first condition. The control set is closed and convex by definition. The righthand side of the state system satisfies Condition III since the state solutions are a priori bounded. The integrand in the objective functional, \(B_{0}(x(t)+z(t))B_{1} y(t)B_{2}({u_{\eta}(t)}^{2}+{u_{\gamma}(t)}^{2})\) is concave on U. Furthermore, \(c_{1}>0\), \(c_{2}>0\), and \(k>1\), so
$$ B_{0}\bigl(x(t)+z(t)\bigr)B_{1} y(t)B_{2}\bigl({u_{\eta}(t)}^{2}+{u_{\gamma}(t)}^{2} \bigr) \leq c_{2}c_{1} \bigl(\bigl u_{\eta}(t)\bigr^{2}+\biglu_{\gamma}(t)\bigr^{2}\bigr). $$
(31)
Therefore, the optimal control exists, since the lefthand side of (31) is bounded; consequently, the states are bounded.
Since there exists an optimal control for maximizing the functional (29) subject to (28), we use Pontryagin’s maximum principle to derive the necessary conditions for this optimal control. Pontryagin’s maximum principle introduces adjoint functions that allow us to attach our state system (of differential equations), to our objective functional. After first showing the existence of optimal controls, this principle can be used to obtain the differential equations for the adjoint variables, corresponding boundary conditions, and the characterization of an optimal control \(u^{\ast}_{\eta}(t)\), \(u^{\ast}_{\gamma}(t)\). This characterization gives a representation of an optimal control in terms of the state and adjoint functions. Also, this principle converts the problem of minimizing the objective functional subject to the state system into minimizing either the Lagrangian or the Hamiltonian with respect to the controls (bounded measurable functions) at each time t.
The Lagrangian is defined as
$$\begin{aligned} L =& B_{0}\bigl(x(t)+z(t)\bigr)B_{1} y(t)B_{2}\bigl({u_{\eta}(t)}^{2}+{u_{\gamma}(t)}^{2}\bigr) +\rho_{1} [1\beta xyx] \\ &{}+\rho_{2} [\beta xy\theta\alpha w u_{\eta}\eta \alpha ww] \\ &{}+\rho_{3} \bigl[\theta\alpha w\lambda(y+z)y(1u_{\gamma}) yy \bigr] \\ &{}+\rho_{4} \bigl[u_{\eta}\eta\alpha w+\lambda(y+z)y+(1u_{\gamma}) yz\bigr] \\ &{}+\omega_{11}\bigl(a_{11}u_{\eta}(t)\bigr) + \omega_{12}\bigl(u_{\eta}(t)b_{11}\bigr) \\ &{}+\omega_{21}\bigl(a_{22}u_{\gamma}(t)\bigr) + \omega_{22}\bigl(u_{\gamma}(t)b_{22}\bigr), \end{aligned}$$
(32)
where \(\omega_{11}\geq0\), \(\omega_{12}\geq0\) are penalty multipliers satisfying \(\omega_{11}(a_{11}u_{\eta}(t)) +\omega_{12}(u_{\eta}(t)b_{11})\) at optimal \(u^{\ast}_{\eta}(t)\), and \(\omega_{21}\geq0\), \(\omega_{22}\geq0\) are penalty multipliers satisfying \(\omega_{21}(a_{22}u_{\gamma}(t)) +\omega_{22}(u_{\gamma}(t)b_{22})\) at optimal \(u^{\ast}_{\gamma}(t)\).
Given optimal controls \(u^{\ast}_{\eta}(t)\) and \(u^{\ast}_{\gamma}(t)\), and solutions of the corresponding state system (28), there exist adjoint variables \(\rho _{i}\), for \(i=1,2,3,4\), satisfying the following equation:
$$ \begin{cases} \frac{d \rho_{1}}{d t }=\frac{\partial L}{\partial x }=B_{0}+(\rho_{1}\rho _{2}) \beta y+\rho_{1},\\ \frac{d \rho_{2}}{d t }=\frac{\partial L}{\partial w }=(\rho_{2}\rho_{3}) \theta\alpha+(\rho_{2}\rho_{4}) u_{\eta}\eta\alpha+\rho_{2},\\ \frac{d \rho_{3}}{d t }=\frac{\partial L}{\partial y }=B_{1}+ (\rho _{1}\rho_{2})\beta x+(\rho_{3}\rho_{4}) (2 \lambda y +\lambda z+(1u_{\gamma})) +\rho_{3},\\ \frac{d \rho_{4}}{d t }=\frac{\partial L}{\partial z }=B_{0}+ (\rho _{3}\rho_{4}) \lambda y +\rho_{4}, \end{cases} $$
(33)
with transversality conditions \(\rho_{i}(t_{f})\), for \(i=1,2,3,4\). To determine the interior maximum of our Lagrangian, we take the partial derivatives of L with respect to \(u_{\eta}(t)\) and \(u_{\gamma}(t)\), respectively, and set them to zero. Thus,
$$ \begin{cases} \frac{\partial L}{\partial u_{\eta}(t)}=2B_{2}u^{\ast}_{\eta}(t)+(\rho _{4}\rho_{2}) \eta\alpha w\omega_{11}+\omega_{12},\\ \frac{\partial L}{\partial u_{\gamma}(t)}=2B_{2}u^{\ast}_{\gamma}(t)+(\rho _{3}\rho_{4}) y\omega_{21}+\omega_{22}. \end{cases} $$
(34)
To determine an explicit expression for our controls \(u^{\ast}_{\eta}(t)\) and \(u^{\ast}_{\gamma}(t)\) (without \(\omega_{11}\), \(\omega_{12}\), \(\omega _{21}\), \(\omega_{22}\)), a standard optimality technique is utilized. The following cases are considered to determine the specific characterization of the optimal control.
Case 1: Optimality of \(u^{\ast}_{\eta}(t)\)

1.
We consider the set \(\{t a_{11}\leq u^{\ast}_{\eta}(t) \leq b_{11} \}\), \(\omega_{11}=\omega_{12}=0\). Hence, the optimal control is
$$ u^{\ast}_{\eta}(t)=\frac{(\rho_{4}\rho_{2}) \eta\alpha w}{2B_{2}}. $$
(35)

2.
We consider the set \(\{t a_{11}=u^{\ast}_{\eta}(t)\}\), \(\omega _{11}=0\). We have
$$ u^{\ast}_{\eta}(t)=\frac{(\rho_{4}\rho_{2}) \eta\alpha w + \omega_{12} }{2B_{2}} $$
(36)
or
$$ u^{\ast}_{\eta}(t)=\frac{(\rho_{4}\rho_{2}) \eta\alpha w }{2B_{2}}\leq a_{11} $$
(37)
since \(\omega_{12}\geq0\).

3.
We consider the set \(\{t b_{11}=u^{\ast}_{\eta}(t)\}\), \(\omega _{12}=0\). We have
$$ u^{\ast}_{\eta}(t)=\frac{(\rho_{4}\rho_{2}) \eta\alpha w  \omega_{11} }{2B_{2}} $$
(38)
or
$$ u^{\ast}_{\eta}(t)=\frac{(\rho_{4}\rho_{2}) \eta\alpha w }{2B_{2}}\geq b_{11}. $$
(39)
Combining all the three subcases in a compact form gives
$$ u^{\ast}_{\eta}(t)=\min\biggl\{ \max \biggl\{ a_{11} \frac{(\rho_{4}\rho_{2}) \eta\alpha w }{2B_{2}} \biggr\} , b_{11} \biggr\} . $$
(40)
Case 2: Optimality of \(u^{\ast}_{\gamma}(t)\)

1.
We consider the set \(\{t a_{22}\leq u^{\ast}_{\gamma}(t) \leq b_{22} \}\), \(\omega_{21}=\omega_{22}=0\). Hence, the optimal control is
$$ u^{\ast}_{\gamma}(t)=\frac{(\rho_{3}\rho_{4}) y}{2B_{2}}. $$
(41)

2.
We consider the set \(\{t a_{22}=u^{\ast}_{\gamma}(t)\}\), \(\omega _{21}=0\). We have
$$ u^{\ast}_{\gamma}(t)=\frac{(\rho_{3}\rho_{4}) y+\omega_{22}}{2B_{2}} $$
(42)
or
$$ u^{\ast}_{\gamma}(t)=\frac{(\rho_{3}\rho_{4}) y+\omega_{22}}{2B_{2}}\leq a_{22} $$
(43)
since \(\omega_{22}\geq0\).

3.
We consider the set \(\{t b_{22}=u^{\ast}_{\gamma}(t)\}\), \(\omega _{22}=0\). We have
$$ u^{\ast}_{\gamma}(t)=\frac{(\rho_{3}\rho_{4}) y\omega_{21}}{2B_{2}} $$
(44)
or
$$ u^{\ast}_{\gamma}(t)=\frac{(\rho_{4}\rho_{2}) \eta\alpha w }{2B_{2}}\geq b_{22}. $$
(45)
Combining all the three subcases in a compact form gives
$$ u^{\ast}_{\gamma}(t)=\min\biggl\{ \max \biggl\{ a_{22}, \frac{(\rho_{4}\rho_{2}) \eta\alpha w }{2B_{2}} \biggr\} , b_{22} \biggr\} . $$
(46)
The optimality system consists of the state system coupled with the adjoint system, with the initial conditions, the transversality conditions and the characterization of the optimal control:
$$ \begin{cases} \frac{dx}{d t}=1\beta xyx,\\ \frac{dw}{d t}=\beta xy\theta\alpha w u_{\eta}\eta \alpha ww,\\ \frac{dy}{d t}=\theta\alpha w\lambda(y+z)y(1u_{\gamma}) yy,\\ \frac{dz}{d t}=u_{\eta}\eta\alpha w+\lambda(y+z)y+(1u_{\gamma}) yz,\\ \frac{d \rho_{1}}{d t }=B_{0}+(\rho_{1}\rho_{2}) \beta y+\rho_{1},\\ \frac{d \rho_{2}}{d t }=(\rho_{2}\rho_{3}) \theta\alpha+(\rho_{2}\rho_{4}) u_{\eta}\eta\alpha+\rho_{2},\\ \frac{d \rho_{3}}{d t }=B_{1}+ (\rho_{1}\rho_{2})\beta x+(\rho_{3}\rho_{4}) (2 \lambda y +\lambda z+(1u_{\gamma})) +\rho_{3},\\ \frac{d \rho_{4}}{d t }=B_{0}+ (\rho_{3}\rho_{4}) \lambda y +\rho_{4}, \end{cases} $$
(47)
where \(u^{\ast}_{\eta}(t)\) and \(u^{\ast}_{\gamma}(t)\), are given by (40) and (46), respectively, with \(x(0)=x_{0}\), \(w(0)=w_{0}\), \(y(0)=y_{0}\), \(z(0)=z_{0}\), and \(\rho_{i}(t_{f})=0\), for \(i = 1,\ldots,4\). Due to the a priori boundedness of the state and adjoint functions and the resulting Lipschitz structure of the ODEs, we obtain the uniqueness of the optimal control for small \([t_{f}]\). The uniqueness of the optimal control follows from the uniqueness of the optimality system. The state system of differential equations and the adjoint system of differential equations together with the control characterization above form the optimality system solved numerically and depicted in the next section. □