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 influenza intervention. Using optimal control theory and numerical simulations, we determine the benefit of vaccination and media coverage when the latter has positive or negative effect on the former.
We will integrate the essential components into one SIVR-type model to accommodate the dynamics of an influenza outbreak determined by population-specific parameters such as the effect of contact reduction when infectious and vaccinated individuals are reported in the media.
be the control variables for vaccination and media coverage respectively. Thus, model (1)-(4) now reads
A balance of multiple intervention methods can differ between populations. A successful mitigation scheme is one which reduces influenza-related deaths with minimal cost. A control scheme is assumed to be optimal if it maximizes the objective functional
The first two terms represent the benefit of the susceptible and vaccinated populations. The parameters B1
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
]. The goal is to maximize the populations of susceptible and vaccinated individuals, minimize the population of infectives, maximize the benefits of media coverage and vaccination, while minimizing the systemic costs of both media coverage and vaccination. The value uv
) = um
) = 1 represents the maximal control due to vaccination and media coverage, respectively. The terms
represent the maximal cost of education, implementation and campaigns on both vaccination and media coverage. S
) and V
) account for the fitness of the susceptible and the vaccinated groups as a result of a reduction in the rate at which the vaccine wanes, and vaccination and treatment efforts are implemented [35
]. We thus seek optimal controls
measurable, 0 ≤ a11
≤ 1, 0 ≤ a22
≤ 1, t
]} is the control set, with t
]. The basic framework of this problem is to characterize the optimal control.
Existence of an optimal control
The existence of an optimal control can be obtained by using a result by Joshi [36
] and Fister et al.
Theorem 3Consider the control problem with the system of Equations (4.1)-(4.4). There exists an optimal controlsuch that max Proof.
To prove this theorem on the existence of an optimal control, we use a result from Fleming and Rishel [38
] (Theorem 4.1 pp. 68-69), where the following properties must be satisfied.
1. The set of controls and corresponding state variables is nonempty.
2. The control set U is closed and convex.
3. The right-hand side of the state system is bounded above by a linear function in the state and control.
4. The integrand of the functional is concave on U and is bounded above by c2 – c1(|uv|k + |um|k), where c1, c2 > 0 and k > 1.
An existence result in Lukes [39
] (Theorem 9.2.1) for the system of equations (6)-(9) for bounded coefficients is used to give the first condition. The control set is closed and convex by definition. The right-hand side of the state system (Equations (4.1)-(4.4)) satisfies Condition 3 since the state solutions are a priori bounded. The integrand in the objective functional,
, is concave on U
. Furthermore, c1
> 0 and k
> 1, so
Therefore, the optimal control exists, since the left-hand side of (11) is bounded; consequently, the states are bounded.
Since there exists an optimal control for maximizing the functional (10) subject to equations (6)-(9), 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 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
. 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
) ≥ 0, w12
) ≥ 0 are penalty multipliers satisfying w11
)) + w12
) – b11
) at the optimal
, and w21
) ≥ 0, w22
) ≥ 0 are penalty multipliers satisfying w21
)) + w22
) – b22
) at the optimal
Given optimal controls
, and solutions of the corresponding state system (6)-(9),
there exist adjoint variables λi
, for i = 1, 2, 3, 4 satisfying the following equations
with transversality conditions λi
] = 0, for i
4. To determine the interior maximum of our Lagrangian, we take the partial derivatives of L with respect to uv
, respectively, and set it to zero. Thus,
To determine an explicit expression for our controls
(without w11, w12
), a standard optimality technique is utilized. The following cases are considered to determine a specific characterization of the optimal control.
1. On the set
. Hence, the optimal control is
2. On the set
since w12 ≥ 0.
3. On the set
Combining all the three sub-cases in a compact form gives
1. On the set
. We have
2. On the set
. We have
since w22 ≥ 0.
3. On the set
Combining all the three sub-cases in a compact form gives
The optimal system
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:
are given by expressions (12) and (13), respectively, with S
(0) = S0
(0) = I0
(0) = V0
(0) = R0
] = 0 for i
= 1,··· ,
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 [tf
]. 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 Figures , , , .
Figure 2 Optimality effect when the weight constraint for the infected population varies and media has a beneficial effect on the vaccine. Graphs of the optimality system when media coverage has a beneficial effect on the vaccination rate and when the weight constraint (more ...)
Figure 3 Optimality effect when the weight constraint for the control varies and media has a beneficial effect on the vaccine. Graphs of the optimality system when media coverage has a beneficial effect on the vaccination rate and when the weight constraint for (more ...)
Figure 4 Optimality effect when the weight constraint for the infected population varies and media has an adverse effect on the vaccine. Graphical representation of the optimality system when media coverage has an adverse effect on the vaccination rate and when (more ...)
Figure 5 Optimality effect when the weight constraint for the control varies and media has an adverse effect on the vaccine. Graphs of the optimality system when media coverage has an adverse effect on the vaccination rate and when the weight constraint for the (more ...)
The model with pulse vaccination
The general model with pulse vaccination is given as
for t ≠ tk, where tk is the time of the kth vaccination. We may have tk+1 – tk either constant or not, as we choose. The impulsive effect is given by
ΔS = –θS
ΔV = θS
is the change in state at the impulse time.
In this model, vaccination occurs at fixed times, not continuously. This is closer to reality, since vaccination centres are only open at certain times, when people may get vaccinated in waves. Similarly, media stories tend to clump together, so that a big news story occurs on one day, which may trigger a short period of intense vaccination. We shall use a simplified version of this model to illustrate the possibility that media may have an adverse effect.
Consider the following scenario. At the onset of the outbreak, the media - and hence the general population - is unaware of the disease and thus nobody gets the vaccine, allowing the disease to spread in its initial stages. At some point, there is a critical number of infected individuals, whereupon people are sufficiently aware of the infection to change their behaviour. We suppose that, initially, new infected people arrive at fixed times.
We further assume that vaccinated people mix more than susceptibles. In this case, people who are vaccinated feel confident enough to mix with the infected, even though they may still have the possibility to contract the virus. This might be the case for health-care workers, for instance, who get vaccinated and then have to tend to the sick.
Mathematically, we have a threshold for the critical number of infectives, Icrit.
, this model would look like
, the model becomes
with β4 – β6 ≥ 0.
However, to illustrate the adverse affect, we shall simplify the model even further. For a short timescale, we can assume recovery is permanent, so σ = 0. Thus, we can ignore the R equation.
, we assume that there is no mixing, but rather that new infectives arrive impulsively into the system at fixed times tk
and in numbers Ii,
where Ii Icrit
. (If the new infectives arrive at irregular times, then the broad results will be unchanged.)
For I >Icrit, fear of the disease keeps susceptibles from mixing with the infected, but the vaccinated will.
= 0. Since Ii Icrit
, we can assume that, for I
, the effects of new infectives are negligible.
for I >Icrit.
Thus, the effects of the media are to trigger a vaccinating panic whenever the number of infectives is large enough. We kept the model with impulse vaccination as simple as possible since even this simplified version shows that media reports could have an adverse effect.
Suppose new infectives appear regularly, so that tk+1
. (If not, the analysis generalizes quite easily.) For tk
, we have
is the value immediately after the k
th impulse. Then, since the period is constant, we have
This is a recursion relation with solution
Thus, if m+
, then eventually the system will switch from model (14)-(17) to model (18)-(20). The endemic equilibrium in model (18)-(20) satisfies
At the endemic equilibrium,
. Thus, we have
The characteristic equation is
It follows that the endemic equilibrium is stable if Î >Icrit. Thus, even in an extremely simplified version of the model, the media may make things significantly worse than if no media effect were included. We kept this model deliberately simple, partly for mathematical tractability and partly to show that the media effects apply even in this idealised scenario.
Note that, in reality, the fluctuations would apply in the upper region as well, making the actual value even
larger. In the lower region, we ignored interaction between susceptibles and infectives (ie we assume β4 = β6 = 0). The effect of including these terms would be to slow the exponential decay between impulses (or possibly cause it to increase). This would only increase the effect seen here.
In summary, a small series of outbreaks that would equilibrate at some maximum level m+ >Icrit will, as a result of the media, instead equilibrate at a much larger value I >m+ >Icrit. The driving factor here is if an imperfect vaccine causes overconfidence, so that people who have been vaccinated mix significantly more with infectives than susceptibles do. If this happens (as would be quite likely; most people who have been vaccinated feel invulnerable, even if the vaccine is not perfect, largely thanks to media oversimplifications), then the media effect is likely to be adverse. A simplified version of the model with pulse vaccination shows that the media can make things worse, if the vaccine is imperfect because the vaccinated mix over-confidently with the infectives.