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# Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth

• * Corresponding author: Hans Josef Pesch
• In this paper an improved SEIR model for an infectious disease is presented which includes logistic growth for the total population. The aim is to develop optimal vaccination strategies against the spread of a generic disease. These vaccination strategies arise from the study of optimal control problems with various kinds of constraints including mixed control-state and state constraints. After presenting the new model and implementing the optimal control problems by means of a first-discretize-then-optimize method, numerical results for six scenarios are discussed and compared to an analytical optimal control law based on Pontrygin's minimum principle that allows to verify these results as approximations of candidate optimal solutions.

Mathematics Subject Classification: Primary: 49K15, 49M25, 90C90, 92D30; Secondary: 34A34.

 Citation: • • Figure 1.  The SEIR model with vaccination; cp. 

Figure 2.  Scenario 0: The progress of the population without control

Figure 3.  cenario 1: The progress of the population for WM = ∞

Figure 4.  Scenario 1: Discrete and verified optimal control for WM = ∞

Figure 5.  Scenario 2: "Almost bang-bang" control for WM = 2 500 when vaccination is stopped

Figure 6.  Scenario 2: The infected and the recovered population. The latter exhibits an "almost kink" when vaccination is stopped

Figure 7.  Scenario 3: The upper picture shows the two competing control candidates according to (33) and (36), the latter for V0 = 125. The optimal control is the boundary control (36) of the mixed control-state constraint (34) almost over the entire time interval. Only at the end, the control (32) in the interior of the admissble set becomes active. In the lower picture the discrete and the verified control values are compared showing again a perfect coincidence. The upright bar marks the switching time.

Figure 8.  Scenario 3: Time behaviour of the infected and recovered population. The terminal value of the recovered population is a bit higher than in the model with a finite amount of vaccines

Figure 9.  Scenario 4: Susceptible population and optimal control for $S_{\max} = 1\,200$. The upright lines mark the switching points from unconstrained boundary arcs on state-constrained ones or vice versa. Three boundary arcs occur here; the last one is extremely short; see the zoom. Note that a touch point cannot exist here . The optimal control is discontinuous at entry and/or exit points due to the jump discontinuities of $\lambda_S$ at these junction points; see 

Figure 11.  Scenario 5: Susceptible and infected population: entry and exit point to the state constrained arc are marked in green resp.red. The maximal allowed values $S_{\max}$ and $V_0/u(t)$ are marked in blue and purple respectively

Figure 10.  Scenario 5: The approximate candidate optimal control: the verification test yields $8.5 \cdot 10^{-8}$, hence indicating again a perfect coincidence

Figure 12.  Scenario 6: Discounted functional. The progress of the population for $W_M = \infty$

Figure 13.  Scenario 6: Discounted functional. Discrete and verified optimal control for $W_M = \infty$

Table 1.  Values from , also chosen in 1

 Parameters Definitions Units Values $b$ natural birth rate unit of time$^{-1}$ 0.525 $d$ natural death rate unit of time$^{-1}$ 0.5 $c$ incidence coefficient $\frac{1}{\mbox{ unit of capita}\,\cdot\,\mbox{unit of time}}$ 0.001 $e$ exposed to infectious rate unit of time$^{-1}$ 0.5 $g$ natural recovery rate unit of time$^{-1}$ 0.1 $a$ disease induced death rate unit of time$^{-1}$ 0.1 $u_{\max}$ maximum vaccination rate unit of time$^{-1}$ 1 $W_M$ maximum available vaccines unit of capita various $V_0$ upper bound in Eq. 34 $\frac{\mbox{ unit of capita}}{\mbox{ unit of time}}$ various $S_{\max}$ upper bound in Eq. 38 unit of capita various $A_1$ weight parameter $\frac{\mbox{ unit of money}}{\mbox{ unit of capita}\, \cdot\,\mbox{ unit of time}}$ 0.1 $A_2$ weight parameter unit of money$\,\cdot\,$unit of time 1 $t_0$ initial time unit of time (years) 0 $T$ final time unit of time (years) 20 $S_0$ initial susceptible population unit of capita 1000 $E_0$ initial exposed population unit of capita 100 $I_0$ initial infected population unit of capita 50 $R_0$ initial recovered population unit of capita 15 $N_0$ initial total population unit of capita 1165 $W_0$ initial vaccinated population unit of capita 0
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