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# On application of optimal control to SEIR normalized models: Pros and cons

• In this work we normalize a SEIR model that incorporates exponential natural birth and death, as well as disease-caused death. We use optimal control to control by vaccination the spread of a generic infectious disease described by a normalized model with $L^1$ cost. We discuss the pros and cons of SEIR normalized models when compared with classical models when optimal control with $L^1$ costs are considered. Our discussion highlights the role of the cost. Additionally, we partially validate our numerical solutions for our optimal control problem with normalized models using the Maximum Principle.

Mathematics Subject Classification: Primary: 49K15, 49N90; Secondary: 49M37.

 Citation: • • Figure 1.  Fluid Analogy of the Normalized SEIR compartmental model.

Figure 2.  Optimal control for $(P)$: parameters in tables 1 and 2. Left: Optimal control different dead rates: in red for $d=0.0099$ and in blue for $d=0.0005$. Right: Optimal control with different initial values. In red for $S_0$, $E_0$, $I_0$ and $R_0$ as in the table 2 in blue for initial conditions $S_0\times 100$, $E_0\times 100$, $I_0\times 100$ and $R_0\times 100$.

Figure 3.  Optimal control for $(P_n)$ with $\rho=500$ in blue. Optimal control calculated for $(P)$ in blue. The parameters are described in tables 1, 3, and 2.

Figure 4.  Case 1: Computed optimal control $u^*$ plotted together with the singular control computed according to (22) and with the switching function $\phi$. During the first five years $\phi(t)>1$ and during the last eight years $\phi(t)<0$.

Figure 5.  Case 1: optimal trajectories (including $r$).

Figure 6.  Case 2: Computed optimal control $u^*$ plotted together with the scaled switching function $\phi$. During the last seventeen years $\phi(t)<0$.

Figure 7.  Case 2: optimal trajectories.

Figure 8.  Case 3: Computed optimal control $u^*$ plotted together with the scaled switching function $\phi$. During the first sixteen years $\phi(t)>0$ and during the last three years $\phi(t)<0$.

Figure 9.  Case 3: Optimal trajectories.

Figure 10.  Optimal vaccinated rate, $u^*$, in red. Approximate control $u_{apr}$, in blue dash. $\rho=500$ and $u \in [0,1]$.

Table 1.  Parameters for SEIR models

 Parameter Description Value b Natural birth rate 0.01 d Death rate 0.0099 c Incidence coefficient 1.1 f Exposed to infectious rate 0.5 g Recovery rate 0.1 a Disease induced death rate 0.2 T Number of years 20

Table 2.  Initial Conditions and cost parameters for problems with classical SEIR model

 Parameter Description Value A weight parameter 1 B weight parameter 2 S0 Initial susceptible population 1000 E0 Initial exposed population 100 I0 Initial infected population 50 R0 Initial recovered population 15 N0 Initial population 1165

Table 3.  Initial Conditions and cost parameters for problems with classical SEIR model normalized model.

 Parameter Description Value s0 Percentage of initial susceptible population 0.858 e0 Percentage of initial exposed population 0.086 i0 Percentage of initial infected population 0.043
• Figures(10)

Tables(3)

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