SEIR Epidemic

SEIR Epidemic
State dimension:	1
Differential states:	4
Discrete control functions:	2

The SEIR epidemic optimal control problem models the spread of an infectious disease through a homogeneous population divided into four compartments: susceptible ( $S$ ), exposed ( $E$ ), infectious ( $I$ ), and recovered ( $R$ ). Two controls are applied simultaneously: a vaccination rate that removes susceptible individuals from the transmission chain, and a social-distancing intensity that reduces the effective contact rate. The objective is to minimise a weighted combination of the cumulative infectious burden and the quadratic costs of both interventions over a finite horizon, yielding a trade-off between disease suppression and resource expenditure. The problem is nonlinear and non-convex and is calibrated to a COVID-19-like baseline scenario.

Mathematical formulation

$\begin{array}{lll} \min_{u_{1}, u_{2}} & \int_{0}^{T} [A I (t) + B u_{1} (t)^{2} + C u_{2} (t)^{2}] d t \\ subject to \\ \dot{S} (t) & = & - β (1 - u_{2} (t)) S (t) I (t) - u_{1} (t) S (t), \\ \dot{E} (t) & = & β (1 - u_{2} (t)) S (t) I (t) - σ E (t), \\ \dot{I} (t) & = & σ E (t) - γ I (t), \\ \dot{R} (t) & = & γ I (t) + u_{1} (t) S (t), \\ x (0) & = & (S_{0}, E_{0}, I_{0}, R_{0})^{T}, \\ u_{1} (t) & \in & [0, u_{1, \max}] \forall t \in [0, T], \\ u_{2} (t) & \in & [0, u_{2, \max}] \forall t \in [0, T], \end{array}$

where $x = (S, E, I, R)^{T}$ . All compartments are fractions of the total population, so $S (t) + E (t) + I (t) + R (t) = 1$ for all $t$ .

The first control $u_{1} (t) \in [0, u_{1, \max}]$ is the vaccination rate: the term $- u_{1} (t) S (t)$ removes susceptible individuals through vaccination, moving them directly to the recovered compartment at rate $u_{1} (t) S (t)$ .

The second control $u_{2} (t) \in [0, 1]$ is the social-distancing intensity: $u_{2} = 0$ corresponds to no intervention and $u_{2} = 1$ to full isolation. It enters multiplicatively as $(1 - u_{2} (t))$ in the force of infection $β (1 - u_{2}) S I$ , reducing effective contacts without directly removing individuals from any compartment.

The quadratic penalties $B u_{1}^{2}$ and $C u_{2}^{2}$ model the increasing marginal cost of each intervention.

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$β$	0.5	Contact (transmission) rate (day⁻¹)
$σ$	0.2	Progression rate from exposed to infectious (day⁻¹)
$γ$	0.1	Recovery rate (day⁻¹)
$A$	1	Weight on infectious burden
$B$	1	Quadratic cost weight on vaccination
$C$	1	Quadratic cost weight on social distancing
$u_{1, \max}$	0.10	Maximum vaccination rate (day⁻¹)
$u_{2, \max}$	1.00	Maximum social-distancing intensity
$S_{0}$	0.990	Initial susceptible fraction
$E_{0}$	0.005	Initial exposed fraction
$I_{0}$	0.005	Initial infectious fraction
$R_{0}$	0	Initial recovered fraction
$T$	100	Planning horizon (days)

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
Compartment trajectories and optimal vaccination and social-distancing schedules for a reference solution.

Miscellaneous and Further Reading

The basic reproduction number for the uncontrolled SEIR model is $ℛ_{0} = β / γ$ . With the parameters above, $ℛ_{0} = 5$ , corresponding to a highly transmissible pathogen. The two-control formulation combines pharmaceutical intervention (vaccination) with non-pharmaceutical intervention (social distancing), reflecting the dual-measure response used during the COVID-19 pandemic. Necessary optimality conditions via Pontryagin's minimum principle lead to a coupled system of state and adjoint ODEs; the optimal control typically exhibits a bang–singular–bang structure for the vaccination arc and a smooth interior arc for social distancing.

References

[1] Behncke, H. (2000). Optimal control of deterministic epidemics. Optimal Control Applications and Methods, 21(6), 269–285.
[2] Perkins, T. A., & España, G. (2020). Optimal control of the COVID-19 pandemic with non-pharmaceutical interventions. Bulletin of Mathematical Biology, 82, 118.