SEIR Epidemic

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

The SEIR epidemic optimal control problem is a classical benchmark in mathematical epidemiology and optimal control of biological systems. This description is taken from [1].

It models the spread of an infectious disease through a homogeneous population divided into four compartments: susceptible ($S$), exposed ($E$), infectious ($I$), and recovered ($R$). The control represents a time-varying vaccination (or isolation) rate applied to the susceptible compartment. The objective is to minimise a weighted combination of the cumulative infectious burden and the cost of the intervention over a finite horizon, yielding a trade-off between disease suppression and resource expenditure. The problem is nonlinear and non-convex, and typically admits a singular arc structure in the optimal control.

Mathematical formulation

$\begin{array}{lll}\underset{u}{\min }& & \underset{0}{\overset{t_{\mathrm{f}}}{\int }}[AI(t)+\frac{1-A}{2}u(t{)}^2]\mathrm{d}t\\ \text{subject to}\\ \dot{S}(t)& =& -\beta S(t)I(t)-u(t)S(t),\\ \dot{E}(t)& =& \beta S(t)I(t)-\sigma E(t),\\ \dot{I}(t)& =& \sigma E(t)-\gamma I(t),\\ \dot{R}(t)& =& \gamma I(t)+u(t)S(t),\\ x(0)& =& (S_0,E_0,I_0,R_0{)}^T,\\ u(t)& \in & [0,u_{\max }]\mathrm{\forall }t\in [0,t_{\mathrm{f}}],\end{array}$

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 term $-u(t)S(t)$ removes susceptible individuals through vaccination (or effective isolation), moving them directly to the recovered compartment at rate $u(t)S(t)$. The quadratic penalty $\frac{1-A}{2}{u}^2$ models the increasing marginal cost of intervention.

Parameters

These fixed values are used within the model:

Symbol	Value	Description
$\beta$	0.5	Transmission rate (day−1)
$\sigma$	0.2	Progression rate from exposed to infectious (day−1)
$\gamma$	0.1	Recovery rate (day−1)
$A$	0.5	Weight on infection burden vs. control cost
$u_{\max }$	0.5	Maximum vaccination/isolation rate (day−1)
$S_0$	0.99	Initial susceptible fraction
$E_0$	0.01	Initial exposed fraction
$I_0$	0	Initial infectious fraction
$R_0$	0	Initial recovered fraction
$t_{\mathrm{f}}$	100	Horizon of the control problem (days)

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  Compartment trajectories and optimal vaccination rate for a reference solution.

Miscellaneous and Further Reading

The basic reproduction number for the uncontrolled SEIR model is ${{ℛ}}_0=\beta /\gamma$. With the parameters above, ${{ℛ}}_0=5$, corresponding to a highly transmissible pathogen. The optimal control problem was studied in detail by Lenhart and Workman [2], where necessary conditions via Pontryagin's minimum principle are derived and adjoint equations are solved numerically. This formulation and further discussion can be found in [1] and [2].

References

[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl
[2] Lenhart, S., & Workman, J. T. (2007). Optimal Control Applied to Biological Models. CRC Press.