RobertLampel at 11:28, 4 September 2026

2026-09-04T11:28:46Z

RobertLampel: /* References */

2026-09-03T11:49:35Z

References

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			[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]
	[[Category:Epidemiological model]]		[[Category:Epidemiological model]]
	[[Category:Biological systems]]		[[Category:Biological systems]]

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 4 |nw = 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 ( $R..."$

2026-09-03T11:43:09Z

Created page with "{{Dimensions |nd = 1 |nx = 4 |nw = 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 (<math>S</math>), exposed (<math>E</math>), infectious (<math>I</math>), and recovered (<math>R..."

New page

{{Dimensions
|nd = 1
|nx = 4
|nw = 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 [[#OCPjl | [1]]].

It models the spread of an infectious disease through a homogeneous population divided into four compartments: susceptible (<math>S</math>), exposed (<math>E</math>), infectious (<math>I</math>), and recovered (<math>R</math>).
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 ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && \int_0^{t_\mathrm{f}} \Bigl[ A\, I(t) + \tfrac{1-A}{2}\, u(t)^2 \Bigr]\, \mathrm{d}t \\[6pt]
\text{subject to} \\
\quad \dot{S}(t) &=& -\beta\, S(t)\, I(t) - u(t)\, S(t), \\
\quad \dot{E}(t) &=& \beta\, S(t)\, I(t) - \sigma\, E(t), \\
\quad \dot{I}(t) &=& \sigma\, E(t) - \gamma\, I(t), \\
\quad \dot{R}(t) &=& \gamma\, I(t) + u(t)\, S(t), \\
\quad x(0) &=& (S_0,\; E_0,\; I_0,\; R_0)^T, \\
\quad u(t) &\in& [0,\, u_{\max}] \quad \forall\, t \in [0, t_\mathrm{f}],
\end{array}
</math>


where <math>x = (S, E, I, R)^T</math>.
All compartments are fractions of the total population, so <math>S(t) + E(t) + I(t) + R(t) = 1</math> for all <math>t</math>.
The term <math>-u(t)\,S(t)</math> removes susceptible individuals through vaccination (or effective isolation), moving them directly to the recovered compartment at rate <math>u(t)\,S(t)</math>.
The quadratic penalty <math>\tfrac{1-A}{2}\,u^2</math> models the increasing marginal cost of intervention.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\beta</math> || align=right | 0.5 || Transmission rate (day−1)
|-
| align=center | <math>\sigma</math> || align=right | 0.2 || Progression rate from exposed to infectious (day−1)
|-
| align=center | <math>\gamma</math> || align=right | 0.1 || Recovery rate (day−1)
|-
| align=center | <math>A</math> || align=right | 0.5 || Weight on infection burden vs. control cost
|-
| align=center | <math>u_{\max}</math> || align=right | 0.5 || Maximum vaccination/isolation rate (day−1)
|-
| align=center | <math>S_0</math> || align=right | 0.99 || Initial susceptible fraction
|-
| align=center | <math>E_0</math> || align=right | 0.01 || Initial exposed fraction
|-
| align=center | <math>I_0</math> || align=right | 0 || Initial infectious fraction
|-
| align=center | <math>R_0</math> || align=right | 0 || Initial recovered fraction
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 100 || Horizon of the control problem (days)
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image: SEIR_Epidemic.png | Compartment trajectories and optimal vaccination rate for a reference solution.
</gallery>

== Miscellaneous and Further Reading ==
The basic reproduction number for the uncontrolled SEIR model is <math>\mathcal{R}_0 = \beta/\gamma</math>.
With the parameters above, <math>\mathcal{R}_0 = 5</math>, corresponding to a highly transmissible pathogen.
The optimal control problem was studied in detail by Lenhart and Workman [[#Lenhart|[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 [[#OCPjl|[1]]] and [[#Lenhart|[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. 

[[Category:ODE model]]
[[Category:Epidemiological model]]
[[Category:Biological systems]]

SEIR Epidemic - Revision history

RobertLampel at 11:28, 4 September 2026

RobertLampel: /* References */