Path integral control of a stochastic multi-risk SIR pandemic model

Abstract

In this paper a Feynman-type path integral control approach is used for a recursive formulation of a health objective function subject to a fatigue dynamics, a forward-looking stochastic multi-risk susceptible-infective-recovered (SIR) model with risk-group's Bayesian opinion dynamics toward vaccination against COVID-19. My main interest lies in solving a minimization of a policy-maker's social cost which depends on some deterministic weight. I obtain an optimal lock-down intensity from a Wick-rotated Schrödinger-type equation which is analogous to a Hamiltonian-Jacobi-Bellman (HJB) equation. My formulation is based on path integral control and dynamic programming tools facilitates the analysis and permits the application of algorithm to obtain numerical solution for pandemic control model.

Keywords: Bayesian opinion network; Feynman-type path integrals; Multi-risk SIR model; Pandemic control; Stochastic differential equations.

Fig. 1

The direction of the spread of an infection of a risk-group where N represents a person is infected and under non-ICU treatment while II indicates an individual is infected and is under ICU care

Fig. 2

Two realizations of COVID-19 spread according to Barabasi–Albert model with 500 vertices

Fig. 3

Two networks of individual risk-group k such as binary and triangular stubs at time $s\in [0,t]$

Fig. 4

Ten realization of fatigue dynamics with ${\sigma }_0=0.2$

Fig. 5

Static relationship between fatigue and lock-down intensity

Fig. 6

One realization of the infection rate dynamics with ${\sigma }_1=0.025$

Fig. 7

Twenty realizations of the infection rate dynamics

Fig. 8

Stochastic SIR model with ${\sigma }_2=0.6$, ${\sigma }_3=0.01$ and ${\sigma }_4=0.01$

Fig. 9

Ten realizations of opinion dynamics of risk group k with ${\sigma }_5=0.1$

Fig. 10

Optimal lock-down intensity over time

Fig. 11

Behavior of the social cost function over time