Maintaining Hard Infection Caps in Epidemics via the Theory of Barriers

Epidemic management with admissible and robust invariant sets

“…• We demonstrate that the MRPI, whose use in epidemics was introduced in our previous research [25], is also useful as a tool to maintain an infection cap for uncertain epidemic models.…”

“…In our previous research, [25], we showed that the theory of barriers may be utilised to describe the viability kernel (also known as the admissible set) as well as the maximal robust positively invariant set (MRPI) the set of states in which the state and input constraints are satisfied for all time regardless of the input of the malaria model considered in [21]. The theory of barriers describes the non-trivial part of these sets' boundaries, made of integral curves of the system that satisfy a minimum-/maximum-like principle, allowing them to easily be constructed.…”

“…Proposition 4.1. The input � b, given by (25), associated with the barrier of the admissible set of the perfect SIR model, (19)-( 21) only switches at isolated points in time.…”

“…Consider the constrained perfect SIR model, ( 19)- (21). The barrier ½@A� À is constituted by the curve x � u generated by (25) with (23), that ends tangentially to G 0 \ P at the point ð g b min ; I max Þ 2 G 0 \ P with g b min þ I max � 1, and is contained in G − at least for all t 2� � t À �; � t½, � > 0.…”

“…The set-based approach to the management of epidemics, as presented in [21,23,25], argues that intervention strategies (for example, the level of fumigation in the control of mosquito-borne diseases) should be chosen based on the location of the state with respect to the computed sets: If the state is located in the admissible set, then it is possible to maintain the infection cap for all time with a suitably chosen input (intervention strategy). If the state is located in the MRPI, then the cap will never be breached and any input may be used, such as, e.g., saving resources or minimising economic damage.…”