Abstract
We present a detailed set-based analysis of the well-known SIR and SEIR epidemic models subjected to hard caps on the proportion of infective individuals, and bounds on the allowable intervention strategies, such as social distancing, quarantining and vaccination. We describe the admissible and maximal robust positively invariant (MRPI) sets of these two models via the theory of barriers. We show how the sets may be used in the management of epidemics, for both perfect and imperfect/uncertain models, detailing how intervention strategies may be specified such that the hard infection cap is never breached, regardless of the basic reproduction number. The results are clarified with detailed examples.
Highlights
There is a large literature on the application of optimal control to epidemiology
In our previous research, [25], we showed that the theory of barriers may be utilised to describe the viability kernel 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]
For the imperfect model case we assume that there is a pre-designed intervention strategy, possibly a feedback, that has been designed and implemented on the system. For this case we only describe the MRPI, which corresponds to states from which the infection cap can always be maintained by the intervention strategy regardless of the modelling uncertainty
Summary
There is a large literature on the application of optimal control to epidemiology. Some of the earliest papers on the topic are [1], which investigates the optimal control of a disease of SIS type (susceptible-infective-susceptible), and [2], which considers an SIR (susceptible-infectiveremoved) model with vaccination to obtain optimal vaccination policies via dynamic programming. We present a set-based analysis of two well-known compartmental epidemic models: the SIR and SEIR models, see for example [17, 18], subjected to constraints on their inputs (social distancing, quarantine rate and/or vaccination rate) and state (a hard cap on the proportion of infectives), utilising the theory of barriers, [19, 20]. To our knowledge, this is the first paper that deals with the problem of maintaining a hard infection cap for the SIR and SEIR epidemic models (under the assumption of both perfect and imperfect modelling), via a set-theoretic approach: the theory of barriers, [19, 20].
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