Volume 29, 2023
|Number of page(s)||23|
|Published online||03 March 2023|
SPDEs with space interactions and application to population modelling
Department of Mathematics, University of Biskra, Algeria
Department of Mathematics, Royal Institute of Technology (KTH), Sweden
Department of Mathematics, Linnaeus University (LNU), Växjö, Sweden
Department of Mathematics, University of Oslo, Norway
* Corresponding author: email@example.com
Accepted: 1 February 2023
We consider optimal control of a new type of non-local stochastic partial differential equations (SPDEs). The SPDEs have space interactions, in the sense that the dynamics of the system at time t and position in space x also depend on the space-mean of values at neighbouring points. This is a model with many applications, e.g. to population growth studies and epidemiology. We prove the existence and uniqueness of strong, smooth solutions of a class of SPDEs with space interactions, and we show that, under some conditions, the solutions are positive for all times if the initial values are. Sufficient and necessary maximum principles for the optimal control of such systems are derived. Finally, we apply the results to study an optimal vaccine strategy problem for an epidemic by modelling the population density as a space-mean stochastic reaction-diffusion equation.
Mathematics Subject Classification: 60H05 / 60H20 / 60J75 / 93E20 / 91G80 / 91B70
Key words: Stochastic partial differential equations (SPDEs) / strong / smooth solutions / space interactions / spacemean dependence / population modelling / maximum principle / backward stochastic partial differential equations (BSPDEs) / space-mean stochastic reaction diffusion equation / optimal vaccination strategy
© The authors. Published by EDP Sciences, SMAI 2023
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