Spatial behavioural responses to the spread of an infectious disease can suppress Turing and Turing–Hopf patterning of the disease

Reducing risky behaviour and/or avoiding sites where the risk of infection is perceived as higher (by social and/or spatial distancing) represent the two main forms of non-pharmaceutical behavioural responses of humans to the threats of infectious diseases. Here we investigate, within a reaction–diffusion setting, a family of new models for an endemic SIR (susceptible–infective–removed) infectious disease for which no vaccine is available and individuals’ responses to the infection threat are entirely based on changes either in their social behaviour or in their mobility behaviour, that is avoiding to visit sites with a large infection prevalence. First, we derive general conditions for the onset of Turing patterns for a general family of spatially inhomogeneous SIR models with a prevalence-dependent contact rate and constant recruitment. Then, we characterize our main family of models where the behavioural response also includes a spatial component, and show the condition bringing to the mitigation, or even the destruction, of Turing patterns. The same conditions can allow the transition from Turing–Hopf spatio-temporal patterns to pure Hopf temporal patterns. The same is also done for two SIS models. These results bring an inference of interest: the reduction of spatial clustering typically observed during the course of an epidemics might be related to a combination of agents’ spontaneous social and spatial distancing. To validate our theoretical results and further explore other spatio-temporal effects of the proposed spatial behavioural responses, numerical simulations of a specific instance of the above-mentioned family of SIR models have been performed.