![]() This is the canonical form of the model where the sub-populations are assumed to be well-mixed so that spatial variations can be ignored over the domain of interest. The extension to inferring mobility via reaction–diffusion systems is in Sect. The results for inferred parameters and forward prediction are presented in Sect. The application of system identification and machine learning to the ODE system are, respectively, in Sects. 2 we review the foundational SIRD ODE model. The temporal resolution by days and spatial resolution by the 85 counties of Michigan has allowed us to apply our methods of Variational System Identification, PDE-constrained optimization and machine learning to these data. To these tasks we have brought the abundance of high-quality, public domain, data on the evolution of the various compartment pertaining to the SIRD model in the US state of Michigan. Our contribution to this aspect of the mathematical treatment is to also allow the diffusivity of the S, I and R sub-populations to vary with time. Such an extension also has been considered-chiefly in the setting of the mathematical analysis of reaction–diffusion systems. As the world went into lockdown, but at different rates and degrees of rigor, and then began to emerge from it, the detection of patterns of mobility in space and time presents a compelling avenue for investigation. Given the prominence that quarantine protocols-adorned with the current-day euphemism of “social distancing”-have played in the COVID-19 Pandemic, it appears natural to seek an extension of the SIRD model to a spatio-temporal PDE model. The most widely known are gravity models (e.g. Population mobility has been addressed through metapopulation models that characterize how diseases move between population hubs, across countries, or even intercontinentally. The second is the fact of a mobile population. This is not necessarily novel, and has been addressed in other work, although perhaps not with the inference approach of Variational System Identification (VSI) and ODE-constrained optimization that we have adopted. ![]() The first extension that we have undertaken is to allow the ODE coefficients to vary in time to reflect the evolving contours of testing, quarantine and treatment protocols. for details of the other SIR model variants. The interested reader is directed to Ref. We have therefore worked with the SIRD model. However, data are not available to us on the exposed and maternally immune-protected sub-populations in the state of Michigan, and the Maternal compartment is not known to be relevant to COVID-19. These models are typically designated as SIS (Susceptible-Infected-Susceptible again, such as in the common cold) MSIR (Maternal-Susceptible-Infected-Recovered, where immunity is derived from the mother in the M compartment) SEIS (Susceptible-Exposed-Infected-Susceptible again, also typical of the common cold) MSEIR and MSEIRD, which combine more of the compartments. The classical SIR model can be extended to compartments additional to the exposed and deceased ones. This choice is based entirely on the nature of the data available on the epidemic in the state of Michigan, where the numbers of deceased are reported on a daily basis. Driven by data that extends the compartments to the deceased (D), we have adopted the SIRD model. Of particular interest to us are two lines of enquiry: The first is that for a rapidly evolving disease such as COVID-19, with its public health, population-based, political, travel and economic manifestations, the classical SIR model of ordinary differential equations (ODEs) with constant coefficients seems inadequate. The present communication is in this spirit, and brings our recent work in large scale computations of partial differential equations (PDEs), system inference and machine learning to this problem. ![]() (Some of this literature also includes agent-based models, which we do not consider here.) During the COVID-19 Pandemic, the widespread availability of data in the public domain has served to attract methods of mathematics, computation and data science to analyzing this information, inferring the disease’s dynamics and making projections. Even before this, however, the baseline Susceptible-Infected-Recovered (SIR) model had been extended to include Exposed (E) and Deceased (D) compartments and applied with considerable success to influenza, ebola, malaria, cholera, tuberculosis and several other infectious diseases. The current COVID-19 Pandemic has brought them into the common parlance. Starting from their origins in the the work of Kermack and McKendrick, the use of differential equation models of the course of infectious diseases has grown to become one of the more accessible instances of the reach of mathematics.
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