Delay differential equations for the spatially resolved simulation of epidemics with specific application to COVID‐19

Abstract

In the wake of the 2020 COVID‐19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic and of disease models generally. Most works follow the susceptible‐infected‐removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID‐19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult‐to‐estimate, compartments (such as exposed individuals) to incorporate time delays. In the present work, we introduce a DDE epidemic model in both an ordinary and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model's ability to reproduce measured data on realistic problems.

Modeling nonlocal behavior in epidemics via a reaction-diffusion system incorporating population movement along a network

“…The movement is defined to be 20% of the population in each region and assumed to be distributed evenly throughout the day. Hence: We define K Ω1,2 as in (15).…”

“…These models have taken many forms, and a comprehensive survey of the literature is beyond the scope of the current work. However, we note that many different approaches have been used to model the epidemic, including machine-learning and data-driven approaches [33,29,17,9,7], models using a classical compartmental approach, together with parameter estimation techniques [11,26], delay differential equations [15], partial differential equations [5,6,31,32], network-based methods [23,21,22,5], as well as agent-based [34,20] and multiscale models [4]. We note that the cited works represent just a small sample of the total literature, and further that the various approaches discussed are not mutually exclusive.…”

Assessing the Impact of Contact Tracing, Quarantine and Red Zone on the Dynamical Evolution of the Covid-19 Pandemic using the Cellular Automata Approach and the Resulting Mean Field System: A Case study in Mauritius

“…Mathematical modelling has figured prominently in decision making in the control and suppression of the Covid-19 spread [14] . Several mathematical models of Covid-19 have been developed with the goal of estimating and analysing the outbreak [13] , [15] , [16] , [17] , [18] . Mauritius was hit by the Covid-19 virus on 18 March 2020, trailed by three imported cases.…”

Well-posedness for a diffusion-reaction compartmental model simulating the spread of COVID-19

“…Following this latter idea, in the present contribution, we consider the PDE-based model introduced in [21,22] (a variation of it with delay terms is discussed in [16]). In particular, the following system has been considered ∂ t s = αn − (1 − A 0 /n)β i si − (1 − A 0 /n)β e se − µs + div(nν s ∇s)…”