Complete dimensional collapse in the continuum limit of a delayed SEIQR network model with separable distributed infecti
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ORIGINAL PAPER
Complete dimensional collapse in the continuum limit of a delayed SEIQR network model with separable distributed infectivity C. P. Vyasarayani
· Anindya Chatterjee
Received: 28 April 2020 / Accepted: 24 June 2020 © Springer Nature B.V. 2020
Abstract We take up a recently proposed compartmental SEIQR model with delays, ignore loss of immunity in the context of a fast pandemic, extend the model to a network structured on infectivity and consider the continuum limit of the same with a simple separable interaction model for the infectivities β. Numerical simulations show that the evolving dynamics of the network is effectively captured by a single scalar function of time, regardless of the distribution of β in the population. The continuum limit of the network model allows a simple derivation of the simpler model, which is a single scalar delay differential equation (DDE), wherein the variation in β appears through an integral closely related to the moment generating function of √ u = β. If the first few moments of u exist, the governing DDE can be expanded in a series that shows a direct correspondence with the original compartmental DDE with a single β. Even otherwise, the new scalar DDE can be solved using either numerical integration over u at each time step, or with the analytical integral if available in some useful form. Our work provides a new academic example of complete dimensional collapse, ties up an underlying continuum model for a pandemic C. P. Vyasarayani (B) Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Sangareddy 502285, India e-mail: [email protected] A. Chatterjee Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India e-mail: [email protected], [email protected]
with a simpler-seeming compartmental model and will hopefully lead to new analysis of continuum models for epidemics. Keywords Pandemic · COVID-19 · Epidemic · Time delay · Reduced order
1 Introduction The global pandemic of COVID-19 has prompted several studies of epidemic models from a dynamic systems point of view. Pandemics can be studied using simple mean-field models or compartmental models, where the entire population is divided into susceptible (S), exposed (E), infectious (I), quarantined (Q), and recovered (R) groups. More groups like hospitalized (H), vaccinated (V), etc., can be added, or groups can be removed, depending on modeling goals. These models are primarily developed based on the original SIR model due to Kermack and McKendrick [1]. Many models include other complexities like prior immunity, temporary immunity transferred at birth, vaccination history, a carrier population that never recovers [2], reinfection due to loss of immunity after recovery [3], exposed but asymptomatic populations [4], a quarantined population [5], and the influence of vital dynamics [6]. The fidelity of such models can be improved by developing structured network models [7–12], where each node in the network is a compartmental model
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