[Paper Review] Modified SIR Model Yielding a Logistic Solution

Paul A. Reiser|arXiv (Cornell University)|Jun 2, 2020

Mathematical and Theoretical Epidemiology and Ecology Models11 references4 citations

TL;DR

This paper proposes a modified SIR model with a fixed delay in recovery to better reflect real-world pandemic dynamics, replacing the instantaneous removal assumption of the classical SIR model. The model yields a logistic solution for the cumulative infection fraction, enabling analytical tractability while improving realism through a delay differential equation that captures the time lag between infection and recovery.

ABSTRACT

The SIR pandemic model suffers from an unrealistic assumption: The rate of removal from the infectious class of individuals is assumed to be proportional to the number of infectious individuals. This means that a change in the rate of infection is simultaneous with an equal change in the rate of removal. A more realistic assumption is that an individual is removed at a certain time interval after having been infected. A simple modified SIR model is proposed which implements this delay, resulting in a single delay differential equation which comprises the model. A solution to this equation which is applicable to a pandemic is of the form A+B L(t) where L(t) is a logistic function, and A and B are constants. While the classical SIR model is often an oversimplification of pandemic behavior, it is instructive in that many of the fundamental dynamics and descriptors of pandemics are clearly and simply defined. The logistic model is generally used descriptively, dealing as it does with only the susceptible and infected classes and the rate of transfer between them. The present model presents a full but modified SIR model with a simpler logistic solution which is more realistic and equally instructive.

Motivation & Objective

• To address the unrealistic assumption in the classical SIR model that removal from the infectious class is instantaneous and proportional to current infectious cases.
• To introduce a fixed time delay $ T_r $ between infection and recovery, reflecting biological reality that individuals remain infectious for a set duration.
• To derive a closed-form solution for the cumulative infection fraction $ n(t) $, showing it follows a logistic function under the delayed model.
• To establish analytical relationships between the pandemic parameters $ f_c $, $ T_r $, and the phenomenological parameters $ f_e $, $ n_p $, and $ n_m $, enabling data fitting and model interpretation.

Proposed method

• Formulate a single delay differential equation (DDE) for the cumulative infection fraction $ n(t) $, where $ n'(t) = f_c [1 - n(t)] [n(t) - n(t - T_r)] $, modeling infection rate as proportional to susceptible and newly infectious individuals.
• Define the compartments as $ S(t) = 1 - n(t) $, $ I(t) = n(t) - n(t - T_r) $, and $ R(t) = n(t - T_r) $, ensuring conservation of population and proper delay implementation.
• Derive the logistic solution $ n(t) = n_m + \frac{n_p - n_m}{1 + e^{-f_e(t - t_h)}} $, which satisfies the DDE under specific parameter constraints.
• Use the Lambert W function to express the time scaling parameter $ f_e $ as $ f_e = f_r \left( R_o + W_0(-R_o e^{-R_o}) \right) $, with $ R_o = f_c T_r (1 - n_m) $.
• Establish inverse relationships: $ f_c = \frac{f_e}{n_p - n_m} $ and $ T_r = \frac{1}{f_e} \ln\left( \frac{1 - n_m}{1 - n_p} \right) $, enabling parameter calibration from data.
• Prove the logistic solution satisfies the DDE by substitution, using identities involving $ \epsilon = e^{-f_e(t - t_h)} $, $ \epsilon_r = e^{f_e T_r} $, and the derived relation $ \epsilon_r = \frac{1 - n_m}{1 - n_p} $.

Experimental results

Research questions

• RQ1 Can a modified SIR model with a fixed delay in recovery produce a closed-form solution that matches empirical pandemic curves?
• RQ2 How does introducing a time delay between infection and recovery alter the dynamics compared to the classical SIR model?
• RQ3 What is the analytical relationship between the phenomenological parameters of the logistic solution ($ f_e $, $ n_p $) and the underlying pandemic parameters ($ f_c $, $ T_r $)?
• RQ4 Under what conditions does the model reduce to the classical SIR model, and how does the delay affect the basic reproduction number $ R_o $?
• RQ5 Can the Lambert W function be used to express the growth rate $ f_e $ in terms of $ f_c $, $ T_r $, and initial immunity $ n_m $?

Key findings

• The modified SIR model with a fixed delay $ T_r $ in recovery yields a logistic solution for the cumulative infection fraction $ n(t) $, which is both analytically tractable and more biologically realistic than the classical SIR model.
• The time scaling parameter $ f_e $ is given by $ f_e = f_r \left( R_o + W_0(-R_o e^{-R_o}) \right) $, where $ R_o = f_c T_r (1 - n_m) $, and $ W_0 $ is the principal branch of the Lambert W function.
• When $ R_o \leq 1 $, $ f_e = 0 $, implying no epidemic spread, and $ n(t) = n_m $, confirming the model's consistency with epidemic threshold theory.
• The final cumulative attack rate $ n_p $ is related to $ f_e $ and $ f_c $ by $ n_p = n_m + \frac{f_e}{f_c} $, showing how transmission and growth rate jointly determine the epidemic's final size.
• The model allows inversion: given phenomenological parameters $ f_e $, $ n_p $, and $ n_m $, the underlying pandemic parameters are $ f_c = \frac{f_e}{n_p - n_m} $ and $ T_r = \frac{1}{f_e} \ln\left( \frac{1 - n_m}{1 - n_p} \right) $, enabling data-driven calibration.
• The solution is proven to satisfy the DDE through substitution and algebraic verification using the exponential parameter $ \epsilon $, confirming the logistic form as an exact solution under the model's assumptions.