A Fundamental Inconsistency in the SIR Model Structure and Proposed Remedies

The susceptible-infectious-removed (SIR) compartmental model structure and its variants are a fundamental modeling tool in epidemiology. As typically used, however, this tool may introduce an inconsistency by assuming that the rate of depletion of a compartment is proportional to the content of that compartment. As mentioned in the seminal SIR work of Kermack and McKendrick, this is an assumption of mathematical convenience rather than realism. As such, it leads to underprediction of the infectious compartment peaks by a factor of about two, a problem of particular importance when dealing with availability of resources during an epidemic. To remedy this problem, we develop the dSIR model structure, comprising a single delay differential equation and associated delay algebraic equations. We show that SIR and dSIR fully agree in assessing stability and long-term values of a population through an epidemic, but differ considerably in the exponential rates of ascent and descent as well as peak values during the epidemic. The novel Pade-SIR structure is also introduced as a approximation of dSIR by ordinary differential equations. We rigorously analyze the properties of these models and present a number of illustrative simulations, particularly in view of the recent coronavirus epidemic. Suggestions for further study are made.