The Hybrid Incidence Susceptible-Transmissible-Removed Model for Pandemics

Abstract

The susceptible-transmissible-removed (STR) model is a deterministic compartment model, based on the susceptible-infected-removed (SIR) prototype. The STR replaces 2 SIR assumptions. SIR assumes that the emigration rate (due to death or recovery) is directly proportional to the infected compartment's size. The STR replaces this assumption with the biologically appropriate assumption that the emigration rate is the same as the immigration rate one infected period ago. This results in a unique delay differential equation epidemic model with the delay equal to the infected period. Hamer's mass action law for epidemiology is modified to resemble its chemistry precursor-the law of mass action. Constructing the model for an isolated population that exists on a surface bounded by the extent of the population's movements permits compartment density to replace compartment size. The STR reduces to a SIR model in a timescale that negates the delay-the transmissible timescale. This establishes that the SIR model applies to an isolated population in the disease's transmissible timescale. Cyclical social interactions will define a rhythmic timescale. It is demonstrated that the geometric mean maps transmissible timescale properties to their rhythmic timescale equivalents. This mapping defines the hybrid incidence (HI). The model validation demonstrates that the HI-STR can be constructed directly from the disease's transmission dynamics. The basic reproduction number ( R 0 ) is an epidemic impact property. The HI-STR model predicts that R 0 ∝ √ n where n is the population density, and is the ratio of time increments in the transmissible-and rhythmic timescales. The model is validated by experimentally verifying the relationship. R 0 's dependence on n is demonstrated for dropletspread SARS in Asian cities, aerosol-spread measles in Europe and non-airborne Ebola in Africa.

Basic reproduction number projection for novel pandemics and variants. Ancestral SARS-CoV2 R0 projection

“…The HI-STR prototype is based on the SIR model but replaces 2 assumptions and is formulated for an isolated population on a surface [43]. Thus…”

“…Hamer’s mass action law [143] assumption is replaced with the law of mass action [43] – its chemistry precursor [144, 145] – such that the probability density function for a single successful transmission is where η is an infectious disease-specific variable that reflects avidity, μ is a function of mode of transmission, κ ( x ) is a function of social behaviour, s ( t ) is the density of susceptible individuals and τ ( t ) is the density of hosts capable of transmitting the pathogen. The total transmissions (including those of secondary hosts) in a population of size N ≫ 1 and population density ρ n over the period that the primary host is transmission capable is shown to be where S ( t ) is the size of the susceptible population, T ( t ) is the size of the transmission-capable population and [43].…”

“…The hybrid-incidence, susceptible-transmissible-removed (HI-STR) [43] model is a deterministic, compartment model prototype constructed to replace two assumptions of Kermack-McKendrick's susceptible-infectious-removed (SIR) prototype [44][45][46]. It replaces the assumption that the removal rate is proportional to the size of the infected compartment with the more biologically appropriate assumption that the transmissible period is fixed and, consequently, the removal rate is the same as the infection rate one transmissible period ago [43]. It also replaces Hamer's mass action law with its chemistry precursor -the law of mass action [47].…”

“…It also replaces Hamer’s mass action law with its chemistry precursor – the law of mass action [47]. The latter allows the derivation of a population density dependent R 0 [21, 22, 43].…”

“…The collective can prophesize complex phenomena that other predictive methods cannot [96]. These models simulate heterogeneity and mixing [97] but the PDEs that they represent are not apparent [43, 98]. CA can reduce to ODEs [99] and for at least one application (computational fluid dynamics) the PDEs that they represent have been derived [100].…”