Time-varying and state-dependent recovery rates in epidemiological models

Abstract

Differential equation models of infectious disease have undergone many theoretical extensions that are invaluable for the evaluation of disease spread. For instance, while one traditionally uses a bilinear term to describe the incidence rate of infection, physically more realistic generalizations exist to account for effects such as the saturation of infection. However, such theoretical extensions of recovery rates in differential equation models have only started to be developed. This is despite the fact that a constant rate often does not provide a good description of the dynamics of recovery and that the recovery rate is arguably as important as the incidence rate in governing the dynamics of a system. We provide a first-principles derivation of state-dependent and time-varying recovery rates in differential equation models of infectious disease. Through this derivation, we demonstrate how to obtain time-varying and state-dependent recovery rates based on the family of Pearson distributions and a power-law distribution, respectively. For recovery rates based on the family of Pearson distributions, we show that uncertainty in skewness, in comparison to other statistical moments, is at least two times more impactful on the sensitivity of predicting an epidemic's peak. In addition, using recovery rates based on a power-law distribution, we provide a procedure to obtain state-dependent recovery rates. For such state-dependent rates, we derive a natural connection between recovery rate parameters with the mean and standard deviation of a power-law distribution, illustrating the impact that standard deviation has on the shape of an epidemic wave.

Testing, tracing and isolation in compartmental models

“…Regardless, we show in the Results section that even when this approximation does not hold, while it results in oscillatory behaviour early on, it still generally adequately describes the overall trends and long term equilibrium. Eq 22 is equivalent to the integral form of an equation for a compartment model [ 70 ]. It can be written in differential form as, where the are the ‘hazard functions’ for the state X .…”

Managing Marek’s disease in the egg industry

“…Thereby, we obtain the viral shedding rates κ for each MDV strain from data on the dust shed from a typical laying-hen (Table S4), and the viral copy number (VCN) of MDV per milligram of dust for vMDV (pathotype MPF57) and vvMDV (pathotype FT158) (Table S3), and over each cohort duration (older chickens will shed more viral particles) (Table S5) (Bell, 2003; Witter et al, 1968). Data on the virulence level of the MDV pathotype and mortality, in addition to the standard assumption of Exponentially distributed parameters that is typical of compartmental models(Greenhalgh and Day, 2017; Hethcote and Tudor, 1980), was used to estimate MDV mortality rates ( ν ) for each MDV strain (Ralapanawe et al, 2016a). We base the viral particle removal rate ( δ ) on the barn ventilation system and estimates of the decay rate of the viral particles (Kennedy et al, 2018).…”

On Equilibria Stability in an Epidemiological SIR Model with Recovery-dependent Infection Rate

“…Several generalizations and modifications of the SIR model have been proposed by other authors, particularly considering non-constant epidemiological rates (see, [1,7,8,12,15,19,21,22]). These kinds of considerations have been recognized as necessary features to model more realistic epidemic situations, like the interaction between human behavior and disease dynamics [13,21].…”