Fully Probabilistic Microbial Inactivation Models: the Markov Chain Reconstruction from Experimental Survival Ratios

Abstract

During microbial inactivation by a lethal agent (ignoring damage/injury, recovery from damage/injury, etc.), an individual spore or cell can be either viable/alive and countable, or already inactivated/dead and uncountable. But since we only count the survivors’ total number at successive times, the fates of the individual microbes remain unknown. This makes the process probabilistic and creates the need for a stochastic model to describe it. The familiar continuous deterministic models such as the loglinear or Weibullian, which are only applicable to large populations, can be viewed as the mathematical limits of underlying discrete stochastic models. When the targeted microbial population is initially small, its survival curve is inherently irregular and irreproducible. But when the targeted population is large, its survival curve is initially smooth and reproducible but inevitably becomes irregular and irreproducible as the number of survivors diminishes. Perhaps the most important difference between the deterministic loglinear model, or the Weibullian with a shape factor > 1, and their fully stochastic versions is that the latter predict complete elimination of the targeted microbe in a realistic finite time. A stochastic survival model is derived from the individual microbes’ Markov chains (or trees) and the character of the underlying survival probability rate’s time-dependence. Various types of such dependencies are presented, and their different manifestations in the stochastic survival curves shapes are demonstrated. Also discussed are ways to extract (estimate) the stochastic model’s parameters from the regular and reproducible part of static experimental survival data, which in some cases requires unconventional regression techniques.