Nonlinear Stochastic Model for Bacterial Disinfection: Analytical Solution and Monte Carlo Simulation

Abstract

This contribution presents a sequel to our previously published nonlinear stochastic model for bacterial disinfection whose intensity function is explicitly proportional to the contact time of the bacteria with the disinfecting agent. In the current model, the intensity function is proportional to the square of the contact time to account for an accelerated rate of a disinfection process. The model gives rise to the process’ master equation whose solution renders it possible to obtain the analytical expressions of the process’ mean, variance (or standard deviation), and coefficient of variation. Moreover, the master equation has been simulated via the Monte Carlo method, thereby yielding the numerical estimates of these quantities. The estimates’ values are compared with those computed via the analytical expressions; they are in excellent accord. They are also compared with the available experimental data as well as with the results obtained from our earlier model.

A generalized model for bacterial disinfection: Stochastic approach

“…Moreover, the analytical and simulated results for a number of non-linear cases of the generalized stochastic model have been validated with the experimental observations for a specific instance of bacterial disinfection [25]. In this regard, a particular non-linear case appears to describe these experimental data more accurately than those linear cases of the generalized stochastic model equivalent to the models elaborated in two of our earlier works [23,24]. In light of these results, it is expected that the generalized stochastic model could be applied to the mathematical description of a variety of instances of bacterial disinfection performed at 6 distinct ambient conditions and settings.…”

“…For comparison, the master equation's linear and non-linear cases have also been solved numerically via simulation by resorting to the Monte Carlo method, often regarded as numerical experiments. The analytical and numerical results obtained for a set of linear cases of the generalized stochastic model have been validated by comparing them to those derived in our earlier works [21,23,24]; the comparison indicates that they are in excellent agreement. Moreover, the analytical and simulated results for a number of non-linear cases of the generalized stochastic model have been validated with the experimental observations for a specific instance of bacterial disinfection [25].…”

“…The system under consideration can be modeled as a pure-death process, a particular instance of the Markovian birth-and-death process [22], as described in our earlier contributions [21,23,24]. The death event of this pure-death process is identified as the elimination of a single bacterium by means of a disinfecting agent.…”

“…Naturally, it is expected that the generalized model's performance in representing any specific set of experimental data could be enhanced in view of the flexible nature of its intensity of transition. In addition, the models developed in our earlier works [21,23,24] are embodied by the generalized stochastic model proposed herein: They presented stochastic models for bacterial disinfection each based on an intensity of transition given by the product of the first power of number concentration of bacteria and a power function of time t whose values were constrained to positive, integer numbers. 5 The proposed model yields the master equation of the process under consideration, which can be linear or non-linear depending on the power of the number concentration of bacteria, i.e., the process' random variable, in the intensity of transition.…”

“…Both linear and non-linear cases of the master equation can be solved analytically, thereby leading to the expressions for the mean, variance (standard deviation), and coefficient of variation of the process' random variable. On one hand, the linear cases of the master equation have been solved analytically via conventional mathematical techniques, such as the probability-generating function [22,24]. On the other hand, the difficulty in obtaining the analytical solution of the non-linear cases of the master equation has been circumvented by means of a rational approximation method, system-size expansion [22].…”