A abordagem do espaço de Fock para pequenos sistemas estocásticos : a cinética de Michaelis-Menten com inibidores e o modelo SEAIR

Abstract

Nature is full of phenomena that occur randomly, such as the spread of viruses and chemical reactions. The unpredictability of these phenomena implies incredible difficulty in describing them as a function of time. Therefore, searching for new mathematical tools that allow us to improve our understanding of them is fundamental. The formalism of Fock space has proved to be a handy tool in describing classical stochastic systems. In this approach, one can map the master equation as a Schrödinger equation and use the second quantization operators to describe the dynamics of small classical stochastic models. The solution to this equation describes the probability of a given system configuration occurring as a function of time. We used this approach to study the chemical kinetics of Michaelis-Menten with inhibitor and the Susceptible – Exposed – Asymptomatic – Infected – Recovered (SEAIR) infectious diseases model. As a function of time, we obtain the average behavior of the substances involved in the chemical kinetics for different amounts of substances. The results demonstrate the appearance of stiffness (equations characterized by their behavior changing rapidly in different time scales) for short times. We studied the behavior of the probability density at the time of the first formation of the reaction product. We also present a relationship between the average time of product formation and the Lineweaver-Burk linearization. The thermodynamic uncertainty relations (RIT) for unidirectional stochastic processes were also analyzed in the context of the Michaelis-Menten reaction, where it was possible to establish the lower limit for the variance of product formation time for each type of inhibition analyzed. For the SEAIR model, we determined the contamination dynamics as a function of time for different numbers of individuals. We studied the dynamics of the number of susceptibles over long periods and observed a possibility of non-contamination of the entire susceptible population. We quantitatively evaluated a contagion containment strategy removing only asymptomatic/infected individuals and compared its effectiveness with more restrictive strategies.