Mathematical Modeling, Computational Simulations for Molecular Thermodynamic Model
Mathematical Modeling for Epidemic Model with Stochastic Process
At the beginning of the industrial revolution in the 19th century, thermodynamics has intensely developed in various fields of science. The productivity in humans has changed from the power of humans, animals, wind, and water to machinery, and the work of machines should be drive by fuel. Gibbs assimilate the chemical processes into the framework of Thermodynamics to comprehend the spontaneity, energy changed during chemical reaction and phase changes. Gibbs also introduce the concept of ensemble to explain that the status in a system could be extremely different in micro-state, and then develop statistical mechanics which apply statistical methods to obtain the microscopic explanations of macroscopic thermodynamic phenomenon from movement of a large number of microscopic particles, and illustrate how the bulk properties emerge from the microscopic system. Hence, Statistical mechanics is also fundamental and important tool to study molecular simulation. In summary, Molecular Thermodynamics is a theory developed based on thermodynamics and statistical mechanics. It describes the process of energy transfer and phase transition by studying the interaction between molecules microscopically. More specifically, the topics of studying molecule thermodynamics are interested as electrolyte solution, chemical activity coefficient, biological ion channel, and Electrical Double Layer (EDL).
In the programing of chemical engineer, in order to achieve the optimization of the production process, simulation and prediction of the physical and chemical properties of molecules is a very important topic. In our research, we apply the Poisson-Bikerman theory to derive the formula for chemical activity coefficients, verify the experimental results and propose theory to predict it. Moreover, we consider the influence of temperature, pressure, and different solvents. The solvents’ properties include volume, concentration, dielectric, coordination number, and the free energy we expected this research can improve the predictability of molecule-related prediction.
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From the Black Death in the 14th century to COVID-19 in the 21st century, we know that epidemics are inseparable from our lives. Because of the impact of globalization, COVID-19 spread to the world within a few months. How to predict the spread of the pathogen, the direction of the epidemic and the reduction of the death rate have become scientific issues that mankind must face urgently.
__Kermack and McKendrick proposed the SIR compartmental model in 1927. It is based on mathematical modeling to predict the trend of virus transmission. Based on the SIR model, various models such as the SIS model and the SEIR model have been derived to deal with more complex situations. To make the model more complete, researchers in many different fields have introduced the vaccinated populations into the initial model. Furthermore, we choose to add different stochastic processes (such as Brownian motion, Lévy process, ...) to get close to the uncertainty of measurement and statistics in the real medical system.
__Our research uses random disturbances to cope with changes in the environment to improve the usefulness of mathematical modeling. So that the stochastic epidemic model built can more effectively conform to the actual data of the disease and predict the trend of the epidemic.