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Team:British Columbia/Modeling
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Reaction Kinetics: - Modelling the production of cinnamaldehyde, vanillin and caffeine production
Population dynamics: - Modelling the population of a bacteria in a batch reactor with phage dependency
\begin{align} \frac{dX}{dt} = \frac{dX_u}{dt} + \frac{dX_i}{dt} + \Gamma \end{align} Where $\frac{dX}{dt}$ is rate of change in bacteria population over time, $\frac{dX_u}{dt}$ and $\frac{dX_i}{dt}$ are the rates of change in the uninfected and infected populations respectively. $\Gamma$ is defined as a stepwise function. \begin{align} \Gamma &= - E(X_i), \ \ \ during \ phage \ caused \ lysis\\ &= 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise \end{align} Where $E(X_i)$ is the statistically expected infected bacterial populations. We define the rate of change in population in () and (). \begin{align} &\frac{dX_u}{dt} = \big(\mu_u + k_{d_u}\big)X_u\\ &\frac{dX_i}{dt} = \big(\mu_i + k_{d_i}\big)X_i \end{align} Where $\mu_u$ and $\mu_i$ are the growth rate of uninfected and infected cells respectively. And $k_{d_u}$, $k_{d_i}$ are the decay rates of the uninfected and infected bacteria. To create a more complete model, we have chosen to model the bacteriophage population as well. \begin{align} P = X_i \beta ^{\frac{\tau}{LT}} \end{align} However, due to the low rate of secretion of our studied populations, we can redefine our phage population as a stepwise function: \begin{align} \frac{dP}{dt} &= E(X_i)\beta, \ \ \ \ for \ lysis\\ &= 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise \end{align}
The multiplicity of infection (MOI) is defined as the ratio of bacteriophages to bacterial cells. \begin{align} MOI = \frac{P}{X} \end{align} And we use the Poisson distribution
