This is my first time using the StackExchange and I was hoping someone could help me find the Jacobian Matrix for the ODEs I am working with. I am working with three populations; Humans, Ticks, and Carriers. The ODEs involving exposed and infected compartments are:
EH/dt = (βHH * IH) + (βCH * IC) + (βTH * IT) - (deltaH + muH)EH
IH/dt = (deltaH * EH) - (alpha + gamma + muH) * IH)
ET/dt = (piT * βTTV * IT) + (βCT * IC) + (BTTH * IT) - (deltaT + muT) * ET
IT/dt = (deltaT * ET) - (muT * IT)
EC/dt = (βTC * IT) - (deltaC + muC) * EC
IC/dt = (deltaC * EC) - (muC * IC)
Where alpha represents the human recovery rate, gamma represents the human death rate from the disease, piT represents the tick birth rate, mu represents the natural death rate respectively, and delta represents the incubation period respectively for humans, ticks, and carriers. Please let me know if there is any clarification needed or any further information that would be useful such as my full model or code. Thank you very much.
Consider the left hand side of your equations "bla/dt" rates and denote them as $\dot{x}_i$ for $i=1,2,...,N$ ($N=5$ in your case). These quantities constitute the components of a vector $\dot{\textbf{x}}$. The right hand side of the equations is another vector $\textbf{X}$ and each component is ${X_i}$. The Jacobian matrix $\textbf{J}$ has components $J_{ij}=\partial X_i/\partial x_j|_{x_c}$, i.e. the jacobian is $N \times N$. Notice that after computing the derivatives, the result is evaluated at $\textbf{x}_c$, which is a "point state" where you know the value of all the variables.
It is possible to think about linear approximations in terms of Taylor expansions. In your system, and for computational purposes $\textbf{X}(\textbf{x}_c+\Delta)=\textbf{X}(\textbf{x}_c)+\text{J} \Delta+hot$, where the higher order terms ($hot$) are ignored and $\Delta$ is what it is used to update the solution in the next iteration.