I have a system of 5 ODEs and I am trying to calculate the basic reproduction number($R_0$) using the method described in the article on next generation matrix.
First I tried it on the simple virus dynamic model with immune response. I know I don't need to use this method to calculate R0 to this simple system, but I wanted to try it to get an idea as to how I can use this next generation matrix.
The system is,
\begin{alignat}{1}
\dot{x} &= λ - dx - βxv,\\
\dot{y} &= βxv - ay-pyz, \\
\dot{v} &= ky - uv,\\
\dot{z}&=cyz-bz
\end{alignat}
where $x$ is the uninfectedd cells, $y$ is the infected cells $v$ is the virus and $z$ describes the immune response.
When the infected subsystem is linearized around the infection free steady state of $({\lambda\over d},0 ,0,0)$ the system is
\begin{alignat}{4}
\dot{y} &=~& - ay &~+~& \frac{λβ}{d} v, \\
\dot{v} &=& ky &~-~& uv, \\
\dot {z}&=&-bz
\end{alignat}
I tried this with and without $\dot z$ for the infected subsystem (those equations of the ODE system that describe the production of new infections and changes in state among infected individuals), but in both cases as Ro I get $R_0={\beta \lambda k\over adu}$ .
1)However, Ro for this system should be $R_0={\beta \lambda k\over (a+p \hat z)du}$ where $\hat z$ is the equilbrium state immune response when CTL response is activated. $\hat z={1\over p}({\lambda \beta c k\over cdu+\beta bk}-a)$ .
How can I obtain this $R_0$ value using next generation matrix?
2) In this viral model because the virus load doesn't get cleared it reaches the steady state with non zero $\hat z$. But, what happens in a model where the viral load will peak and declines to zero and at the end the model reaches the infection free state with $\hat z$=0.
In this case how to include the effect from immune response (z) into $R_0$ as if I let $\hat z=0$, then it will not be included in $R_0$. I don't understand what this $\hat z$ should represent?
I am trying to find $R_0$ in a model similar to the model in this article
where $B_s,B_r$ are two types of bacteria and $I,P$ are innate and adaptive immune response and $R$ is resource.
3) In a model where the viral load increase to reach a peak and then declines, does R0 have to change over the increasing and declining phase? What I mean is in the viral growth phase should $R_0>1$ and in the viral decline phase $R_0<1$? So, is the growth and decline are governed by two different $R_0$ values?