I am reading on SIR models and I found this article
In the article it has three groups as one without vaccination, one with only whole cell(wP) vaccination, and one with only acelluar(aP) vaccination. These groups are non-overlapping.
In the article it says,
We assume those vaccinated with wP are completely immune to infection . Those vaccinated with aP move into a vaccinated class where they can become asymptomatically infected (that is, they incur a direct benefit from vaccination because they will not develop symptomatic disease). Unvaccinated individuals become infected with B. pertussis at rate β and become symptomatic with probability σ, and aP vaccinated individuals become asymptomatically infected at rate β. We assume no difference in transmissibility between symptomatic and asymptomatic individuals
Here, unlike wP vaccinated some of the aP vaccinated individuals can become asymptomatically infected.
The equations are
$\begin{array}{@{}rcl@{}} {}S'(t) &=& \mu \cdot (1-wP-aP)-\beta [I_{s}(t)+I_{a}(t)] S(t) {}\\ &&+ \omega R(t) -\nu S(t) \end{array}$ $\begin{array}{@{}rcl@{}} {}I_{s}'(t) &=& \beta \sigma [I_{s}(t)+I_{a}(t)] S(t)-\gamma_{s} I_{s}(t) - \nu I_{s}(t) \end{array}$
$\begin{array}{@{}rcl@{}} {}I_{a}'(t) &=& \beta (1-\sigma) [I_{s}(t)+I_{a}(t)] S(t)+\beta [I_{s}(t)+I_{a}(t)] \\&&V(t)-\gamma_{a} I_{a}(t) - \nu I_{a}(t) \end{array}$
$\begin{array}{@{}rcl@{}} {}V'(t) &=& \mu \cdot aP-\beta [I_{s}(t)+I_{a}(t)] V(t)- \nu V(t) \end{array}$
$\begin{array}{@{}rcl@{}} {}R'(t) &=& \mu \cdot wP +\gamma_{s} I_{s}(t)+\gamma_{a} I_{a}(t) - \omega R(t) - \nu R(t) \end{array}$
My question is is it reasonable to take the same rate $\beta$ for both the rate of unvaccinated getting infected and for the rate in which aP vaccinated become asymptomatically infected.
Also how are these equations generated?Are there any specific methods to do them?
I would appreciate if someone can recommend good books for a beginner in mathematical biology to learn on SIR models, on how to develop ODE's and obtain basic reproduction number.