For the simple SIS model $$ \begin{aligned} \mathsf{S}′ &= \Lambda − \beta \mathsf{SI} − \mu \mathsf{S} + \gamma \mathsf{I} \\ \mathsf{I}′ &= \beta \mathsf{SI} −(\mu+\gamma)\mathsf{I} \end{aligned} $$ The basic reproduction number $R_0$ is $R_0 = \beta K/(\mu+\gamma)$.
For the model $$ \begin{aligned} \mathsf{S} ′ &= \Lambda(\mathsf{N}) − \beta(\mathsf{N})\mathsf{SI} − (\mu+\phi)\mathsf{S} + \gamma \mathsf{I} + \theta \mathsf{V} \\ \mathsf{I}′ &= \beta(\mathsf{N})\mathsf{SI} +\sigma\beta(\mathsf{N})\mathsf{VI} −(\mu+\gamma)\mathsf{I} \\ \mathsf{V}′ &= \phi \mathsf{S}−\sigma\beta(\mathsf{N})\mathsf{VI} −(\mu+\theta)\mathsf{V} \end{aligned} $$ As $\mathsf{N}=\mathsf{S}+\mathsf{I}+\mathsf{V}$ , this is rewritten as: $$ \begin{aligned} \mathsf{I}′ &= \beta[K−\mathsf{I}−(1−\sigma)\mathsf{V}]\mathsf{I} −(\mu+\gamma)\mathsf{I} \\ \mathsf{V}′ &= \phi[K−\mathsf{I}]−\sigma\beta \mathsf{V}\mathsf{I} −(\mu+\theta+\phi)\mathsf{V} \end{aligned} $$ And the basic reproduction number $R(\phi)$ is $$R(\phi)= \frac{\beta K(\mu+\theta+\sigma\phi)}{(\mu+\gamma)(\mu+\theta+\phi)}\text{.}$$ With a backward bifurcation at $R(\phi)=1$ then the critical reproduction number $R_c$ satisfies
$R_c=\dfrac{\mu+\theta+\sigma\phi}{\mu+\theta+\phi} \cdot \left( \dfrac{\sigma(\mu+\gamma)+2\sqrt{\sigma(1−\sigma)(\mu+\gamma)\phi}−(\mu+\theta+\sigma\phi)}{\sigma(\mu+\gamma)\phi} \right)$
Where $\sigma\beta_c K = \sigma(\mu+\gamma)+2\sqrt{\sigma(1−\sigma)(\mu+\gamma)\phi}−(\mu+\theta+\sigma\phi)$
Now assume we have $\theta=0$. Given this,
- Show how to choose $\phi$ to make $R_c$ < $R_0$, assuming that all parameters other than $\phi$ are kept fixed.
- Is it possible to improve the vaccine (decrease $\sigma$) enough to make $R_c < R_0$ , assuming all parameters other than $\sigma$ are kept fixed?
As for part 1), I rewrote $$ R_c= \frac{(\mu+\sigma\phi) R_0}{(\mu+\phi) \phi}$$ to solve the inequality $\frac{\mu+\sigma\phi}{\phi\mu+\phi^2} <1$ and then used quadratic form and so on, but I’m not sure about part 2) so help is appreciated. And also I’d like to verify that the procedure in 1) is as I wrote.
Thanks in advance