Okay so this is quite a long question involving a lot of background work I have done myself but there are quite a few holes I need filling in so I'll start from the beginning.
I'm writing a report on the SIR model for infectious diseases. My first task was to reduce a system of equations in $S$ and $I$ to one single ODE as follows:
$$\frac{dS}{dt}=-\beta SI$$ $$\frac{dI}{dt}=\beta SI-\gamma I$$
I did this by the following means:
$$\frac{dS}{dI}=\frac{\frac{dS}{dt}}{\frac{dI}{dt}}$$ $$\frac{dS}{dI}=\frac{-\beta SI}{\beta SI-\gamma I}=\frac{-\beta S}{\beta S-\gamma}$$ $$\int-1+\frac{\gamma}{\beta S}\ dS=\int dI$$
With initial conditions $S(0)=S_0$ and $I(0)=I_0$ hence $I(S_0)=I_0$
$$\int_{S_0}^{S}1+\frac{\gamma}{\beta S}\ dS=\int_{I_0}^{I(S)} dI$$ When solves leads to: $$I(S)=-S+\frac{\gamma\ln (S)}{\beta}+I_0 +S_0-\frac{\gamma\ln (S_0)}{\beta}$$
I need to use this solution to relate to numerical solutions I have earlier produced for the number of susceptible individuals but I'm not sure how this equation relates to it? or graphically this:
Following from this I was asked to show whether or not the infection will spread to a larger value of individuals than the initial value $I_0$ depends on the ratio $\frac{\beta S_0}{\gamma}$ and find the threshold for this ratio by relating it to the ODE for $I$. I did some research and found this ratio to be the reproductive ratio or $R_0$ and found the following:
Considering: $\frac{dI}{dt}\ge0$ assuming $S(0)$ $\rightarrow S_0$:
$$\frac{dI}{dt}\ge0\Rightarrow\beta S(0)C(0)-\gamma C(0)\ge0\Rightarrow\frac{\beta S(0)}{\gamma}\ge 1 \Rightarrow R_0 =\frac{\beta S_0}{\gamma} $$
The two possible thresholds are: $R_0>1$ and $R_0<1$
If $R_0<1$ then $\frac{dI}{dt}<0$ and each susceptible will infect fewer than one person and the disease will die out. Conversely if $R_0>1$ then $\frac{dI}{dt}>0$ and epidemic will occur. If I have misinterpreted any of the above please let me know and an explanation would be appreciated.
Lastly I have be asked to assume the input of an instantaneous vaccination and a proportion $p$ of susceptibles are given this initially and asked to find the size of $p$ for $C_0$ to not be exceeded. My assumptions are instead of the initial condition $S_0$ I can use the proportion $p$ at an instant (not time dependent) allowing me to reform the susceptible equation:
$$\frac{dS}{dt}=-\beta SI-p$$
I have a feeling the solution to this will somehow relate to the following solution involving the reproductive ratio but I'm struggling to come up with something. Does this mean that $\frac{dI}{dt}$ can be consider at the same instant? My lecturer informed me that I can use the same condition for not increasing $(\frac{dI}{dt}<0)$ as I have previously (reproductive ratio) but I fail to see what to do. If anybody with a background in this area could shed some light on this it would be great.