This is my assignment I've been stuck on for a good part of the day. I'm confused because example's I've looked at would do part 3 before part 1 because they use substitution of $$u=S/N$$ $$v=I/N $$ $$w=R/N$$ and then substitute part 3 into the equations before writing them in maple. Maple says I'm recursive when I try to write the equations. Also why are we replacing the equation N for R; when it's the birth rate in S we don't like; wouldn't we want to replace S? Ugh any input would help just frustrating not even getting equations into Maple at this point and don't want to do it wrong because if part 1 is wrong parts 2-9 will be too.
The SIR model takes a certain proportion of the infectives per time period and removes them. This removal can come about from being cured (and hence becoming immune) or being dead. Examples of such diseases are smallpox, measles and chicken pox. Usually we think of SIR models as modelling epidemics, but we can also have endemic disease states if we include a fixed birthrate B (not per capita, but rather a fixed number ). Finally, we have a death rate d on the whole population N and this can be apportioned to each part of the model and a disease-related death rate 0 < c on infectives. (As usual, all parameters are non-negative.) The model is $$(d/dt)* S(t) = B − β I(t) S(t) − d S(t)$$
$$(d/dt)* I(t) = β I(t) S(t) − γ I(t) − c I(t) − d I(t) $$
$$(d/dt)* R(t) = γ I(t) − d R(t) $$ where R(t) is the number of removeds. Note that S + I + R = N, but we cannot non-dimensionalize here because of the birthrate. However, if we add the equations, we obtain $$(d/dt)* N(t) = B − c I(t) − d N(t)$$ and this can be used in place of the removed equation. Again, since the population is open, we cannot reduce to two equations. (Use something like F (for “fectives”) instead of I since Maple reserves I for the imaginary number i.)
(1) Write the system of equations for S, F, N in Maple.
(2) Find the fixed (or equilibrium) points for the system. Note what the disease-free state is and what the endemic state (i.e. coexistence state) is.
(3) Show that the basic reproductive ratio is $R_0 =(β*B)/(γ + c + d)d$
. (Hint: Look at what you get for N in the disease-free state and use the fact that $ 1/(γ + c + d)$ is the time spent infected.)