Find the equilibria of the system and sketch the direction

Question

If we assume that the hospital is always full, we can conserve the system by letting Λ = δ_sS(t) + δ_HH/(t) + δ_cC(t). In this case + C(t) + H(t) = N for all r (assuming you start with a population of size N).

(a) Show that this assumption simplifies the above system of equations to

dH/dt = (β_H/N)(N-C-M)H - (δ_H + α_H)H.

dc/dt = (β_c/N)(N-C-M)C - (δ_c + α_c)C.

S is then determined by the equation S(t) = N - H(t) - C(t).

Parameter values obtained from the Beth Israel Deaconess Medical Center. Plug these values into the model and then complete the following problems.

(b) Find the three equilibria (critical points) of the system (1).

(c) Using a computer, sketch the direction field for the system (I).

(d) Which trajectory configuration exists near each critical point (node, spiral, saddle, or center)? What do they represent in terms of how many patients are susceptible, colo­nized with HA-MRSA, and colonized with CA-MRSA over time?

(e) Examining the direction field, do you think CA-MRSA will overtake HA-MRSA in the hospital?