Write a research paper about( study the stability of the equilibrium points in( the basic SIR model , SIR with birth and death, herd immune and vaccination model, The SIRS model) + provide examples of each case and phase portrait.
abstract
The SIR model
introduction (basic SIR model)
define virable and parameters
use this SIR model+ Ro
equations governing the disease can be modeled as
dS/dt = -αS!
dI/dt = αSl - β1
dR/dt = β1
Remark. Since the total population is assumed to be constant, the third equation can be derived from the first two. Basically we study the first two in detail.
It turns out that the epidemic occurs if > 0, it doesn't if I < 0. So for the epidemic to occur we have to have aS > 13 implying S > 1. For the epidemic to terminate the rate of change of I has to be negative, this implies that
S < β/a
The phase portrait Figure shows this too.
FIGURE. The Phase Portrait of SIR model
Definition 2 (Basic Reproductive Number). The basic reproductive number Re(the average number of persons infected by one case in a totally susceptible population in absence of interventions aimed at con¬trolling the infection). Since S = 1 initially, the ratio αS/β= α/β = Ro.
This is one of the most important parameters in the SIR modeling of any epidemic. Re is especially important in this case as it will inform one as to when an epidemic is in progress. So if Ro > 1 an epidemic will occur and if Ro < 1 there will be no epidemic.
The values of Ro are known for various diseases. For example for Swine flu, it is reported to be 1.3 - 1.6 in 111
the stability of equilibrium.
Linearization
Jacobian Matrix
discuss the stability of the equilibrium points when Epidemic, "endemic" and not the phase portrait of the each case
Examples with phase portrait(for Epidemic case and not with discussion) other info that be useful here.