Starting with the SIR model with births and deaths (where is the birth and death rate, is the infection rate, and is the recovery rate), adding immunization at a rate that moves individuals from the susceptible to the recovered class leads to the model: dtdSdtdIdtdR=SNI+(NS)S=SNIII=I(NS(+ ))=IR+S where all other variables and parameters are as defined in class. Note that dSdN=0, so the total population size N=S+I+R is constant, and we can focus on dtdS and dtdI, ignoring dtdR. All parameters (,,, and ) are positive. (a) (5 pts) What are the units of the new parameter ? (b) (15 pts) Non-dimensionalize both population size and time. Start by providing dimensionless expressions for the proportion of susceptibles x, the proportion of infecteds y, and time in disease generations based on the total rate of departure from the infected population. Then provide expressions for ddx,ddy,R0 (infection rate in disease generations), (birth/death rate in disease generations), and a new parameter (rate of immunization expressed in disease generations) such that all variables and parameters are dimensionless. (c) (10 pts) Find the equilibrium value of x at the disease-free equilibrium with y=0. (d) (10 pts) Mathematically determine whether, or under what conditions, the diseasefree equilibrium found in (c) is locally stable. (e) (10 pts) Given your analysis, what is the rate of immunization that would prevent an epidemic? (Express your answer mathematically.).

1. Starting with the SIR model with births and deaths (where is the birth and death rate, is the
infection rate, and is the recovery rate), adding immunization at a rate that moves individuals from
the susceptible to the recovered class leads to the model: dtdSdtdIdtdR=SNI+(NS)S=SNIII=I(NS(+
))=IR+S where all other variables and parameters are as defined in class. Note that dSdN=0, so
the total population size N=S+I+R is constant, and we can focus on dtdS and dtdI, ignoring dtdR.
All parameters (,,, and ) are positive. (a) (5 pts) What are the units of the new parameter ? (b) (15
pts) Non-dimensionalize both population size and time. Start by providing dimensionless
expressions for the proportion of susceptibles x, the proportion of infecteds y, and time in disease
generations based on the total rate of departure from the infected population. Then provide
expressions for ddx,ddy,R0 (infection rate in disease generations), (birth/death rate in disease
generations), and a new parameter (rate of immunization expressed in disease generations) such
that all variables and parameters are dimensionless. (c) (10 pts) Find the equilibrium value of x at
the disease-free equilibrium with y=0. (d) (10 pts) Mathematically determine whether, or under
what conditions, the diseasefree equilibrium found in (c) is locally stable. (e) (10 pts) Given your
analysis, what is the rate of immunization that would prevent an epidemic? (Express your answer
mathematically.)