Different mathematical models have been used to predict the fraction of a population that will be infected by a disease under different conditions, one such model is called the Kermack-McKendrick model. This model gives the following equation when solved under certain conditions: ρe−qA =1−A (2) where A is the fraction of the population ultimately affected, q is a measure of disease transmissibility, and ρ is a measure of the fraction of the population that is initially susceptible to infection. How does the outbreak size, A change with an increase in transmissibility q, mathematically this is symbolised by the derivative dA ? dq 1 Solve the above problem through the following steps: (a) Equation 2 can’t be solved for A and fortunately we don’t have to solve this equation explicitly dA dA in order to find dq . What method of differentiation can we use to solve for dq .[1] dA (b) Apply this method of differentiation to equation 2, keeping in mind that A is a function of q, that dA is A (q) and solve for dq .[9] (c) State in English what you understand by the term dq .[2]
Disease Outbreak Size [12]
Different mathematical models have been used to predict the fraction of a population that will be infected by a disease under different conditions, one such model is called the Kermack-McKendrick model. This model gives the following equation when solved under certain conditions:
ρe−qA =1−A (2)
where A is the fraction of the population ultimately affected, q is a measure of disease transmissibility, and ρ is a measure of the fraction of the population that is initially susceptible to infection. How does the outbreak size, A change with an increase in transmissibility q, mathematically this is symbolised
by the derivative dA ? dq
1
Solve the above problem through the following steps:
(a) Equation 2 can’t be solved for A and fortunately we don’t have to solve this equation explicitly
dA dA
in order to find dq . What method of differentiation can we use to solve for dq .[1]
dA
(b) Apply this method of differentiation to equation 2, keeping in mind that A is a function of q, that dA
is A (q) and solve for dq .[9]
(c) State in English what you understand by the term dq .[2]
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