I appreciate any answer. Thank you. For each of the following description!) formulate a
mathematical model as a system of differential equations. In each case give a suitable
compartment diagram and define any parameters or symbols that you introduce that were not
mentioned as part of the question. Consider a model for the spread of a disease where lifelong
immunity is attained after catching the disease. The susceptibles are continuously vaccinated at a
per-capita rate mu against the disease. Develop differential equations for the number of
susceptibles S(t). the number of infectives I(t). the number vaccinated V(t) and the number
recovered. R(t). assuming all who recovered from the infection become immune for life. The
infectious disease Dengue fever is spread by infected mosquitoes that transmit the disease to
humans when they bite susceptible humans. Similarly, when a susceptible mosquito bites an
infected human the mosquito becomes infected Assume that humans cannot directly infect other
humans nor infected mosquitoes infect other mosquitoes. Let Sm(t) and Im(t) denote the number
of susceptible and infected mosquitoes respectively and let S h(t) and I h(t) denote the member of
susceptible and infected humans. Assume that there are no birth or deaths of humans over the
time-scale of interest but that there is a per-capita birth-rate of mosquitoes 6 and per-capita death
rate of mosquitoes of a. Also, assume that humans are infected for a period of gamma-1 and then
recover with immunity. Mosquitoes do not recover. (Note: typically gamma-1 is about. 2 weeks
for dengue fever).
Solution
The number of Vaccinated people (V(t)) is a simple one.
dV/dt = mu * S(t), where S(t) is the number of susceptable people.
A certain percentage of the people who need to be vaccinated are vaccinated every year, and this
will slow down as less people need to be vaccinated.
Susceptable people are decreasing over time, once you have been vaccinated, become sick, or
recover you are set for life.
S(t) = 1 - V(t) - R(t) - I(t)
The growth of the infective population depends on how infective the disease is, but it is also
dependant on how many people can be infected still. Let\'s say every infective person infects
beta percent of other vulnerable people, and they recover x days later.
dI/dt = [beta * I(t) * S(t)] - I(t-x)
The number of recovered people will follow the number of infective people but unlike infective
people they remain recovered
dR/dt = I(t-x)
All of these equations use a total population size of one, you are modelling the percentage of the
population that has fallen under these conditions. Eventually, the equations should reach these
final end conditions when t = infinity:
S(t) = 0, V(t) + R(t) = 1, and I(t) = 0.
Part two is just using these equations and applying them..
I appreciate any answer. Thank you. For each of the following descri.pdf
1. I appreciate any answer. Thank you. For each of the following description!) formulate a
mathematical model as a system of differential equations. In each case give a suitable
compartment diagram and define any parameters or symbols that you introduce that were not
mentioned as part of the question. Consider a model for the spread of a disease where lifelong
immunity is attained after catching the disease. The susceptibles are continuously vaccinated at a
per-capita rate mu against the disease. Develop differential equations for the number of
susceptibles S(t). the number of infectives I(t). the number vaccinated V(t) and the number
recovered. R(t). assuming all who recovered from the infection become immune for life. The
infectious disease Dengue fever is spread by infected mosquitoes that transmit the disease to
humans when they bite susceptible humans. Similarly, when a susceptible mosquito bites an
infected human the mosquito becomes infected Assume that humans cannot directly infect other
humans nor infected mosquitoes infect other mosquitoes. Let Sm(t) and Im(t) denote the number
of susceptible and infected mosquitoes respectively and let S h(t) and I h(t) denote the member of
susceptible and infected humans. Assume that there are no birth or deaths of humans over the
time-scale of interest but that there is a per-capita birth-rate of mosquitoes 6 and per-capita death
rate of mosquitoes of a. Also, assume that humans are infected for a period of gamma-1 and then
recover with immunity. Mosquitoes do not recover. (Note: typically gamma-1 is about. 2 weeks
for dengue fever).
Solution
The number of Vaccinated people (V(t)) is a simple one.
dV/dt = mu * S(t), where S(t) is the number of susceptable people.
A certain percentage of the people who need to be vaccinated are vaccinated every year, and this
will slow down as less people need to be vaccinated.
Susceptable people are decreasing over time, once you have been vaccinated, become sick, or
recover you are set for life.
S(t) = 1 - V(t) - R(t) - I(t)
The growth of the infective population depends on how infective the disease is, but it is also
2. dependant on how many people can be infected still. Let's say every infective person infects
beta percent of other vulnerable people, and they recover x days later.
dI/dt = [beta * I(t) * S(t)] - I(t-x)
The number of recovered people will follow the number of infective people but unlike infective
people they remain recovered
dR/dt = I(t-x)
All of these equations use a total population size of one, you are modelling the percentage of the
population that has fallen under these conditions. Eventually, the equations should reach these
final end conditions when t = infinity:
S(t) = 0, V(t) + R(t) = 1, and I(t) = 0.
Part two is just using these equations and applying them.
![dependant on how many people can be infected still. Let's say every infective person infects
beta percent of other vulnerable people, and they recover x days later.
dI/dt = [beta * I(t) * S(t)] - I(t-x)
The number of recovered people will follow the number of infective people but unlike infective
people they remain recovered
dR/dt = I(t-x)
All of these equations use a total population size of one, you are modelling the percentage of the
population that has fallen under these conditions. Eventually, the equations should reach these
final end conditions when t = infinity:
S(t) = 0, V(t) + R(t) = 1, and I(t) = 0.
Part two is just using these equations and applying them.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)