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Differential equations with a parametric form
$begingroup$
I have been exploring mathematical epidemiology for a school project and have decided to model epidemics using the SIR model (Susceptibles, Infected and Recovered), so $x(t), y(t), z(t)$ are functions describing the number of individuals belonging to each group. The model is described by a system of differential equations:
begin{equation}
frac{dx}{dt}=-beta x(t)y(t)end{equation}
begin{equation}
frac{dy}{dt}=beta x(t)y(t) - gamma y(t)
end{equation}
begin{equation}
frac{dz}{dt} = gamma y(t)
end{equation}
Where, $ beta, gamma $ are infection and recovery rates.
I have been reading a research paper showing how the system can be solved https://arxiv.org/abs/1403.2160, however, I have some difficulty with it.
(I'm only in the $10^{th}$ grade and have little understanding of differential equations}
This system can be reduced to an equation(I understand how):
begin{equation}
z''=-x_{0}beta z'e^{-frac{beta}{gamma}z}-gamma z
end{equation}
Where $x_{0}$ is the susceptible population at $t=0$. Then in the paper they introduce a function
begin{equation}
u(t)=e^{-frac{beta}{gamma}z(t)}
end{equation}
Which at $t_0$ takes the value
begin{equation}
u_{0}=e^{-frac{beta}{gamma}z}
end{equation}
Substitution of this equation into the previous one leads to:
begin{equation}
ufrac{d^{2}u}{dt^{2}}-bigg(frac{du}{dt}bigg)^{2}+bigg(gamma - x_{0}beta ubigg)u frac{du}{dt} = 0
end{equation}
Then they introduce an equation:
begin{equation}
phi=frac{dt}{du}
end{equation}
By substituting it into the previous equation they get:
begin{equation}
frac{dphi}{du}+frac{1}{u} phi = (gamma-x_{0}beta u)phi^{2}
end{equation}
The solution of this equation is:
begin{equation}
phi = frac{1}{u(C_{1}-gamma ln u +x_{0}beta u)}
end{equation}
Where $C_{1}$ is an arbitrary integration constant. Up to this point I have no problems, however, then they proceed with saying in view of equation $phi=frac{dt}{du}$ and the previous equation, they get:
begin{equation}
t-t_{0} = int_{u_0}^{u}frac{dxi}{xi(C_{1}-gamma lnxi + x_{0}beta xi}
end{equation}
where one can choose $t_{0}=0$ without loss of generality. Then they proceed by saying hence we have obtained the complete exact solution of the original system, given in parametric form by:
begin{equation}
x=x_{0}u,
end{equation}
begin{equation}
y=frac{gamma}{beta}ln u -x_{0}u - frac{C_{1}}{beta},
end{equation}
begin{equation}
z = -frac{gamma}{beta}ln u
end{equation}
and the next equation (which is derived from the fact that $x+y+z=N$ where $N$ is the whole population which stays constant during the epidemic) understand is:
begin{equation}
C_{1} = -beta N
end{equation}
Could someone please explain how they acquired the last few equations and what they mean, moreover, if I wanted to graph them how could I do it? Thank you
calculus integration ordinary-differential-equations parametric
asked Jan 9 at 17:27
nipohc88nipohc88
163
$endgroup$
add a comment |
$begingroup$
I have been exploring mathematical epidemiology for a school project and have decided to model epidemics using the SIR model (Susceptibles, Infected and Recovered), so $x(t), y(t), z(t)$ are functions describing the number of individuals belonging to each group. The model is described by a system of differential equations:
begin{equation}
frac{dx}{dt}=-beta x(t)y(t)end{equation}
begin{equation}
frac{dy}{dt}=beta x(t)y(t) - gamma y(t)
end{equation}
begin{equation}
frac{dz}{dt} = gamma y(t)
end{equation}
Where, $ beta, gamma $ are infection and recovery rates.
I have been reading a research paper showing how the system can be solved https://arxiv.org/abs/1403.2160, however, I have some difficulty with it.
(I'm only in the $10^{th}$ grade and have little understanding of differential equations}
This system can be reduced to an equation(I understand how):
begin{equation}
z''=-x_{0}beta z'e^{-frac{beta}{gamma}z}-gamma z
end{equation}
Where $x_{0}$ is the susceptible population at $t=0$. Then in the paper they introduce a function
begin{equation}
u(t)=e^{-frac{beta}{gamma}z(t)}
end{equation}
Which at $t_0$ takes the value
begin{equation}
u_{0}=e^{-frac{beta}{gamma}z}
end{equation}
Substitution of this equation into the previous one leads to:
begin{equation}
ufrac{d^{2}u}{dt^{2}}-bigg(frac{du}{dt}bigg)^{2}+bigg(gamma - x_{0}beta ubigg)u frac{du}{dt} = 0
end{equation}
Then they introduce an equation:
begin{equation}
phi=frac{dt}{du}
end{equation}
By substituting it into the previous equation they get:
begin{equation}
frac{dphi}{du}+frac{1}{u} phi = (gamma-x_{0}beta u)phi^{2}
end{equation}
The solution of this equation is:
begin{equation}
phi = frac{1}{u(C_{1}-gamma ln u +x_{0}beta u)}
end{equation}
Where $C_{1}$ is an arbitrary integration constant. Up to this point I have no problems, however, then they proceed with saying in view of equation $phi=frac{dt}{du}$ and the previous equation, they get:
begin{equation}
t-t_{0} = int_{u_0}^{u}frac{dxi}{xi(C_{1}-gamma lnxi + x_{0}beta xi}
end{equation}
where one can choose $t_{0}=0$ without loss of generality. Then they proceed by saying hence we have obtained the complete exact solution of the original system, given in parametric form by:
begin{equation}
x=x_{0}u,
end{equation}
begin{equation}
y=frac{gamma}{beta}ln u -x_{0}u - frac{C_{1}}{beta},
end{equation}
begin{equation}
z = -frac{gamma}{beta}ln u
end{equation}
and the next equation (which is derived from the fact that $x+y+z=N$ where $N$ is the whole population which stays constant during the epidemic) understand is:
begin{equation}
C_{1} = -beta N
end{equation}
Could someone please explain how they acquired the last few equations and what they mean, moreover, if I wanted to graph them how could I do it? Thank you
calculus integration ordinary-differential-equations parametric
asked Jan 9 at 17:27
nipohc88nipohc88
163
$endgroup$
add a comment |
$begingroup$
I have been exploring mathematical epidemiology for a school project and have decided to model epidemics using the SIR model (Susceptibles, Infected and Recovered), so $x(t), y(t), z(t)$ are functions describing the number of individuals belonging to each group. The model is described by a system of differential equations:
begin{equation}
frac{dx}{dt}=-beta x(t)y(t)end{equation}
begin{equation}
frac{dy}{dt}=beta x(t)y(t) - gamma y(t)
end{equation}
begin{equation}
frac{dz}{dt} = gamma y(t)
end{equation}
Where, $ beta, gamma $ are infection and recovery rates.
I have been reading a research paper showing how the system can be solved https://arxiv.org/abs/1403.2160, however, I have some difficulty with it.
(I'm only in the $10^{th}$ grade and have little understanding of differential equations}
This system can be reduced to an equation(I understand how):
begin{equation}
z''=-x_{0}beta z'e^{-frac{beta}{gamma}z}-gamma z
end{equation}
Where $x_{0}$ is the susceptible population at $t=0$. Then in the paper they introduce a function
begin{equation}
u(t)=e^{-frac{beta}{gamma}z(t)}
end{equation}
Which at $t_0$ takes the value
begin{equation}
u_{0}=e^{-frac{beta}{gamma}z}
end{equation}
Substitution of this equation into the previous one leads to:
begin{equation}
ufrac{d^{2}u}{dt^{2}}-bigg(frac{du}{dt}bigg)^{2}+bigg(gamma - x_{0}beta ubigg)u frac{du}{dt} = 0
end{equation}
Then they introduce an equation:
begin{equation}
phi=frac{dt}{du}
end{equation}
By substituting it into the previous equation they get:
begin{equation}
frac{dphi}{du}+frac{1}{u} phi = (gamma-x_{0}beta u)phi^{2}
end{equation}
The solution of this equation is:
begin{equation}
phi = frac{1}{u(C_{1}-gamma ln u +x_{0}beta u)}
end{equation}
Where $C_{1}$ is an arbitrary integration constant. Up to this point I have no problems, however, then they proceed with saying in view of equation $phi=frac{dt}{du}$ and the previous equation, they get:
begin{equation}
t-t_{0} = int_{u_0}^{u}frac{dxi}{xi(C_{1}-gamma lnxi + x_{0}beta xi}
end{equation}
where one can choose $t_{0}=0$ without loss of generality. Then they proceed by saying hence we have obtained the complete exact solution of the original system, given in parametric form by:
begin{equation}
x=x_{0}u,
end{equation}
begin{equation}
y=frac{gamma}{beta}ln u -x_{0}u - frac{C_{1}}{beta},
end{equation}
begin{equation}
z = -frac{gamma}{beta}ln u
end{equation}
and the next equation (which is derived from the fact that $x+y+z=N$ where $N$ is the whole population which stays constant during the epidemic) understand is:
begin{equation}
C_{1} = -beta N
end{equation}
Could someone please explain how they acquired the last few equations and what they mean, moreover, if I wanted to graph them how could I do it? Thank you
calculus integration ordinary-differential-equations parametric
asked Jan 9 at 17:27
nipohc88nipohc88
163
$endgroup$
I have been exploring mathematical epidemiology for a school project and have decided to model epidemics using the SIR model (Susceptibles, Infected and Recovered), so $x(t), y(t), z(t)$ are functions describing the number of individuals belonging to each group. The model is described by a system of differential equations:
begin{equation}
frac{dx}{dt}=-beta x(t)y(t)end{equation}
begin{equation}
frac{dy}{dt}=beta x(t)y(t) - gamma y(t)
end{equation}
begin{equation}
frac{dz}{dt} = gamma y(t)
end{equation}
Where, $ beta, gamma $ are infection and recovery rates.
I have been reading a research paper showing how the system can be solved https://arxiv.org/abs/1403.2160, however, I have some difficulty with it.
(I'm only in the $10^{th}$ grade and have little understanding of differential equations}
This system can be reduced to an equation(I understand how):
begin{equation}
z''=-x_{0}beta z'e^{-frac{beta}{gamma}z}-gamma z
end{equation}
Where $x_{0}$ is the susceptible population at $t=0$. Then in the paper they introduce a function
begin{equation}
u(t)=e^{-frac{beta}{gamma}z(t)}
end{equation}
Which at $t_0$ takes the value
begin{equation}
u_{0}=e^{-frac{beta}{gamma}z}
end{equation}
Substitution of this equation into the previous one leads to:
begin{equation}
ufrac{d^{2}u}{dt^{2}}-bigg(frac{du}{dt}bigg)^{2}+bigg(gamma - x_{0}beta ubigg)u frac{du}{dt} = 0
end{equation}
Then they introduce an equation:
begin{equation}
phi=frac{dt}{du}
end{equation}
By substituting it into the previous equation they get:
begin{equation}
frac{dphi}{du}+frac{1}{u} phi = (gamma-x_{0}beta u)phi^{2}
end{equation}
The solution of this equation is:
begin{equation}
phi = frac{1}{u(C_{1}-gamma ln u +x_{0}beta u)}
end{equation}
Where $C_{1}$ is an arbitrary integration constant. Up to this point I have no problems, however, then they proceed with saying in view of equation $phi=frac{dt}{du}$ and the previous equation, they get:
begin{equation}
t-t_{0} = int_{u_0}^{u}frac{dxi}{xi(C_{1}-gamma lnxi + x_{0}beta xi}
end{equation}
where one can choose $t_{0}=0$ without loss of generality. Then they proceed by saying hence we have obtained the complete exact solution of the original system, given in parametric form by:
begin{equation}
x=x_{0}u,
end{equation}
begin{equation}
y=frac{gamma}{beta}ln u -x_{0}u - frac{C_{1}}{beta},
end{equation}
begin{equation}
z = -frac{gamma}{beta}ln u
end{equation}
and the next equation (which is derived from the fact that $x+y+z=N$ where $N$ is the whole population which stays constant during the epidemic) understand is:
begin{equation}
C_{1} = -beta N
end{equation}
Could someone please explain how they acquired the last few equations and what they mean, moreover, if I wanted to graph them how could I do it? Thank you
calculus integration ordinary-differential-equations parametric
calculus integration ordinary-differential-equations parametric
asked Jan 9 at 17:27
nipohc88nipohc88
163
asked Jan 9 at 17:27
nipohc88nipohc88
163
asked Jan 9 at 17:27
nipohc88nipohc88
163
asked Jan 9 at 17:27
nipohc88nipohc88
163
asked Jan 9 at 17:27
nipohc88nipohc88
163
163
add a comment |
add a comment |
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