daviddoji/CalculusWithJuliaNotes.jl
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jverzani
2025-04-16 14:31:16 -04:00
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quarto/ODEs/differential_equations.qmd
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@@ -79,7 +79,7 @@ R(0) &= 0\\
\end{align*}
$$
In `Julia` we define these, `N` to model the total population, and `u0` to be the proportions.
In `Julia` we define these parameter values and `N` to model the total population and `u0` to be represent the proportions.
```{julia}
@@ -94,7 +94,7 @@ An *estimated* set of values for $k$ and $b$ are $k=1/3$, coming from the averag
Okay, the mathematical modeling is done; now we try to solve for the unknown functions using `DifferentialEquations`.
To warm up, if $b=0$ then $i'(t) = -k \cdot i(t)$ describes the infected. (There is no circulation of people in this case.) The solution would be achieved through:
To warm up, if $b=0$ then $i'(t) = -k \cdot i(t)$ describes the infected. (There is no circulation of people in this case.) This is a single ODE. The solution would be achieved through:
```{julia}
@@ -102,10 +102,12 @@ To warm up, if $b=0$ then $i'(t) = -k \cdot i(t)$ describes the infected. (The
k = 1/3
f(u,p,t) = -k * u # solving u′(t) = - k u(t)
uᵢ0= I0/N
time_span = (0.0, 20.0)
prob = ODEProblem(f, I0/N, time_span)
sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)
prob = ODEProblem(f, uᵢ0, time_span)
sol = solve(prob, Tsit5(); reltol=1e-8, abstol=1e-8)
plot(sol)
```
@@ -120,7 +122,7 @@ $$
\frac{di}{dt} = -k \cdot i(t) = F(i(t), k, t)
$$
where $F$ depends on the current value ($i$), a parameter ($k$), and the time ($t$). We did not utilize $p$ above for the parameter, as it was easy not to, but could have, and will in the following. The time variable $t$ does not appear by itself in our equation, so only `f(u, p, t) = -k * u` was used, `u` the generic name for a solution which in this case is $i$.
where $F$ depends on the current value ($i$), a parameter ($k$), and the time ($t$). We did not utilize $p$ above for the parameter, as it was easy not to, but could have, and will in the following. The time variable $t$ does not appear by itself in our equation, so only `f(u, p, t) = -k * u` was used, `u` the generic name for a solution which in this case was labeled with an $i$.
The problem we set up needs an initial value (the $u0$) and a time span to solve over. Here we want time to model real time, so use floating point values.
@@ -167,7 +169,7 @@ The `sir!` function has the trailing `!` indicating – by convention – it *mu
:::
With the update function defined, the problem is setup and a solution found with in the same manner:
With the update function defined, the problem is setup and a solution is found using the same manner as before:
```{julia}
@@ -193,7 +195,7 @@ p = (k=1/2, b=2) # change b from 1/2 to 2 -- more daily contact
prob = ODEProblem(sir!, u0, time_span, p)
sol = solve(prob, Tsit5())
plot(sol)
plot(sol; legend=:right)
```
The graphs are somewhat similar, but the steady state is reached much more quickly and nearly everyone became infected.
@@ -252,7 +254,7 @@ end
p
```
The 3-dimensional graph with `plotly` can have its viewing angle adjusted with the mouse. When looking down on the $x-y$ plane, which code `b` and `k`, we can see the rapid growth along a line related to $b/k$.
(A 3-dimensional graph with `plotly` or `Makie` can have its viewing angle adjusted with the mouse. When looking down on the $x-y$ plane, which code `b` and `k`, we can see the rapid growth along a line related to $b/k$.)
Smith and Moore point out that $k$ is roughly the reciprocal of the number of days an individual is sick enough to infect others. This can be estimated during a breakout. However, they go on to note that there is no direct way to observe $b$, but there is an indirect way.
@@ -382,7 +384,7 @@ SOL = solve(trajectory_problem, Tsit5(); p = ps, callback=cb)
plot(t -> SOL(t)[1], t -> SOL(t)[2], TSPAN...; legend=false)
```
Finally, we note that the `ModelingToolkit` package provides symbolic-numeric computing. This allows the equations to be set up symbolically, as in `SymPy` before being passed off to `DifferentialEquations` to solve numerically. The above example with no wind resistance could be translated into the following:
Finally, we note that the `ModelingToolkit` package provides symbolic-numeric computing. This allows the equations to be set up symbolically, as has been illustrated with `SymPy`, before being passed off to `DifferentialEquations` to solve numerically. The above example with no wind resistance could be translated into the following:
```{julia}