revising models
This commit is contained in:
mhjensen
@@ -172,7 +172,7 @@ will introduce various machine learning algorithm s to make fits of
|
||||
the data data and predictions. We move thereafter to more interesting
|
||||
cases such as the simulation of financial transactions or disease
|
||||
models. These are examples where we can easily set up the data and
|
||||
then use machine learning algorithms using included in for example _scikit_learn_. Another model we
|
||||
then use machine learning algorithms using included in for example _scikit-learn_. Another model we
|
||||
will consider is the so-called Ising model. Here we will use this
|
||||
model to produce data for selected spin configurations and attempt to classify the data.
|
||||
Finally, our last example consists of economic data from the OECD.
|
||||
@@ -180,6 +180,7 @@ Finally, our last example consists of economic data from the OECD.
|
||||
|
||||
!split
|
||||
===== Software and needed installations =====
|
||||
|
||||
We will make intensive use of python as programming language and the myriad of available libraries.
|
||||
Furthermore, you will find IPython/Jupyter notebooks invaluable in your work.
|
||||
You can run _R_ codes in the Jupyter/IPython notebooks, with the immediate benefit of visualizing your data.
|
||||
@@ -321,7 +322,7 @@ plt.show()
|
||||
|
||||
|
||||
!split
|
||||
===== Simple regression model, now using scikit=learn =====
|
||||
===== Simple regression model, now using _scikit-learn_ =====
|
||||
Add info about the equations
|
||||
!bc pycod
|
||||
# Importing various packages
|
||||
@@ -748,7 +749,6 @@ text(0, 0.5, paste("y =x^ (", power, " +/- ", power.se, ")", sep = ""), pos = 4)
|
||||
!eblock
|
||||
|
||||
|
||||
!split
|
||||
|
||||
!split
|
||||
===== Example: ecoli lab experiment =====
|
||||
@@ -846,613 +846,6 @@ Change `r` in the program and play around to make a better fit!
|
||||
|
||||
|
||||
|
||||
!split
|
||||
===== We shall model a very complex phenomenon by simple math.... =====
|
||||
|
||||
!bblock Assumptions:
|
||||
* We consider a perfectly mixed population in a confined area
|
||||
* No spatial transport, just temporal evolution
|
||||
* We do not consider individuals, just a grand mix of them<linebreak>
|
||||
(cf. statistical mechanics vs thermodynamics)
|
||||
!eblock
|
||||
|
||||
!bpop
|
||||
!bblock (small)
|
||||
We consider very simple models, but these can be extended to full
|
||||
models that are used world-wide by health authorities. Typical
|
||||
diseases modeled are flu, measles, swine flu, HIV, ...
|
||||
!eblock
|
||||
!epop
|
||||
|
||||
|
||||
!split
|
||||
===== We keep track of 3 categories in the SIR model =====
|
||||
|
||||
!bblock
|
||||
* _S_: susceptibles - who can get the disease
|
||||
* _I_: infected - who have developed the disease and infect susceptibles
|
||||
* _R_: recovered - who have recovered and become immune
|
||||
!eblock
|
||||
|
||||
!bblock Mathematical quantities:
|
||||
$S(t)$, $I(t)$, $R(t)$: no of people in each category
|
||||
!eblock
|
||||
|
||||
!bblock Goal:
|
||||
Find and solve equations for $S(t)$, $I(t)$, $R(t)$
|
||||
!eblock
|
||||
|
||||
FIGURE: [fig/categories_SIR, width=400 frac=0.5]
|
||||
|
||||
!split
|
||||
===== The traditional modeling approach is very mathematical - our idea is to model, program and experiment =====
|
||||
|
||||
!bblock
|
||||
* Numerous books on mathematical biology treat the SIR model
|
||||
* Quick modeling step (max 2 pages)
|
||||
* Nonlinear differential equation model
|
||||
* Cannot solve the equations, so focus is on discussing
|
||||
stability (eigenvalues), qualitative properties, etc.
|
||||
* Very few extensions of the model to real-life situations
|
||||
!eblock
|
||||
|
||||
|
||||
!split
|
||||
===== Dynamics in a time interval $\Delta t$: $\Delta t\,\beta SI$ people move from S to I =====
|
||||
|
||||
!bblock S-I interaction:
|
||||
* In a mix of S and I people, there are $SI$ possible pairs
|
||||
* A certain fraction $\Delta t\,\beta$ of $SI$ meet in a (small)
|
||||
time interval $\Delta t$, with the result that the infected
|
||||
``successfully'' infects the susceptible
|
||||
* The loss $\Delta t\,\beta SI$ in the S catogory is a corresponding
|
||||
gain in the I category
|
||||
!eblock
|
||||
|
||||
!bpop
|
||||
!bblock (small) Remark
|
||||
It is reasonable that the fraction depends on $\Delta t$
|
||||
(twice as many infected in $2\Delta t$ as in $\Delta t$).
|
||||
$\beta$ is some unknown parameter we must measure, supposed to not
|
||||
depend on $\Delta t$, but maybe time $t$.
|
||||
$\beta$ lumps *a lot* of biological and sociological effects into
|
||||
one number.
|
||||
!eblock
|
||||
!epop
|
||||
|
||||
!split
|
||||
===== For practical calculations, we must express the S-I interaction with symbols =====
|
||||
|
||||
Loss in $S(t)$ from time $t$ to $t+\Delta t$:
|
||||
|
||||
!bt
|
||||
\[ S(t+\Delta t) = S(t) - \Delta t\,\beta S(t)I(t)\]
|
||||
!et
|
||||
|
||||
Gain in $I(t)$:
|
||||
|
||||
!bt
|
||||
\[ I(t+\Delta t) = I(t) + \Delta t\,\beta S(t)I(t)\]
|
||||
!et
|
||||
|
||||
!split
|
||||
===== Modeling the interaction between R and I =====
|
||||
|
||||
!bblock R-I interaction:
|
||||
* After some days, the infected has recovered and moves to the R category
|
||||
* A simple model: in a small time $\Delta t$ (say 1 day),
|
||||
a fraction $\Delta t\,\nu$ of the infected are removed
|
||||
($\nu$ must be measured)
|
||||
!eblock
|
||||
|
||||
We must subtract this fraction in the balance equation for $I$:
|
||||
|
||||
!bt
|
||||
\[ I(t+\Delta t) = I(t) + \Delta t\,\beta S(t)I(t) -\Delta t\,\nu I(t) \]
|
||||
!et
|
||||
|
||||
The loss $\Delta t\,\nu I$ is a gain in $R$:
|
||||
|
||||
!bt
|
||||
\[ R(t+\Delta t) = R(t) + \Delta t\,\nu I(t)\]
|
||||
!et
|
||||
|
||||
|
||||
!split
|
||||
===== We have three equations for $S$, $I$, and $R$ =====
|
||||
|
||||
!bt
|
||||
\begin{align}
|
||||
S(t+\Delta t) &= S(t) - \Delta t\,\beta S(t)I(t)
|
||||
label{SIR1:S}\\
|
||||
I(t+\Delta t) &= I(t) + \Delta t\,\beta S(t)I(t) -\Delta t\nu I(t)
|
||||
label{SIR1:I}\\
|
||||
R(t+\Delta t) &= R(t) + \Delta t\,\nu I(t)
|
||||
label{SIR1:R}
|
||||
\end{align}
|
||||
!et
|
||||
|
||||
FIGURE: [fig/categories_SIR, width=400 frac=0.5]
|
||||
|
||||
Before we can compute with these, we must
|
||||
|
||||
* know $\beta$ and $\nu$
|
||||
* know $S(0)$ (many), $I(0)$ (few), $R(0)$ (0?)
|
||||
* choose $\Delta t$
|
||||
|
||||
!split
|
||||
===== The computation involves just simple arithmetics =====
|
||||
|
||||
* Set $\Delta t=6$ minutes
|
||||
* Set $\beta =0.0013$, $\nu =0.8333$
|
||||
* Set $S(0)=50$, $I(0)=1$, $R(0)=0$
|
||||
|
||||
!bt
|
||||
\begin{align*}
|
||||
S(\Delta t) &= S(0) - \Delta t\,\beta S(0)I(0)\approx 49.99\\
|
||||
I(\Delta t) &= I(0) + \Delta t\,\beta S(0)I(0) -\Delta t\,\nu I(0)\approx 1.002\\
|
||||
R(\Delta t) &= R(0) + \Delta t\,\nu I(0)\approx 0.0008333
|
||||
\end{align*}
|
||||
!et
|
||||
|
||||
!bpop
|
||||
* In reality, $S$, $I$, $R$ are integers and events are discrete (meet, get sick)
|
||||
* In the model, we work with real numbers and continuous events
|
||||
* Reasonable approximation in a not too small population
|
||||
!epop
|
||||
|
||||
!split
|
||||
===== And we can continue... =====
|
||||
|
||||
!bt
|
||||
\begin{align*}
|
||||
S(2\Delta t) &= S(\Delta t) - \Delta t\,\beta S(\Delta t)I(\Delta t)\approx 49.87\\
|
||||
I(2\Delta t) &= I(\Delta t) + \Delta t\,\beta S(\Delta t)I(\Delta t) -\Delta t\,\nu I(\Delta t)\approx 1.011\\
|
||||
R(2\Delta t) &= R(\Delta t) + \Delta t\,\nu I(\Delta t)\approx 0.00167
|
||||
\end{align*}
|
||||
!et
|
||||
|
||||
Repeat...
|
||||
|
||||
!bt
|
||||
\begin{align*}
|
||||
S(3\Delta t) &= S(2\Delta t) - \Delta t\,\beta S(2\Delta t)I(2\Delta t)\approx 49.98\\
|
||||
I(3\Delta t) &= I(2\Delta t) + \Delta t\,\beta S(2\Delta t)I(2\Delta t) -\Delta t\,\nu I(2\Delta t)\approx 1.017\\
|
||||
R(3\Delta t) &= R(2\Delta t) + \Delta t\,\nu I(2\Delta t)\approx 0.0025
|
||||
\end{align*}
|
||||
!et
|
||||
|
||||
!bpop
|
||||
But this is getting boring! Let's ask a computer to do the work!
|
||||
!epop
|
||||
|
||||
|
||||
|
||||
!split
|
||||
===== First, some handy notation =====
|
||||
|
||||
!bt
|
||||
\[ S^n = S(n\Delta t),\quad I^n = I(n\Delta t),\quad R^n = R(n\Delta t)\]
|
||||
!et
|
||||
|
||||
!bt
|
||||
\[ S^{n+1} = S((n+1)\Delta t),\quad I^{n+1} = I((n+1)\Delta t),\quad R^{n+1} = R((n+1)\Delta t)\]
|
||||
!et
|
||||
|
||||
The equations can now be written more compactly (and computer friendly):
|
||||
|
||||
!bt
|
||||
\begin{align}
|
||||
S^{n+1} &= S^n - \Delta t\,\beta S^nI^n
|
||||
label{SIR1:Sc}\\
|
||||
I^{n+1} &= I^n + \Delta t\,\beta S^nI^n -\Delta t\,\nu I^n
|
||||
label{SIR1:Ic}\\
|
||||
R^{n+1} &= R^n + \Delta t\,\nu I^n
|
||||
label{SIR1:Rc}
|
||||
\end{align}
|
||||
!et
|
||||
|
||||
!split
|
||||
===== With variables, arrays, and a loop we can program =====
|
||||
|
||||
Suppose we want to compute until $t=N\Delta t$, i.e., for $n=0,1,\ldots,N-1$.
|
||||
We can store $S^0, S^1, S^2, \ldots, S^N$ in an array (or list).
|
||||
|
||||
Python (Matlab):
|
||||
|
||||
!bc pycod
|
||||
t = linspace(0, N*dt, N+1) # all time points
|
||||
S = zeros(N+1)
|
||||
I = zeros(N+1)
|
||||
R = zeros(N+1)
|
||||
|
||||
for n in range(N):
|
||||
S[n+1] = S[n] - dt*beta*S[n]*I[n]
|
||||
I[n+1] = I[n] + dt*beta*S[n]*I[n] - dt*nu*I[n]
|
||||
R[n+1] = R[n] + dt*nu*I[n]
|
||||
!ec
|
||||
|
||||
!split
|
||||
===== Here is the complete program =====
|
||||
|
||||
!bc pycod
|
||||
beta = 0.0013
|
||||
nu =0.8333
|
||||
dt = 0.1 # 6 min (time measured in hours)
|
||||
D = 30 # simulate for D days
|
||||
N = int(D*24/dt) # corresponding no of hours
|
||||
|
||||
from numpy import zeros, linspace
|
||||
t = linspace(0, N*dt, N+1)
|
||||
S = zeros(N+1)
|
||||
I = zeros(N+1)
|
||||
R = zeros(N+1)
|
||||
|
||||
for n in range(N):
|
||||
S[n+1] = S[n] - dt*beta*S[n]*I[n]
|
||||
I[n+1] = I[n] + dt*beta*S[n]*I[n] - dt*nu*I[n]
|
||||
R[n+1] = R[n] + dt*nu*I[n]
|
||||
|
||||
# Plot the graphs
|
||||
from matplotlib.pyplot import *
|
||||
plot(t, S, 'k-', t, I, 'b-', t, R, 'r-')
|
||||
legend(['S', 'I', 'R'], loc='lower right')
|
||||
xlabel('hours')
|
||||
show()
|
||||
!ec
|
||||
|
||||
!split
|
||||
===== We have predicted a disease! =====
|
||||
|
||||
FIGURE: [fig/SIR1, width=800]
|
||||
|
||||
|
||||
!split
|
||||
===== How much math and programming did we use? =====
|
||||
|
||||
!bblock Math:
|
||||
* Plain arithmetics
|
||||
* The concept of a graph (i.e., discrete function in time)
|
||||
* Units
|
||||
* Greek letters
|
||||
!eblock
|
||||
|
||||
!bblock Programming:
|
||||
* Variable
|
||||
* Array
|
||||
* Loop
|
||||
* Plotting
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== Detour: The standard mathematical approach =====
|
||||
|
||||
We had from intuition established
|
||||
|
||||
!bt
|
||||
\begin{align*}
|
||||
S(t+\Delta t) &= S(t) - \Delta t\,\beta S(t)I(t)\\
|
||||
I(t+\Delta t) &= I(t) + \Delta t\,\beta S(t)I(t) -\Delta t\,\nu I(t)\\
|
||||
R(t+\Delta t) &= R(t) + \Delta t\,\nu R(t)
|
||||
\end{align*}
|
||||
!et
|
||||
|
||||
The mathematician will now make *differential equations*.
|
||||
Divide by $\Delta t$ and rearrange:
|
||||
|
||||
!bt
|
||||
\begin{align*}
|
||||
\frac{S(t+\Delta t) - S(t)}{\Delta t} &= - \beta S(t)I(t)\\
|
||||
\frac{I(t+\Delta t) - I(t)}{\Delta t} &= \beta t S(t)I(t) -\nu I(t)\\
|
||||
\frac{R(t+\Delta t) - R(t)}{\Delta t} &= \nu R(t)
|
||||
\end{align*}
|
||||
!et
|
||||
|
||||
!split
|
||||
===== A derivative arises as $\Delta t\rightarrow 0$ =====
|
||||
|
||||
In any calculus book, the derivative of $S$ at $t$ is defined as
|
||||
|
||||
!bt
|
||||
\[ S'(t) = \lim_{t\rightarrow 0}\frac{S(t+\Delta t) - S(t)}{\Delta t}\]
|
||||
!et
|
||||
|
||||
If we let $\Delta t\rightarrow 0$, we get derivatives on the left-hand side:
|
||||
|
||||
!bt
|
||||
\begin{align*}
|
||||
S'(t) &= - \beta S(t)I(t)\\
|
||||
I'(t) &= \beta t S(t)I(t) -\nu I(t)\\
|
||||
R'(t) &= \nu R(t)
|
||||
\end{align*}
|
||||
!et
|
||||
|
||||
This is a 3x3 system of differential equations for the functions
|
||||
$S(t)$, $I(t)$, $R(t)$. For a unique solution, we need
|
||||
$S(0)$, $I(0)$, $R(0)$.
|
||||
|
||||
!split
|
||||
===== Bad news: we cannot solve these equations! =====
|
||||
|
||||
!bblock Time to ask a numerical methods expert:
|
||||
Replace the derivative with a *finite difference*, e.g.,
|
||||
|
||||
!bt
|
||||
\[ S'(t) \approx \frac{S(t+\Delta t) - S(t)}{\Delta t}\]
|
||||
!et
|
||||
which is accurate for small $\Delta t$.
|
||||
!eblock
|
||||
|
||||
This brings us back to the first model, which we can solve
|
||||
on a computer!
|
||||
|
||||
% if EXTRA:
|
||||
!split
|
||||
===== SIR is an ideal model for teaching modeling =====
|
||||
|
||||
!bquestion
|
||||
``I believe genes are important for spreading of diseases. It's not
|
||||
included in the model.''
|
||||
!equestion
|
||||
% endif
|
||||
|
||||
!split
|
||||
===== Parameter estimation is needed for predictive modeling =====
|
||||
|
||||
* Any small $\Delta t$ will do
|
||||
* One can reason about $\nu$ and say that $1/\nu$ is the mean
|
||||
recovery time for the disease (e.g., 1 week for a flu)
|
||||
* $\beta$ must in some way be measured, but we don't know what it means...
|
||||
|
||||
!bblock So, what if we don't know $\beta$?
|
||||
* Can still learn about the *dynamics* of diseases
|
||||
* Can find the sensitivity to and influence of $\beta$
|
||||
* Can apply *parameter estimation* procedures to fit $\beta$ to data
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== Let us extend the model: no life-long immunity =====
|
||||
|
||||
!bblock Assumption
|
||||
After some time, people in the R category lose the immunity.
|
||||
In a small time $\Delta t$ this gives a leakage $\Delta t\,\gamma R$
|
||||
to the S category. ($1/\gamma$ is the mean time for immunity.)
|
||||
!eblock
|
||||
|
||||
FIGURE: [fig/categories_SIR_feedback, width=400 frac=0.5]
|
||||
|
||||
!bt
|
||||
\begin{align}
|
||||
S^{n+1} &= S^n - \Delta t\,\beta S^nI^n + {\color{red}\Delta t\,\gamma R^n}
|
||||
label{SIR2:S}\\
|
||||
I^{n+1} &= I^n + \Delta t\,\beta S^nI^n -\Delta t\,\nu I^n
|
||||
label{SIR2:I}\\
|
||||
R^{n+1} &= R^n + \Delta t\,\nu R^n - {\color{red}\Delta t\,\gamma R^n}
|
||||
label{SIR2:R}
|
||||
\end{align}
|
||||
!et
|
||||
|
||||
No complications in the computational model!
|
||||
|
||||
!split
|
||||
===== The effect of loss of immunity =====
|
||||
|
||||
$1/\gamma = 50$ days. $\beta$ reduced by 2 and 4 (left and right, resp.):
|
||||
|
||||
FIGURE: [fig/SIR2, width=950]
|
||||
|
||||
!split
|
||||
===== What is the effect of vaccination? =====
|
||||
|
||||
!bblock Assumptions
|
||||
A fraction $p$ of the S category, per time unit, is vaccinated with
|
||||
success. Then in time $\Delta t$, $p\Delta t S$ will move to a
|
||||
vaccinated category, V. This does not affect the I and R categories.
|
||||
!eblock
|
||||
|
||||
FIGURE: [fig/categories_SIRV, width=400 frac=0.3]
|
||||
|
||||
!bt
|
||||
\begin{align}
|
||||
S^{n+1} &= S^n - \Delta t\,\beta S^nI^n + \Delta t\,\gamma R^n - {\color{red}p\Delta t S^n}
|
||||
label{SIR3:S}\\
|
||||
V^{n+1} &= V^n + {\color{red}p\Delta t S^n}
|
||||
label{SIR3:V}\\
|
||||
I^{n+1} &= I^n + \Delta t\,\beta S^nI^n -\Delta t\,\nu I^n
|
||||
label{SIR3:I}\\
|
||||
R^{n+1} &= R^n + \Delta t\,\nu R^n - \Delta t\,\gamma R^n
|
||||
label{SIR3:R}
|
||||
\end{align}
|
||||
!et
|
||||
|
||||
# #if FORMAT not in ("latex", "pdflatex")
|
||||
# Too much for Beamer...
|
||||
Implementation: Just add array for $V^n$ and add equation.
|
||||
# #endif
|
||||
|
||||
!split
|
||||
===== Many possibilities for adjusting the model... =====
|
||||
|
||||
The effect of vaccination decreases over time, so we may move people back to
|
||||
the S category (term proportional to $\Delta t V$).
|
||||
|
||||
FIGURE: [fig/categories_SIRV_feedback, width=400 frac=0.5]
|
||||
|
||||
|
||||
!split
|
||||
===== Effect of adding vaccination =====
|
||||
|
||||
FIGURE: [fig/SIRV1, width=800 frac=0.8]
|
||||
|
||||
($p=0.0005$)
|
||||
|
||||
|
||||
!split
|
||||
===== What is the effect of an intensive vaccination campaign? =====
|
||||
|
||||
10 times more intense vaccination for 10 days, 6 days after outbreak:
|
||||
|
||||
!bt
|
||||
\begin{equation*} p(t) = \left\lbrace\begin{array}{ll}
|
||||
0.005,& 6\leq t\leq 15,\\
|
||||
0,& \hbox{otherwise} \end{array}\right.\end{equation*}
|
||||
!et
|
||||
|
||||
Implementation: Let $p^n$ be an array as $V^n$. Set $p^n=0.05$ for
|
||||
$n=6\cdot 24/0.1,\ldots, 15\cdot 24/0.1$ ($\mbox{days}\cdot 24 /\Delta t$, 24 is hours per day).
|
||||
|
||||
FIGURE: [fig/p_discont, width=400 frac=0.5]
|
||||
|
||||
!split
|
||||
===== Effect of vaccination campaign =====
|
||||
|
||||
FIGURE: [fig/SIRV2, width=500 frac=0.6]
|
||||
|
||||
Note:
|
||||
|
||||
* Mathematicians would be scared by the cusps on the curves...
|
||||
* Could now let the computer run a lot of cases and find the optimal
|
||||
vaccination period
|
||||
|
||||
!split
|
||||
===== We can experiment with other campaigns =====
|
||||
|
||||
!bslidecell 00 0.3
|
||||
FIGURE: [fig/disease2.jpg, width=400]
|
||||
!eslidecell
|
||||
|
||||
!bslidecell 01 0.7
|
||||
Wearing masks lowers $\beta$:
|
||||
|
||||
!bt
|
||||
\begin{equation*} \beta(t) = \left\lbrace\begin{array}{ll}
|
||||
\beta_1,& 0\leq t < 5,\\
|
||||
\beta_2 < \beta_1,& t \geq 5\end{array}\right.
|
||||
\end{equation*}
|
||||
!et
|
||||
|
||||
Very easy to implement. (Used to be complicated in differential
|
||||
equation models...)
|
||||
!eslidecell
|
||||
|
||||
!split
|
||||
===== And now for something similar: zombification! =====
|
||||
|
||||
|
||||
FIGURE: [fig/zombie1, width=900]
|
||||
|
||||
_Zombification_: The disease that turns you into a zombie.
|
||||
|
||||
!split
|
||||
===== Zombie modeling is almost the same as SIR modeling =====
|
||||
|
||||
!bblock Categories
|
||||
o S: susceptible humans who can become zombies
|
||||
o I: infected humans, being bitten by zombies
|
||||
o Z: zombies
|
||||
o R: removed individuals, either conquered zombies or dead humans
|
||||
!eblock
|
||||
|
||||
Mathematical quantities: $S(t)$, $I(t)$, $Z(t)$, $R(t)$
|
||||
|
||||
Zombie movie: *The Night of the Living Dead*, Geoerge A. Romero, 1968
|
||||
|
||||
!split
|
||||
===== Dynamics of the zombie SIZR model =====
|
||||
|
||||
FIGURE: [fig/categories_SIZR, width=380 frac=0.4]
|
||||
|
||||
!bpop
|
||||
o Susceptibles are infected by zombies: $-\Delta t\beta SZ$ in time $\Delta t$ (cf. the $\Delta t\,\beta SI$ term in the SIR model).
|
||||
o Susceptibles die naturally or get killed and then enter the removed category. The no of deaths in time $\Delta t$ is $\Delta t\delta_S S$.
|
||||
o We also allow new humans to enter the area with zombies (necessary in a war on zombies): $\Delta t\Sigma$ during a time $\Delta t$.
|
||||
o Some infected turn into zombies (Z): $\Delta t\rho I$, while others die (R): $\delta_I\Delta t I$.
|
||||
o Nobody from R can turn into Z (important - otherwise zombies win).
|
||||
o Killed zombies go to R: $\Delta t\alpha SZ$.
|
||||
!epop
|
||||
|
||||
!split
|
||||
===== The four equations in the SIZR model for zombification =====
|
||||
|
||||
!bt
|
||||
\begin{align*}
|
||||
S^{n+1} &= S^n + \Delta t\,\Sigma - \Delta t\,\beta S^nZ - \Delta t\,\delta_S S^n\\
|
||||
I^{n+1} &= I^n + \Delta t\,\beta S^nZ^n - \Delta t\,\rho I^n - \Delta t\,\delta_I I^n\\
|
||||
Z^{n+1} &= Z^n + \Delta t\,\rho I^n - \Delta t\,\alpha S^nZ^n\\
|
||||
R^{n+1} &= R^n + \Delta t\,\delta_S S^n + \Delta t\,\delta_I I^n +
|
||||
\Delta t\,\alpha S^nZ^n
|
||||
\end{align*}
|
||||
!et
|
||||
|
||||
!bblock (small) Interpretation of parameters:
|
||||
|
||||
* $\Sigma$: no of new humans brought into the zombified area per unit time.
|
||||
* $\beta$: the probability that a theoretically possible human-zombie pair actually meets physically, during a unit time interval, with the result that the human is infected.
|
||||
* $\delta_S$: the probability that a susceptible human is killed or dies, in a unit time interval.
|
||||
* $\delta_I$: the probability that an infected human is killed or dies, in a unit time interval.
|
||||
* $\rho$: the probability that an infected human is turned into a zombie, during a unit time interval.
|
||||
* $\alpha$: the probability that, during a unit time interval, a theoretically possible human-zombie pair fights and the human kills the zombie.
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== Simulate a zombie movie! =====
|
||||
|
||||
!bslidecell 00 0.6
|
||||
!bblock Three fundamental phases
|
||||
o The initial phase (4 h)
|
||||
o The hysteric phase (24 h)
|
||||
o The counter attack phase (5 h)
|
||||
!eblock
|
||||
!eslidecell
|
||||
|
||||
!bslidecell 01 0.4
|
||||
FIGURE: [fig/TNotLD, width=300]
|
||||
!eslidecell
|
||||
|
||||
!bpop
|
||||
How do we do this? As $p$ in the vaccination campaign - the parameters
|
||||
take on different constant values in different time intervals.
|
||||
!epop
|
||||
|
||||
!bpop
|
||||
H. P. Langtangen, K.-A. Mardal and P. Røtnes:
|
||||
Escaping the Zombie Threat by Mathematics, in
|
||||
A. Whelan et al.: *Zombies in the Academy - Living Death in Higher Education*,
|
||||
University of Chicago Press, 2013
|
||||
!epop
|
||||
|
||||
!split
|
||||
===== Effective war on zombies =====
|
||||
|
||||
Introduce attacks on zombies at selected times $T_0, T_1, \ldots, T_m$.
|
||||
|
||||
Model: Replace $\alpha$ by
|
||||
|
||||
!bt
|
||||
\[ \alpha_0 + \omega (t),\]
|
||||
!et
|
||||
where $\alpha_0$ is constant and $\omega(t)$ is a series of
|
||||
Gaussian functions (peaks) in time:
|
||||
|
||||
!bt
|
||||
\[ \omega(t) = a\sum_{i=0}^m \exp{\left(-\frac{1}{2}\left({t - T_i\over\sigma}\right)\right)}
|
||||
\]
|
||||
!et
|
||||
|
||||
Must experiment with values of $a$ (strength), $\sigma$ (duration is $6\sigma$),
|
||||
point of attacks ($T_i$) - with proper values humans beat the zombies!
|
||||
|
||||
!split
|
||||
===== Summary =====
|
||||
|
||||
* A complex spreading of diseases can be modeled by intuitive, simple
|
||||
accounting of movement between categories
|
||||
* Such models are knowns as *compartment models*
|
||||
* Result: difference equations that are easy to simulate on a computer
|
||||
* (Can let $\Delta t\rightarrow 0$ and get differential equations)
|
||||
* Easy to add new effects (vaccination, campaigns, zombification)
|
||||
|
||||
|
||||
|
||||
!split
|
||||
===== Simulating financial transcations =====
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
Reference in New Issue
Block a user