Added De Leo scaling model

SUMMARY.md

@@ -15,6 +15,10 @@
     * [Javascript using Observable](notebooks/sir/js_observable.md)
   * [SIS model](notebooks/sis/intro.md)
     * [Javascript using Observable](notebooks/sis/js_observable.md)
+  * [Scaling model](notebooks/deleo1996/intro.md)
+    * [Julia](notebooks/deleo1996/julia.ipynb)
+    * [Python](notebooks/deleo1996/python.ipynb)
+    * [R using deSolve](notebooks/deleo1996/r_desolve.ipynb)
 * [Simple stochastic models](notebooks/stochastic.md)
   * [Continuous time SIR](notebooks/sir-stochastic-discretestate-continuoustime/intro.md)
     * [R](notebooks/sir-stochastic-discretestate-continuoustime/r.ipynb)

_chapters/deleo1996/intro.md

@@ -0,0 +1,33 @@
+---
+title: 'Scaling model'
+permalink: 'chapters/deleo1996/intro'
+previouschapter:
+  url: chapters/sis/js_observable
+  title: 'Javascript using Observable'
+nextchapter:
+  url: chapters/deleo1996/julia
+  title: 'Julia'
+redirect_from:
+  - 'chapters/deleo1996/intro'
+---
+
+## Basic microparasite model
+
+*Author*: Christopher Davis
+
+*Date*: 2018-10-02
+
+### Description
+
+A basic microparasite model of susceptibles and infecteds with the force of infection density dependent. $\beta_{\text{min}}$ is the minimum value of the transmission rate $\beta$, such that the disease will spread.
+
+### Equations
+
+$$
+\frac{dS(t)}{dt}  = (\nu - \mu)\left(1- \frac{S(t)}{K}\right) S(t)- \beta S(t) I(t)\\
+\frac{dI(t)}{dt}  = \beta S(t) I(t)- (\mu +\ \alpha) I(t)\\
+$$
+
+### References
+
+- [De Leo GA, Dobson AP (February, 1996). "Allometry and simple epidemic models for microparasites". Nature. 379(6567):720](https://doi.org/10.1038/379720a0)

_chapters/deleo1996/julia.md

@@ -0,0 +1,130 @@
+---
+interact_link: notebooks/deleo1996/julia.ipynb
+title: 'Julia'
+permalink: 'chapters/deleo1996/julia'
+previouschapter:
+  url: chapters/deleo1996/intro
+  title: 'Scaling model'
+nextchapter:
+  url: chapters/deleo1996/python
+  title: 'Python'
+redirect_from:
+  - 'chapters/deleo1996/julia'
+---
+
+### De Leo et al. scaling model in Julia
+
+*Author*: Christopher Davis
+
+*Date*: 2018-10-02
+
+
+{:.input_area}
+```julia
+using DifferentialEquations
+```
+
+
+{:.input_area}
+```julia
+micro_1 = @ode_def Micro1 begin
+    dS = r*(1-S/K)*S - β*S*I
+    dI = β*S*I-(μ+α)*I
+    end β r μ K α
+```
+
+
+
+
+{:.output_data_text}
+```
+(::Micro1) (generic function with 4 methods)
+```
+
+
+
+
+{:.input_area}
+```julia
+w = 1;
+m = 10;
+β = 0.0247*m*w^0.44;
+r = 0.6*w^-0.27;
+μ = 0.4*w^-0.26;
+K = 16.2*w^-0.7;
+α = (m-1)*μ;
+```
+
+
+{:.input_area}
+```julia
+parms = [β,r,μ,K,α];
+init = [K,1.];
+tspan = (0.0,10.0);
+```
+
+
+{:.input_area}
+```julia
+sir_prob = ODEProblem(micro_1,init,tspan,parms)
+```
+
+
+
+
+{:.output_data_text}
+```
+DiffEqBase.ODEProblem with uType Array{Float64,1} and tType Float64. In-place: true
+timespan: (0.0, 10.0)
+u0: [16.2, 1.0]
+```
+
+
+
+
+{:.input_area}
+```julia
+sir_sol = solve(sir_prob);
+```
+
+#### Visualisation
+
+
+{:.input_area}
+```julia
+using Plots
+```
+
+
+{:.input_area}
+```julia
+plot(sir_sol,xlabel="Time",ylabel="Number")
+```
+
+
+
+
+![svg](../../images/chapters/deleo1996/julia_10_0.svg)
+
+
+
+#### Threshold criterion for transmission rate
+
+
+{:.input_area}
+```julia
+m = [5,10,20,40]
+ws = 10.^linspace(-3,3,601)
+βs = zeros(601,4)
+for i = 1:4
+    βs[:,i] = 0.0247*m[i]*ws.^0.44
+end
+plot(ws,βs,xlabel="Weight",ylabel="\\beta_min", xscale=:log10,yscale=:log10, label=["m = 5" "m = 10" "m = 20" "m = 40"],lw=3)
+```
+
+
+
+
+![svg](../../images/chapters/deleo1996/julia_12_0.svg)
+
+

_chapters/deleo1996/python.md

@@ -0,0 +1,126 @@
+---
+interact_link: notebooks/deleo1996/python.ipynb
+title: 'Python'
+permalink: 'chapters/deleo1996/python'
+previouschapter:
+  url: chapters/deleo1996/julia
+  title: 'Julia'
+nextchapter:
+  url: chapters/deleo1996/r_desolve
+  title: 'R using deSolve'
+redirect_from:
+  - 'chapters/deleo1996/python'
+---
+
+### Python using SciPy
+
+*Author*: Christopher Davis
+
+*Date*: 2018-10-02
+
+
+{:.input_area}
+```python
+import numpy as np
+from scipy.integrate import ode, solve_ivp
+```
+
+
+{:.input_area}
+```python
+def micro_1(times,init,parms):
+    beta, r, mu, K, alpha = parms
+    S,I = init
+    # ODEs
+    dS = r*(1-S/K)*S - beta*S*I
+    dI = beta*S*I-(mu + alpha)*I
+    return [dS,dI]
+```
+
+
+{:.input_area}
+```python
+w = 1
+m = 10
+beta = 0.0247*m*w**0.44
+r = 0.6*w**-0.27
+mu = 0.4*w**-0.26
+K = 16.2*w**-0.7
+alpha = (m-1)*mu
+```
+
+
+{:.input_area}
+```python
+parms = [beta,r,mu,K,alpha]
+init = [K,1.]
+times = np.linspace(0,10,101)
+```
+
+
+{:.input_area}
+```python
+sir_sol = solve_ivp(fun=lambda t, y: micro_1(t, y, parms), t_span=[min(times),max(times)], y0=init, t_eval=times)
+```
+
+#### Visualisation
+
+
+{:.input_area}
+```python
+import matplotlib.pyplot as plt
+```
+
+
+{:.input_area}
+```python
+plt.plot(sir_sol.t,sir_sol.y[0],color="red",linewidth=2, label = "S(t)")
+plt.plot(sir_sol.t,sir_sol.y[1],color="blue",linewidth=2, label = "I(t)")
+plt.xlabel("Time")
+plt.ylabel("Number")
+plt.legend()
+```
+
+
+
+
+{:.output_data_text}
+```
+<matplotlib.legend.Legend at 0x7f776a919198>
+```
+
+
+
+
+![png](../../images/chapters/deleo1996/python_9_1.png)
+
+
+#### Threshold criterion for transmission rate
+
+
+{:.input_area}
+```python
+m = [5,10,20,40]
+ws = 10**np.linspace(-3,3,601)
+betas = np.zeros((601,4))
+for i in range(4):
+    betas[:,i] = 0.0247*m[i]*ws**0.44
+plt.loglog(ws,betas)
+plt.xlabel("Weight")
+plt.ylabel(r'$\beta_{min}$')
+plt.legend(("m = 5", "m = 10", "m = 20", "m = 40"))
+```
+
+
+
+
+{:.output_data_text}
+```
+<matplotlib.legend.Legend at 0x7f7769b707f0>
+```
+
+
+
+
+![png](../../images/chapters/deleo1996/python_11_1.png)
+

_chapters/sis/js_observable.md

@@ -5,8 +5,8 @@ previouschapter:
   url: chapters/sis/intro
   title: 'SIS model'
 nextchapter:
-  url: chapters/stochastic
-  title: 'Simple stochastic models'
+  url: chapters/deleo1996/intro
+  title: 'Scaling model'
 redirect_from:
   - 'chapters/sis/js-observable'
 ---

_chapters/stochastic.md

@@ -2,8 +2,8 @@
 title: 'Simple stochastic models'
 permalink: 'chapters/stochastic'
 previouschapter:
-  url: chapters/sis/js_observable
-  title: 'Javascript using Observable'
+  url: chapters/deleo1996/r_desolve
+  title: 'R using deSolve'
 nextchapter:
   url: chapters/sir-stochastic-discretestate-continuoustime/intro
   title: 'Continuous time SIR'

_data/textbook.yml

@@ -46,6 +46,18 @@ chapters:
   - class: level_2
     title: Javascript using Observable
     url: chapters/sis/js_observable
+  - class: level_1
+    title: Scaling model
+    url: chapters/deleo1996/intro
+  - class: level_2
+    title: Julia
+    url: chapters/deleo1996/julia
+  - class: level_2
+    title: Python
+    url: chapters/deleo1996/python
+  - class: level_2
+    title: R using deSolve
+    url: chapters/deleo1996/r_desolve
   class: level_0
   title: Simple deterministic models
   url: chapters/simple