Create image of resource allocation transitions #2

docs/ResourceAllocation.tex

@@ -0,0 +1,31 @@
+\documentclass{article}
+
+% External references
+\usepackage{pgf}
+\usepackage{tikz}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{IEEEtrantools}
+\usepackage{mathrsfs}
+
+% For the Markov chain diagrams
+\usetikzlibrary{arrows,automata}
+
+\title{Resource Allocation Models}
+\author{Aaron Geoffrey Sheldon}
+\date{\today}
+\begin{document}
+  \pagenumbering{gobble}
+  \begin{tikzpicture}[->,thick,node distance=3cm]
+    \node[state] (RA)                     {RA};
+    \node[state] (RR) [below right of=RA] {RR};
+    \node[state] (RO) [above right of=RR] {RO};
+    \node[state] (RC) [below of=RR]       {RC};
+    \path
+      (RA) edge [bend left] node [above] {$\displaystyle\int_0^\infty \gamma\beta_\text{hos} e^{\gamma\alpha_\text{hos} a} n_\text{I} da$} (RO)
+      (RR) edge [bend left] node [below] {$\beta_\text{ref}$} (RA)
+      (RO) edge [bend left] node [right] {$\displaystyle\int_0^\infty \left(\gamma\beta_\text{age} e^{\gamma\alpha_\text{age} a} + \gamma\beta_\text{dis} e^{-\gamma\alpha_\text{dis} a}\right) n_{H} da$} (RR)
+      (RR) edge (RC);
+  \end{tikzpicture}
+\end{document}

src/vonFoersterHazards.ipynb

@@ -746,7 +746,7 @@
     "\\end{array}\n",
     "$$\n",
     "\n",
-    "The scattering rates are normalized to be either greater than one $\\hat{\\sigma} \\ge 1$ or less than one $\\hat{\\sigma} \\le 1$ for all ages and can be of any form such as logistic or Gompertz. The hazard rate matrix $H$ can be a constant, representing the endogenous self interaction within a single age cohort, such as with ageing. It is from the modest observation that we construct the updating function inside the main loop of our engine.\n",
+    "The scattering rates are normalized to be either greater than one $\\hat{\\sigma} \\ge 1$ or less than one $\\hat{\\sigma} \\le 1$ for all ages and can be of any form such as logistic or exponential. The hazard rate matrix $H$ can be a constant, representing the endogenous self interaction within a single age cohort, such as with ageing. It is from the modest observation that we construct the updating function inside the main loop of our engine.\n",
     "\n",
     "By accumulating all the endogenous self interactions into a single matrix we can exploit the symmetry in age-age scattering cross section to write the exogenous term as a single product so that to first order the terms in our minimal age dependent communicable disease model are:\n",
     "\n",
@@ -793,8 +793,8 @@
     "    & - \\gamma\\beta_\\text{hos} e^{\\gamma\\alpha_\\text{hos} a} \\int_0^\\infty n_\\text{RA} da & 0 & 0 & 0 & 0 & \\\\\n",
     "    & \\gamma\\beta_\\text{hos} e^{\\gamma\\alpha_\\text{hos} a} \\int_0^\\infty n_\\text{RA} da & - \\gamma\\beta_\\text{age} e^{\\gamma\\alpha_\\text{age} a} - \\gamma\\beta_\\text{dis} e^{-\\gamma\\alpha_\\text{dis} a} & 0 & 0 & 0 & \\\\\n",
     "    & 0 & 0 & -\\int_0^\\infty \\gamma\\beta_\\text{hos} e^{\\gamma\\alpha_\\text{hos} a} n_\\text{I} da & 0 & \\beta_{ref} & \\\\\n",
-    "    & 0 & 0 & \\int_0^\\infty \\gamma\\beta_\\text{hos} e^{\\gamma\\alpha_\\text{hos} a} n_\\text{I} da & - \\int_0^\\infty \\gamma\\beta_\\text{age} e^{\\gamma\\alpha_\\text{age} a} - \\gamma\\beta_\\text{dis} e^{-\\gamma\\alpha_\\text{dis} a} n_{H} da & 0 & \\\\\n",
-    "    & 0 & 0 & 0 & \\int_0^\\infty \\gamma\\beta_\\text{age} e^{\\gamma\\alpha_\\text{age} a} - \\gamma\\beta_\\text{dis} e^{-\\gamma\\alpha_\\text{dis} a} n_{H} da & -\\beta_{ref} & \\\\\n",
+    "    & 0 & 0 & \\int_0^\\infty \\gamma\\beta_\\text{hos} e^{\\gamma\\alpha_\\text{hos} a} n_\\text{I} da & - \\int_0^\\infty \\left( \\gamma\\beta_\\text{age} e^{\\gamma\\alpha_\\text{age} a} + \\gamma\\beta_\\text{dis} e^{-\\gamma\\alpha_\\text{dis} a} \\right) n_{H} da & 0 & \\\\\n",
+    "    & 0 & 0 & 0 & \\int_0^\\infty \\left(\\gamma\\beta_\\text{age} e^{\\gamma\\alpha_\\text{age} a} + \\gamma\\beta_\\text{dis} e^{-\\gamma\\alpha_\\text{dis} a} \\right) n_{H} da & -\\beta_{ref} & \\\\\n",
     "    & & & & & & \\ddots\n",
     "\\end{bmatrix}\n",
     "$$\n",

src/vonFoersterHazards.jl

@@ -95,7 +95,7 @@ struct evolve{R<:population, S<:Real, T<:Real}
     gestation::T
     count::Int64
 end
-Base.eltype(::Type{evolve{S, T}}) = S
+Base.eltype(::Type{evolve{R, S, T}}) = R
 Base.length(E::evolve) = E.count
 Base.size(E::evolve, d=1) = ((d==1) ? length(E) : 1)
 function Base.iterate(E::evolve)