Networks in epidemic modelling.ipynb

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    "# Networks (or graphs)\n",
    "- In our simulations, agents move and touch each other\n",
    "- Contact via touch\n",
    "- Can abstract this by having contact represented _implicitly_\n",
    "- \"__Network__\" of contacts: who I may transmit infection to\n",
    "- E.g. Family, friends, people on campus, people in bank / nigthclub..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How can we represent this on the computer?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
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   "source": [
    "L = 20\n",
    "\n",
    "M = zeros(Int, L, L);"
   ]
  },
  {
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   "execution_count": 9,
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       "0.1"
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   "source": [
    "ρ = 0.1 # density of agents"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
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       "20×20 Matrix{Int64}:\n",
       " 0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  1  0  0  0  0  1  0  0  0  0  1  0  0  1\n",
       " 0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0\n",
       " 1  0  0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0\n",
       " 0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  1  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  1  1  0  0  1  0  0  0  0  0  0  1\n",
       " 1  0  1  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  1  0  1  0  0\n",
       " 0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  1  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  1  0  1  0  1  0  1  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0\n",
       " 0  0  1  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  1  0"
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   "source": [
    "M .= rand(L, L) .< ρ"
   ]
  },
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    "heatmap(M, ratio=1)"
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   "execution_count": 14,
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     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ = 0.2\n",
    "M .= rand(L, L) .< ρ\n",
    "heatmap(M, ratio = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ = 0.45\n",
    "M .= rand(L, L) .< ρ\n",
    "heatmap(M, ratio = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ρ = 0.7\n",
    "M .= rand(L, L) .< ρ\n",
    "heatmap(M, ratio = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Who are the neighbours of agent 1?  \n",
    "E.g. agent 1 is in contact with agents 2, 7, 73"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any}()"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "contacts = Dict()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Int64}:\n",
       "  2\n",
       "  7\n",
       " 73"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "contacts[1] = [2, 7, 73]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Set{Int64} with 3 elements:\n",
       "  4\n",
       "  7\n",
       "  73"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "contacts[2] = Set([4, 7, 73])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3 in contacts[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "false"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 ∈ contacts[2] "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Adjacency matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10×10 Matrix{Int64}:\n",
       " 1  0  0  0  0  1  1  0  0  0\n",
       " 0  0  1  1  0  0  0  0  1  1\n",
       " 0  0  0  0  1  0  1  1  1  0\n",
       " 1  0  1  0  1  0  1  1  0  0\n",
       " 1  1  1  0  1  0  1  0  1  1\n",
       " 1  0  0  0  1  1  0  1  1  0\n",
       " 0  0  1  0  1  1  0  0  0  1\n",
       " 0  1  1  1  1  0  1  0  1  1\n",
       " 0  1  0  1  0  0  1  0  0  0\n",
       " 1  1  1  0  0  1  1  0  1  0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = rand(0:1, 10, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.6/Project.toml`\n",
      " \u001b[90m [093fc24a] \u001b[39m\u001b[92m+ LightGraphs v1.3.5\u001b[39m\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.6/Manifest.toml`\n"
     ]
    }
   ],
   "source": [
    "# using Pkg; Pkg.add(\"LightGraphs\")\n",
    "using LightGraphs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0, 0} undirected simple Int64 graph"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = Graph()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "add_vertices!(g, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{5, 0} undirected simple Int64 graph"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "add_edge!(g, 1, 2)\n",
    "add_edge!(g, 2, 3)\n",
    "add_edge!(g, 1, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{5, 3} undirected simple Int64 graph"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "# using Pkg; Pkg.add(\"GraphPlot\")\n",
    "using GraphPlot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
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      ],
      "text/plain": [
       "Compose.Context(BBox{l,t,r,b,w,h = 0.0w,0.0h, 1.0w,1.0h, 1.0w,1.0h}, Compose.UnitBox{Float64, Float64, Float64, Float64}(-1.2, -1.2, 2.4, 2.4, 0.0mm, 0.0mm, 0.0mm, 0.0mm), nothing, nothing, nothing, List([Compose.Context(BBox{l,t,r,b,w,h = 0.0w,0.0h, 1.0w,1.0h, 1.0w,1.0h}, nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.LinePrimitive}(Compose.LinePrimitive[Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.0834741633365655cx, 0.9997519755001513cy), (-0.12945337440072352cx, 0.9995787244701894cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.16823986521316645cx, 0.8908456509575625cy), (-0.1329540481312568cx, 0.7492619957796042cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.04432895035164421cx, 0.8914261787948214cy), (-0.07923896368000524cx, 0.7493507679720046cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.3416407864998738mm)]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(BBox{l,t,r,b,w,h = 0.0w,0.0h, 1.0w,1.0h, 1.0w,1.0h}, nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.3416407864998738mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(BBox{l,t,r,b,w,h = 0.0w,0.0h, 1.0w,1.0h, 1.0w,1.0h}, nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(BBox{l,t,r,b,w,h = 0.0w,0.0h, 1.0w,1.0h, 1.0w,1.0h}, nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}}(Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}[Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.1952767685250314cx, 0.9993306999703406cy), 0.044721359549995794w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.01765076921225761cx, 1.0cy), 0.044721359549995794w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.10591714481939185cx, 0.6407769467668261cy), 0.044721359549995794w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-1.0cx, -1.0cy), 0.044721359549995794w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((1.0cx, -0.6280132852713225cy), 0.044721359549995794w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(BBox{l,t,r,b,w,h = 0.0w,0.0h, 1.0w,1.0h, 1.0w,1.0h}, nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gplot(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{30, 134} undirected simple Int64 graph"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = erdos_renyi(30, 0.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: both LightGraphs and Plots export \"grid\"; uses of it in module Main must be qualified\n"
     ]
    }
   ],
   "source": [
    "gplot(g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gplot(LightGraphs.grid((3, 4)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Int64}:\n",
       "  4\n",
       " 11\n",
       " 18\n",
       " 29"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "neighbors(g, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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