Exponential and logistic growth.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Introduction to Modelling\n",
    "## Exponential and Logistic Growth\n",
    "\n",
    "Goals:\n",
    "- Simple models of dynamics of infection\n",
    "- Exponential and logistic growth\n",
    "- **Simulate** to produce **data**\n",
    "- Visualize for understanding"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dynamics\n",
    "\n",
    "*how system changes (**evolves**) over time"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Modelling\n",
    "\n",
    "### What is a **model**?\n",
    "\n",
    "*Mathematical/computational description of the world*\n",
    "\n",
    "- Isolate effects of interest\n",
    "- Start simple, understand behaviour\n",
    "- Make more compliocated (add additional effects), understand how results change\n",
    "- Repeat!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Modeling Infections\n",
    "- Let's try to model infection spreading thourgh campus\n",
    "- Need to make assumptions to start modelling process\n",
    "- Assumptions must be **explicit**\n",
    " \n",
    "\n",
    "- **Suppose** no influence from outside\n",
    "- **Suppose** no births or deaths\n",
    "- **Suppose** infection is transmitted by proximity\n",
    "- **Suppose** stay infected once infected"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Mathematical Model\n",
    "- **Mathematical model**: mathematical description of assumptions\n",
    "- Choose a model in **discrete time**: *n* = days\n",
    "- Notation: $I_n$ is number of **infectious** people on *n*th day\n",
    "- $S_n$ is number of **susceptible** people on *n*th day\n",
    " \n",
    " \n",
    " **Suppose** that everybody is either susceptible or infected:\n",
    "  \n",
    "  $S_n + I_n = N$ \n",
    " \n",
    " and total population $N$ is constant\n",
    " \n",
    " **Initial conditions**\n",
    " - Need to specify initial values at time *n* = 0 \n",
    "  \n",
    "      E.g. $I_0 = 1$ and $S_0 = N - 1$\n",
    "  \n",
    " - We will need hypotheses on dynamics of infection:\n",
    "  Specify how $I_{n+1}$ depends on $I_0, ..., I_n$ \n",
    "   \n",
    "   E.g. $I_{n+1} = f(I_n)$ &rarr; **deterministic** model\n",
    "    \n",
    "   or $I_{n+1} = f(I_n) + \\xi$ &rarr; **stochastic** model with random fluctuatinos $\\xi$\n",
    "   \n",
    "**Output of model**\n",
    "- The output of model will be a **time series**\n",
    " \n",
    " i.e. vectors of values of $S_n$ and $I_n$ at times $n = 0, ..., T$\n",
    "- Up to some final time $T$ \n",
    " \n",
    "- *Possible goal: Obtain time series that \"mimics\" real-world data*\n",
    "- *Posiible goal: Understand what effects must be included in model and which ones may not be relevant*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simple model of infection\n",
    "Each individual infects *c* new people each day."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$I_{n+1} = I_n + (c I_n) = I_n(1 + c)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$I_{n+1} = \\lambda I_n$, where $\\lambda$ := 1 + c -- **growth rate**\n",
    " \n",
    "and $I_0 = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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       "1"
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     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "I₀ = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.01"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = 0.01 # Every individual infects 0.01 new people every day"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.01"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "λ = 1 + c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.01"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I₁ = λ * I₀"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0201"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I₂ = λ * I₁"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.030301"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I₃ = λ * I₂"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Arrays"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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     "data": {
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       "10-element Vector{Float64}:\n",
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       " 0.0"
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     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = 10 # final time\n",
    "\n",
    "I = zeros(T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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       "0.0"
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   "source": [
    "I[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       " 0.0\n",
       " 0.0\n",
       " 0.0"
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     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "I[1:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Vector{Int64}:\n",
       " 0\n",
       " 0\n",
       " 0\n",
       " 0\n",
       " 0\n",
       " 0\n",
       " 0\n",
       " 0\n",
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       " 0"
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     "execution_count": 14,
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   ],
   "source": [
    "I = zeros(Int64, T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Automate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Vector{Float64}:\n",
       " 0.0\n",
       " 0.0\n",
       " 0.0\n",
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     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "I = zeros(T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
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     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "I[1] = I₀"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "(n, I[n]) = (1, 1.0)\n",
      "(n, I[n]) = (2, 1.01)\n",
      "(n, I[n]) = (3, 1.0201)\n",
      "(n, I[n]) = (4, 1.030301)\n",
      "(n, I[n]) = (5, 1.04060401)\n",
      "(n, I[n]) = (6, 1.0510100501)\n",
      "(n, I[n]) = (7, 1.061520150601)\n",
      "(n, I[n]) = (8, 1.0721353521070098)\n",
      "(n, I[n]) = (9, 1.08285670562808)\n"
     ]
    }
   ],
   "source": [
    "for n in 1:T-1\n",
    "    I[n+1] = λ * I[n]\n",
    "    @show n, I[n]\n",
    "end\n",
    "\n",
    "# for loops do not return anything"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "plot(I, m=:o, label=\"I[n]\", legend=:topleft)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = 20\n",
    "\n",
    "I = zeros(T)\n",
    "I[1] = I₀\n",
    "\n",
    "for n in 1:T-1\n",
    "    I[n+1] = λ * I[n]\n",
    "#     @show n, I[n]\n",
    "end\n",
    "\n",
    "plot(I)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "run_infection (generic function with 2 methods)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function run_infection(T=20)    \n",
    "    I = zeros(T)\n",
    "    I[1] = I₀\n",
    "    \n",
    "    for n in 1:T-1\n",
    "        I[n+1] = λ * I[n]\n",
    "    end\n",
    "    \n",
    "    return(I)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20-element Vector{Float64}:\n",
       " 1.0\n",
       " 1.01\n",
       " 1.0201\n",
       " 1.030301\n",
       " 1.04060401\n",
       " 1.0510100501\n",
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       " 1.1268250301319698\n",
       " 1.1380932804332895\n",
       " 1.1494742132376223\n",
       " 1.1609689553699987\n",
       " 1.1725786449236986\n",
       " 1.1843044313729356\n",
       " 1.196147475686665\n",
       " 1.2081089504435316"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "run_infection()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "# 2 methods for generic function <b>run_infection</b>:<ul><li> run_infection() in Main at In[32]:1</li> <li> run_infection(T) in Main at In[32]:1</li> </ul>"
      ],
      "text/plain": [
       "# 2 methods for generic function \"run_infection\":\n",
       "[1] run_infection() in Main at In[32]:1\n",
       "[2] run_infection(T) in Main at In[32]:1"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "methods(run_infection)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Vector{Float64}:\n",
       " 1.0\n",
       " 1.01\n",
       " 1.0201\n",
       " 1.030301\n",
       " 1.04060401\n",
       " 1.0510100501\n",
       " 1.061520150601\n",
       " 1.0721353521070098\n",
       " 1.08285670562808\n",
       " 1.0936852726843609"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_result = run_infection(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
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     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "plot(I_result)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(run_infection(20), m=:o) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "                '<div style=\"padding: 1em; background-color: #f8d6da; border: 1px solid #f5c6cb\">' +\n",
       "                '<p><strong>WebIO not detected.</strong></p>' +\n",
       "                '<p>Please read ' +\n",
       "                '<a href=\"https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/\" target=\"_blank\">the troubleshooting guide</a> ' +\n",
       "                'for more information on how to resolve this issue.</p>' +\n",
       "                '<p><a href=\"https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/\" target=\"_blank\">https://juliagizmos.github.io/WebIO.jl/latest/troubleshooting/not-detected/</a></p>' +\n",
       "                '</div>'\n",
       "            );\n",
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(WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"14721937422247612075\\\",\\\"id\\\":\\\"18421728926803028525\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"14721937422247612075\\\",\\\"id\\\":\\\"13697678738429543450\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"index\\\"]=false}),self)]\\n        \\n    }\\n    self.model = new AppViewModel();\\n    self.valueFromJulia = {};\\n    for (var key in json_data) {\\n        self.valueFromJulia[key] = false;\\n    }\\n    ko.applyBindings(self.model, self.dom);\\n}\\n);\\n    (WebIO.importBlock({\\\"data\\\":[{\\\"name\\\":\\\"knockout\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/4e15ea8f38f5d33bf0b2930cb09c8089f78b114e-knockout.js\\\"},{\\\"name\\\":\\\"knockout_punches\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/6967e46b2259ea2e5c469b61c410326b52955db7-knockout_punches.js\\\"}],\\\"type\\\":\\\"async_block\\\"})).then((imports) => handler.apply(this, imports));\\n}\\n\")])], Dict{Symbol, Any}(:className => \"field interact-widget\")), Observable{Any} with 0 listeners. Value:\n",
       "Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Plot{Plots.GRBackend() n=1}], Dict{Symbol, Any}(:className => \"interact-flex-row interact-widget\"))], Dict{Symbol, Any}())"
      ]
     },
     "execution_count": 45,
     "metadata": {
      "@webio": {
       "kernelId": "4fe35d73-4234-4d9b-9302-78f461ba68d7"
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "end_T = 1000\n",
    "@manipulate for T in slider(1:end_T, value=1) \n",
    "    I_result = run_infection(T)\n",
    "    \n",
    "    plot(I_result, m=:o)\n",
    "    \n",
    "    xlims!(0,end_T)\n",
    "    ylims!(0,10)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exponential Growth"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  886.322,1444.72 888.28,1444.71 890.238,1444.7 892.196,1444.69 894.154,1444.68 896.112,1444.67 898.07,1444.65 900.028,1444.64 901.986,1444.63 903.944,1444.62 \n",
       "  905.902,1444.61 907.86,1444.6 909.818,1444.59 911.775,1444.57 913.733,1444.56 915.691,1444.55 917.649,1444.54 919.607,1444.52 921.565,1444.51 923.523,1444.5 \n",
       "  925.481,1444.49 927.439,1444.47 929.397,1444.46 931.355,1444.45 933.313,1444.43 935.271,1444.42 937.229,1444.41 939.187,1444.39 941.144,1444.38 943.102,1444.37 \n",
       "  945.06,1444.35 947.018,1444.34 948.976,1444.32 950.934,1444.31 952.892,1444.29 954.85,1444.28 956.808,1444.26 958.766,1444.25 960.724,1444.23 962.682,1444.22 \n",
       "  964.64,1444.2 966.598,1444.19 968.556,1444.17 970.513,1444.15 972.471,1444.14 974.429,1444.12 976.387,1444.1 978.345,1444.09 980.303,1444.07 982.261,1444.05 \n",
       "  984.219,1444.04 986.177,1444.02 988.135,1444 990.093,1443.98 992.051,1443.97 994.009,1443.95 995.967,1443.93 997.925,1443.91 999.882,1443.89 1001.84,1443.87 \n",
       "  1003.8,1443.85 1005.76,1443.83 1007.71,1443.81 1009.67,1443.79 1011.63,1443.78 1013.59,1443.76 1015.55,1443.73 1017.5,1443.71 1019.46,1443.69 1021.42,1443.67 \n",
       "  1023.38,1443.65 1025.34,1443.63 1027.29,1443.61 1029.25,1443.59 1031.21,1443.57 1033.17,1443.54 1035.13,1443.52 1037.08,1443.5 1039.04,1443.48 1041,1443.45 \n",
       "  1042.96,1443.43 1044.91,1443.41 1046.87,1443.38 1048.83,1443.36 1050.79,1443.33 1052.75,1443.31 1054.7,1443.28 1056.66,1443.26 1058.62,1443.23 1060.58,1443.21 \n",
       "  1062.54,1443.18 1064.49,1443.16 1066.45,1443.13 1068.41,1443.1 1070.37,1443.08 1072.33,1443.05 1074.28,1443.02 1076.24,1443 1078.2,1442.97 1080.16,1442.94 \n",
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       "  1101.7,1442.61 1103.65,1442.58 1105.61,1442.55 1107.57,1442.51 1109.53,1442.48 1111.48,1442.45 1113.44,1442.42 1115.4,1442.38 1117.36,1442.35 1119.32,1442.31 \n",
       "  1121.27,1442.28 1123.23,1442.24 1125.19,1442.21 1127.15,1442.17 1129.11,1442.14 1131.06,1442.1 1133.02,1442.06 1134.98,1442.03 1136.94,1441.99 1138.9,1441.95 \n",
       "  1140.85,1441.91 1142.81,1441.87 1144.77,1441.83 1146.73,1441.8 1148.69,1441.76 1150.64,1441.72 1152.6,1441.67 1154.56,1441.63 1156.52,1441.59 1158.48,1441.55 \n",
       "  1160.43,1441.51 1162.39,1441.47 1164.35,1441.42 1166.31,1441.38 1168.26,1441.33 1170.22,1441.29 1172.18,1441.24 1174.14,1441.2 1176.1,1441.15 1178.05,1441.11 \n",
       "  1180.01,1441.06 1181.97,1441.01 1183.93,1440.97 1185.89,1440.92 1187.84,1440.87 1189.8,1440.82 1191.76,1440.77 1193.72,1440.72 1195.68,1440.67 1197.63,1440.62 \n",
       "  1199.59,1440.57 1201.55,1440.51 1203.51,1440.46 1205.47,1440.41 1207.42,1440.36 1209.38,1440.3 1211.34,1440.25 1213.3,1440.19 1215.26,1440.13 1217.21,1440.08 \n",
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       "  1238.75,1439.42 1240.71,1439.36 1242.67,1439.29 1244.62,1439.23 1246.58,1439.16 1248.54,1439.09 1250.5,1439.03 1252.46,1438.96 1254.41,1438.89 1256.37,1438.82 \n",
       "  1258.33,1438.75 1260.29,1438.68 1262.25,1438.61 1264.2,1438.54 1266.16,1438.47 1268.12,1438.39 1270.08,1438.32 1272.04,1438.25 1273.99,1438.17 1275.95,1438.09 \n",
       "  1277.91,1438.02 1279.87,1437.94 1281.82,1437.86 1283.78,1437.78 1285.74,1437.7 1287.7,1437.62 1289.66,1437.54 1291.61,1437.46 1293.57,1437.37 1295.53,1437.29 \n",
       "  1297.49,1437.21 1299.45,1437.12 1301.4,1437.03 1303.36,1436.95 1305.32,1436.86 1307.28,1436.77 1309.24,1436.68 1311.19,1436.59 1313.15,1436.5 1315.11,1436.4 \n",
       "  1317.07,1436.31 1319.03,1436.21 1320.98,1436.12 1322.94,1436.02 1324.9,1435.92 1326.86,1435.83 1328.82,1435.73 1330.77,1435.63 1332.73,1435.52 1334.69,1435.42 \n",
       "  1336.65,1435.32 1338.61,1435.21 1340.56,1435.11 1342.52,1435 1344.48,1434.89 1346.44,1434.78 1348.39,1434.67 1350.35,1434.56 1352.31,1434.45 1354.27,1434.34 \n",
       "  1356.23,1434.22 1358.18,1434.11 1360.14,1433.99 1362.1,1433.87 1364.06,1433.75 1366.02,1433.63 1367.97,1433.51 1369.93,1433.39 1371.89,1433.26 1373.85,1433.14 \n",
       "  1375.81,1433.01 1377.76,1432.89 1379.72,1432.76 1381.68,1432.63 1383.64,1432.49 1385.6,1432.36 1387.55,1432.23 1389.51,1432.09 1391.47,1431.96 1393.43,1431.82 \n",
       "  1395.39,1431.68 1397.34,1431.54 1399.3,1431.39 1401.26,1431.25 1403.22,1431.1 1405.17,1430.96 1407.13,1430.81 1409.09,1430.66 1411.05,1430.51 1413.01,1430.36 \n",
       "  1414.96,1430.2 1416.92,1430.05 1418.88,1429.89 1420.84,1429.73 1422.8,1429.57 1424.75,1429.41 1426.71,1429.24 1428.67,1429.08 1430.63,1428.91 1432.59,1428.74 \n",
       "  1434.54,1428.57 1436.5,1428.4 1438.46,1428.23 1440.42,1428.05 1442.38,1427.87 1444.33,1427.69 1446.29,1427.51 1448.25,1427.33 1450.21,1427.15 1452.17,1426.96 \n",
       "  1454.12,1426.77 1456.08,1426.58 1458.04,1426.39 1460,1426.2 1461.95,1426 1463.91,1425.8 1465.87,1425.6 1467.83,1425.4 1469.79,1425.2 1471.74,1424.99 \n",
       "  1473.7,1424.78 1475.66,1424.57 1477.62,1424.36 1479.58,1424.15 1481.53,1423.93 1483.49,1423.71 1485.45,1423.49 1487.41,1423.27 1489.37,1423.04 1491.32,1422.82 \n",
       "  1493.28,1422.59 1495.24,1422.35 1497.2,1422.12 1499.16,1421.88 1501.11,1421.64 1503.07,1421.4 1505.03,1421.16 1506.99,1420.91 1508.95,1420.66 1510.9,1420.41 \n",
       "  1512.86,1420.16 1514.82,1419.9 1516.78,1419.64 1518.73,1419.38 1520.69,1419.12 1522.65,1418.85 1524.61,1418.58 1526.57,1418.31 1528.52,1418.04 1530.48,1417.76 \n",
       "  1532.44,1417.48 1534.4,1417.2 1536.36,1416.91 1538.31,1416.62 1540.27,1416.33 1542.23,1416.04 1544.19,1415.74 1546.15,1415.44 1548.1,1415.13 1550.06,1414.83 \n",
       "  1552.02,1414.52 1553.98,1414.21 1555.94,1413.89 1557.89,1413.57 1559.85,1413.25 1561.81,1412.92 1563.77,1412.59 1565.73,1412.26 1567.68,1411.93 1569.64,1411.59 \n",
       "  1571.6,1411.25 1573.56,1410.9 1575.51,1410.55 1577.47,1410.2 1579.43,1409.84 1581.39,1409.49 1583.35,1409.12 1585.3,1408.76 1587.26,1408.39 1589.22,1408.01 \n",
       "  1591.18,1407.63 1593.14,1407.25 1595.09,1406.87 1597.05,1406.48 1599.01,1406.08 1600.97,1405.69 1602.93,1405.29 1604.88,1404.88 1606.84,1404.47 1608.8,1404.06 \n",
       "  1610.76,1403.64 1612.72,1403.22 1614.67,1402.8 1616.63,1402.37 1618.59,1401.93 1620.55,1401.49 1622.51,1401.05 1624.46,1400.6 1626.42,1400.15 1628.38,1399.69 \n",
       "  1630.34,1399.23 1632.3,1398.77 1634.25,1398.3 1636.21,1397.82 1638.17,1397.34 1640.13,1396.86 1642.08,1396.37 1644.04,1395.88 1646,1395.38 1647.96,1394.87 \n",
       "  1649.92,1394.36 1651.87,1393.85 1653.83,1393.33 1655.79,1392.81 1657.75,1392.28 1659.71,1391.74 1661.66,1391.2 1663.62,1390.66 1665.58,1390.1 1667.54,1389.55 \n",
       "  1669.5,1388.98 1671.45,1388.42 1673.41,1387.84 1675.37,1387.26 1677.33,1386.68 1679.29,1386.09 1681.24,1385.49 1683.2,1384.89 1685.16,1384.28 1687.12,1383.66 \n",
       "  1689.08,1383.04 1691.03,1382.42 1692.99,1381.78 1694.95,1381.14 1696.91,1380.5 1698.86,1379.84 1700.82,1379.18 1702.78,1378.52 1704.74,1377.84 1706.7,1377.17 \n",
       "  1708.65,1376.48 1710.61,1375.79 1712.57,1375.09 1714.53,1374.38 1716.49,1373.67 1718.44,1372.94 1720.4,1372.22 1722.36,1371.48 1724.32,1370.74 1726.28,1369.99 \n",
       "  1728.23,1369.23 1730.19,1368.46 1732.15,1367.69 1734.11,1366.91 1736.07,1366.12 1738.02,1365.32 1739.98,1364.52 1741.94,1363.71 1743.9,1362.89 1745.86,1362.06 \n",
       "  1747.81,1361.22 1749.77,1360.37 1751.73,1359.52 1753.69,1358.66 1755.64,1357.79 1757.6,1356.91 1759.56,1356.02 1761.52,1355.12 1763.48,1354.21 1765.43,1353.3 \n",
       "  1767.39,1352.37 1769.35,1351.44 1771.31,1350.49 1773.27,1349.54 1775.22,1348.58 1777.18,1347.61 1779.14,1346.63 1781.1,1345.63 1783.06,1344.63 1785.01,1343.62 \n",
       "  1786.97,1342.6 1788.93,1341.57 1790.89,1340.53 1792.85,1339.47 1794.8,1338.41 1796.76,1337.34 1798.72,1336.25 1800.68,1335.16 1802.64,1334.05 1804.59,1332.93 \n",
       "  1806.55,1331.8 1808.51,1330.66 1810.47,1329.51 1812.42,1328.35 1814.38,1327.18 1816.34,1325.99 1818.3,1324.79 1820.26,1323.58 1822.21,1322.36 1824.17,1321.13 \n",
       "  1826.13,1319.88 1828.09,1318.62 1830.05,1317.35 1832,1316.07 1833.96,1314.77 1835.92,1313.46 1837.88,1312.13 1839.84,1310.8 1841.79,1309.45 1843.75,1308.09 \n",
       "  1845.71,1306.71 1847.67,1305.32 1849.63,1303.91 1851.58,1302.49 1853.54,1301.06 1855.5,1299.61 1857.46,1298.15 1859.42,1296.68 1861.37,1295.19 1863.33,1293.68 \n",
       "  1865.29,1292.16 1867.25,1290.62 1869.21,1289.07 1871.16,1287.5 1873.12,1285.92 1875.08,1284.32 1877.04,1282.71 1878.99,1281.08 1880.95,1279.43 1882.91,1277.77 \n",
       "  1884.87,1276.09 1886.83,1274.39 1888.78,1272.67 1890.74,1270.94 1892.7,1269.2 1894.66,1267.43 1896.62,1265.65 1898.57,1263.84 1900.53,1262.03 1902.49,1260.19 \n",
       "  1904.45,1258.33 1906.41,1256.46 1908.36,1254.56 1910.32,1252.65 1912.28,1250.72 1914.24,1248.77 1916.2,1246.8 1918.15,1244.81 1920.11,1242.8 1922.07,1240.77 \n",
       "  1924.03,1238.72 1925.99,1236.65 1927.94,1234.56 1929.9,1232.45 1931.86,1230.31 1933.82,1228.16 1935.77,1225.98 1937.73,1223.78 1939.69,1221.56 1941.65,1219.32 \n",
       "  1943.61,1217.06 1945.56,1214.77 1947.52,1212.46 1949.48,1210.13 1951.44,1207.77 1953.4,1205.39 1955.35,1202.99 1957.31,1200.56 1959.27,1198.11 1961.23,1195.63 \n",
       "  1963.19,1193.13 1965.14,1190.6 1967.1,1188.05 1969.06,1185.47 1971.02,1182.87 1972.98,1180.24 1974.93,1177.58 1976.89,1174.9 1978.85,1172.19 1980.81,1169.46 \n",
       "  1982.77,1166.69 1984.72,1163.9 1986.68,1161.08 1988.64,1158.24 1990.6,1155.36 1992.55,1152.46 1994.51,1149.53 1996.47,1146.56 1998.43,1143.57 2000.39,1140.55 \n",
       "  2002.34,1137.5 2004.3,1134.41 2006.26,1131.3 2008.22,1128.15 2010.18,1124.98 2012.13,1121.77 2014.09,1118.53 2016.05,1115.26 2018.01,1111.95 2019.97,1108.61 \n",
       "  2021.92,1105.24 2023.88,1101.84 2025.84,1098.4 2027.8,1094.92 2029.76,1091.42 2031.71,1087.87 2033.67,1084.29 2035.63,1080.68 2037.59,1077.03 2039.55,1073.34 \n",
       "  2041.5,1069.61 2043.46,1065.85 2045.42,1062.05 2047.38,1058.22 2049.33,1054.34 2051.29,1050.43 2053.25,1046.47 2055.21,1042.48 2057.17,1038.45 2059.12,1034.37 \n",
       "  2061.08,1030.26 2063.04,1026.1 2065,1021.91 2066.96,1017.67 2068.91,1013.39 2070.87,1009.06 2072.83,1004.7 2074.79,1000.29 2076.75,995.83 2078.7,991.331 \n",
       "  2080.66,986.786 2082.62,982.197 2084.58,977.561 2086.54,972.878 2088.49,968.149 2090.45,963.373 2092.41,958.549 2094.37,953.677 2096.33,948.756 2098.28,943.785 \n",
       "  2100.24,938.765 2102.2,933.695 2104.16,928.574 2106.12,923.402 2108.07,918.179 2110.03,912.903 2111.99,907.574 2113.95,902.192 2115.9,896.756 2117.86,891.266 \n",
       "  2119.82,885.72 2121.78,880.12 2123.74,874.463 2125.69,868.75 2127.65,862.98 2129.61,857.152 2131.57,851.265 2133.53,845.32 2135.48,839.316 2137.44,833.251 \n",
       "  2139.4,827.126 2141.36,820.939 2143.32,814.691 2145.27,808.38 2147.23,802.006 2149.19,795.568 2151.15,789.066 2153.11,782.499 2155.06,775.866 2157.02,769.167 \n",
       "  2158.98,762.401 2160.94,755.567 2162.9,748.665 2164.85,741.694 2166.81,734.653 2168.77,727.541 2170.73,720.359 2172.68,713.105 2174.64,705.778 2176.6,698.378 \n",
       "  2178.56,690.904 2180.52,683.355 2182.47,675.731 2184.43,668.03 2186.39,660.253 2188.35,652.398 2190.31,644.464 2192.26,636.451 2194.22,628.357 2196.18,620.183 \n",
       "  2198.14,611.927 2200.1,603.589 2202.05,595.167 2204.01,586.66 2205.97,578.069 2207.93,569.392 2209.89,560.628 2211.84,551.777 2213.8,542.837 2215.76,533.807 \n",
       "  2217.72,524.687 2219.68,515.477 2221.63,506.174 2223.59,496.777 2225.55,487.287 2227.51,477.702 2229.46,468.022 2231.42,458.244 2233.38,448.369 2235.34,438.395 \n",
       "  2237.3,428.321 2239.25,418.146 2241.21,407.87 2243.17,397.491 2245.13,387.008 2247.09,376.42 2249.04,365.726 2251,354.926 2252.96,344.017 2254.92,333 \n",
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      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(run_infection(1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
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       "  922.878,983.622 924.953,982.263 927.028,980.904 929.104,979.545 931.179,978.186 933.255,976.827 935.33,975.468 937.405,974.109 939.481,972.749 941.556,971.39 \n",
       "  943.631,970.031 945.707,968.672 947.782,967.313 949.858,965.954 951.933,964.595 954.008,963.236 956.084,961.877 958.159,960.518 960.235,959.158 962.31,957.799 \n",
       "  964.385,956.44 966.461,955.081 968.536,953.722 970.612,952.363 972.687,951.004 974.762,949.645 976.838,948.286 978.913,946.927 980.988,945.567 983.064,944.208 \n",
       "  985.139,942.849 987.215,941.49 989.29,940.131 991.365,938.772 993.441,937.413 995.516,936.054 997.592,934.695 999.667,933.336 1001.74,931.976 1003.82,930.617 \n",
       "  1005.89,929.258 1007.97,927.899 1010.04,926.54 1012.12,925.181 1014.19,923.822 1016.27,922.463 1018.35,921.104 1020.42,919.745 1022.5,918.385 1024.57,917.026 \n",
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       "  1068.15,888.485 1070.23,887.126 1072.31,885.767 1074.38,884.408 1076.46,883.049 1078.53,881.69 1080.61,880.331 1082.68,878.972 1084.76,877.613 1086.83,876.253 \n",
       "  1088.91,874.894 1090.98,873.535 1093.06,872.176 1095.13,870.817 1097.21,869.458 1099.29,868.099 1101.36,866.74 1103.44,865.381 1105.51,864.022 1107.59,862.662 \n",
       "  1109.66,861.303 1111.74,859.944 1113.81,858.585 1115.89,857.226 1117.96,855.867 1120.04,854.508 1122.11,853.149 1124.19,851.79 1126.27,850.431 1128.34,849.071 \n",
       "  1130.42,847.712 1132.49,846.353 1134.57,844.994 1136.64,843.635 1138.72,842.276 1140.79,840.917 1142.87,839.558 1144.94,838.199 1147.02,836.84 1149.09,835.48 \n",
       "  1151.17,834.121 1153.25,832.762 1155.32,831.403 1157.4,830.044 1159.47,828.685 1161.55,827.326 1163.62,825.967 1165.7,824.608 1167.77,823.249 1169.85,821.889 \n",
       "  1171.92,820.53 1174,819.171 1176.07,817.812 1178.15,816.453 1180.23,815.094 1182.3,813.735 1184.38,812.376 1186.45,811.017 1188.53,809.658 1190.6,808.298 \n",
       "  1192.68,806.939 1194.75,805.58 1196.83,804.221 1198.9,802.862 1200.98,801.503 1203.05,800.144 1205.13,798.785 1207.21,797.426 1209.28,796.067 1211.36,794.708 \n",
       "  1213.43,793.348 1215.51,791.989 1217.58,790.63 1219.66,789.271 1221.73,787.912 1223.81,786.553 1225.88,785.194 1227.96,783.835 1230.03,782.476 1232.11,781.117 \n",
       "  1234.19,779.757 1236.26,778.398 1238.34,777.039 1240.41,775.68 1242.49,774.321 1244.56,772.962 1246.64,771.603 1248.71,770.244 1250.79,768.885 1252.86,767.526 \n",
       "  1254.94,766.166 1257.01,764.807 1259.09,763.448 1261.17,762.089 1263.24,760.73 1265.32,759.371 1267.39,758.012 1269.47,756.653 1271.54,755.294 1273.62,753.935 \n",
       "  1275.69,752.575 1277.77,751.216 1279.84,749.857 1281.92,748.498 1283.99,747.139 1286.07,745.78 1288.15,744.421 1290.22,743.062 1292.3,741.703 1294.37,740.344 \n",
       "  1296.45,738.984 1298.52,737.625 1300.6,736.266 1302.67,734.907 1304.75,733.548 1306.82,732.189 1308.9,730.83 1310.97,729.471 1313.05,728.112 1315.13,726.753 \n",
       "  1317.2,725.393 1319.28,724.034 1321.35,722.675 1323.43,721.316 1325.5,719.957 1327.58,718.598 1329.65,717.239 1331.73,715.88 1333.8,714.521 1335.88,713.162 \n",
       "  1337.95,711.802 1340.03,710.443 1342.11,709.084 1344.18,707.725 1346.26,706.366 1348.33,705.007 1350.41,703.648 1352.48,702.289 1354.56,700.93 1356.63,699.571 \n",
       "  1358.71,698.212 1360.78,696.852 1362.86,695.493 1364.93,694.134 1367.01,692.775 1369.09,691.416 1371.16,690.057 1373.24,688.698 1375.31,687.339 1377.39,685.98 \n",
       "  1379.46,684.621 1381.54,683.261 1383.61,681.902 1385.69,680.543 1387.76,679.184 1389.84,677.825 1391.91,676.466 1393.99,675.107 1396.07,673.748 1398.14,672.389 \n",
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       "  1898.31,344.846 1900.38,343.487 1902.46,342.128 1904.53,340.769 1906.61,339.41 1908.69,338.05 1910.76,336.691 1912.84,335.332 1914.91,333.973 1916.99,332.614 \n",
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       "  1939.82,317.664 1941.89,316.305 1943.97,314.946 1946.04,313.587 1948.12,312.228 1950.19,310.868 1952.27,309.509 1954.34,308.15 1956.42,306.791 1958.49,305.432 \n",
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     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(run_infection(1000), yscale=:log10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
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       "  401.557,1336.99 403.613,1335.63 405.669,1334.27 407.725,1332.91 409.782,1331.55 411.838,1330.19 413.894,1328.83 415.95,1327.47 418.006,1326.12 420.062,1324.76 \n",
       "  422.118,1323.4 424.174,1322.04 426.231,1320.68 428.287,1319.32 430.343,1317.96 432.399,1316.6 434.455,1315.24 436.511,1313.88 438.567,1312.52 440.623,1311.16 \n",
       "  442.679,1309.81 444.736,1308.45 446.792,1307.09 448.848,1305.73 450.904,1304.37 452.96,1303.01 455.016,1301.65 457.072,1300.29 459.128,1298.93 461.185,1297.57 \n",
       "  463.241,1296.21 465.297,1294.86 467.353,1293.5 469.409,1292.14 471.465,1290.78 473.521,1289.42 475.577,1288.06 477.634,1286.7 479.69,1285.34 481.746,1283.98 \n",
       "  483.802,1282.62 485.858,1281.26 487.914,1279.91 489.97,1278.55 492.026,1277.19 494.083,1275.83 496.139,1274.47 498.195,1273.11 500.251,1271.75 502.307,1270.39 \n",
       "  504.363,1269.03 506.419,1267.67 508.475,1266.31 510.532,1264.96 512.588,1263.6 514.644,1262.24 516.7,1260.88 518.756,1259.52 520.812,1258.16 522.868,1256.8 \n",
       "  524.924,1255.44 526.981,1254.08 529.037,1252.72 531.093,1251.36 533.149,1250.01 535.205,1248.65 537.261,1247.29 539.317,1245.93 541.373,1244.57 543.43,1243.21 \n",
       "  545.486,1241.85 547.542,1240.49 549.598,1239.13 551.654,1237.77 553.71,1236.41 555.766,1235.06 557.822,1233.7 559.878,1232.34 561.935,1230.98 563.991,1229.62 \n",
       "  566.047,1228.26 568.103,1226.9 570.159,1225.54 572.215,1224.18 574.271,1222.82 576.327,1221.46 578.384,1220.11 580.44,1218.75 582.496,1217.39 584.552,1216.03 \n",
       "  586.608,1214.67 588.664,1213.31 590.72,1211.95 592.776,1210.59 594.833,1209.23 596.889,1207.87 598.945,1206.51 601.001,1205.16 603.057,1203.8 605.113,1202.44 \n",
       "  607.169,1201.08 609.225,1199.72 611.282,1198.36 613.338,1197 615.394,1195.64 617.45,1194.28 619.506,1192.92 621.562,1191.56 623.618,1190.21 625.674,1188.85 \n",
       "  627.731,1187.49 629.787,1186.13 631.843,1184.77 633.899,1183.41 635.955,1182.05 638.011,1180.69 640.067,1179.33 642.123,1177.97 644.18,1176.61 646.236,1175.26 \n",
       "  648.292,1173.9 650.348,1172.54 652.404,1171.18 654.46,1169.82 656.516,1168.46 658.572,1167.1 660.629,1165.74 662.685,1164.38 664.741,1163.02 666.797,1161.66 \n",
       "  668.853,1160.31 670.909,1158.95 672.965,1157.59 675.021,1156.23 677.077,1154.87 679.134,1153.51 681.19,1152.15 683.246,1150.79 685.302,1149.43 687.358,1148.07 \n",
       "  689.414,1146.71 691.47,1145.35 693.526,1144 695.583,1142.64 697.639,1141.28 699.695,1139.92 701.751,1138.56 703.807,1137.2 705.863,1135.84 707.919,1134.48 \n",
       "  709.975,1133.12 712.032,1131.76 714.088,1130.4 716.144,1129.05 718.2,1127.69 720.256,1126.33 722.312,1124.97 724.368,1123.61 726.424,1122.25 728.481,1120.89 \n",
       "  730.537,1119.53 732.593,1118.17 734.649,1116.81 736.705,1115.45 738.761,1114.1 740.817,1112.74 742.873,1111.38 744.93,1110.02 746.986,1108.66 749.042,1107.3 \n",
       "  751.098,1105.94 753.154,1104.58 755.21,1103.22 757.266,1101.86 759.322,1100.5 761.379,1099.15 763.435,1097.79 765.491,1096.43 767.547,1095.07 769.603,1093.71 \n",
       "  771.659,1092.35 773.715,1090.99 775.771,1089.63 777.828,1088.27 779.884,1086.91 781.94,1085.55 783.996,1084.2 786.052,1082.84 788.108,1081.48 790.164,1080.12 \n",
       "  792.22,1078.76 794.276,1077.4 796.333,1076.04 798.389,1074.68 800.445,1073.32 802.501,1071.96 804.557,1070.6 806.613,1069.25 808.669,1067.89 810.725,1066.53 \n",
       "  812.782,1065.17 814.838,1063.81 816.894,1062.45 818.95,1061.09 821.006,1059.73 823.062,1058.37 825.118,1057.01 827.174,1055.65 829.231,1054.3 831.287,1052.94 \n",
       "  833.343,1051.58 835.399,1050.22 837.455,1048.86 839.511,1047.5 841.567,1046.14 843.623,1044.78 845.68,1043.42 847.736,1042.06 849.792,1040.7 851.848,1039.35 \n",
       "  853.904,1037.99 855.96,1036.63 858.016,1035.27 860.072,1033.91 862.129,1032.55 864.185,1031.19 866.241,1029.83 868.297,1028.47 870.353,1027.11 872.409,1025.75 \n",
       "  874.465,1024.4 876.521,1023.04 878.578,1021.68 880.634,1020.32 882.69,1018.96 884.746,1017.6 886.802,1016.24 888.858,1014.88 890.914,1013.52 892.97,1012.16 \n",
       "  895.027,1010.8 897.083,1009.45 899.139,1008.09 901.195,1006.73 903.251,1005.37 905.307,1004.01 907.363,1002.65 909.419,1001.29 911.475,999.931 913.532,998.572 \n",
       "  915.588,997.213 917.644,995.854 919.7,994.495 921.756,993.136 923.812,991.777 925.868,990.418 927.924,989.059 929.981,987.699 932.037,986.34 934.093,984.981 \n",
       "  936.149,983.622 938.205,982.263 940.261,980.904 942.317,979.545 944.373,978.186 946.43,976.827 948.486,975.468 950.542,974.109 952.598,972.749 954.654,971.39 \n",
       "  956.71,970.031 958.766,968.672 960.822,967.313 962.879,965.954 964.935,964.595 966.991,963.236 969.047,961.877 971.103,960.518 973.159,959.158 975.215,957.799 \n",
       "  977.271,956.44 979.328,955.081 981.384,953.722 983.44,952.363 985.496,951.004 987.552,949.645 989.608,948.286 991.664,946.927 993.72,945.567 995.777,944.208 \n",
       "  997.833,942.849 999.889,941.49 1001.94,940.131 1004,938.772 1006.06,937.413 1008.11,936.054 1010.17,934.695 1012.23,933.336 1014.28,931.976 1016.34,930.617 \n",
       "  1018.39,929.258 1020.45,927.899 1022.51,926.54 1024.56,925.181 1026.62,923.822 1028.67,922.463 1030.73,921.104 1032.79,919.745 1034.84,918.385 1036.9,917.026 \n",
       "  1038.96,915.667 1041.01,914.308 1043.07,912.949 1045.12,911.59 1047.18,910.231 1049.24,908.872 1051.29,907.513 1053.35,906.154 1055.4,904.794 1057.46,903.435 \n",
       "  1059.52,902.076 1061.57,900.717 1063.63,899.358 1065.68,897.999 1067.74,896.64 1069.8,895.281 1071.85,893.922 1073.91,892.563 1075.97,891.203 1078.02,889.844 \n",
       "  1080.08,888.485 1082.13,887.126 1084.19,885.767 1086.25,884.408 1088.3,883.049 1090.36,881.69 1092.41,880.331 1094.47,878.972 1096.53,877.613 1098.58,876.253 \n",
       "  1100.64,874.894 1102.69,873.535 1104.75,872.176 1106.81,870.817 1108.86,869.458 1110.92,868.099 1112.98,866.74 1115.03,865.381 1117.09,864.022 1119.14,862.662 \n",
       "  1121.2,861.303 1123.26,859.944 1125.31,858.585 1127.37,857.226 1129.42,855.867 1131.48,854.508 1133.54,853.149 1135.59,851.79 1137.65,850.431 1139.71,849.071 \n",
       "  1141.76,847.712 1143.82,846.353 1145.87,844.994 1147.93,843.635 1149.99,842.276 1152.04,840.917 1154.1,839.558 1156.15,838.199 1158.21,836.84 1160.27,835.48 \n",
       "  1162.32,834.121 1164.38,832.762 1166.43,831.403 1168.49,830.044 1170.55,828.685 1172.6,827.326 1174.66,825.967 1176.72,824.608 1178.77,823.249 1180.83,821.889 \n",
       "  1182.88,820.53 1184.94,819.171 1187,817.812 1189.05,816.453 1191.11,815.094 1193.16,813.735 1195.22,812.376 1197.28,811.017 1199.33,809.658 1201.39,808.298 \n",
       "  1203.44,806.939 1205.5,805.58 1207.56,804.221 1209.61,802.862 1211.67,801.503 1213.73,800.144 1215.78,798.785 1217.84,797.426 1219.89,796.067 1221.95,794.708 \n",
       "  1224.01,793.348 1226.06,791.989 1228.12,790.63 1230.17,789.271 1232.23,787.912 1234.29,786.553 1236.34,785.194 1238.4,783.835 1240.46,782.476 1242.51,781.117 \n",
       "  1244.57,779.757 1246.62,778.398 1248.68,777.039 1250.74,775.68 1252.79,774.321 1254.85,772.962 1256.9,771.603 1258.96,770.244 1261.02,768.885 1263.07,767.526 \n",
       "  1265.13,766.166 1267.18,764.807 1269.24,763.448 1271.3,762.089 1273.35,760.73 1275.41,759.371 1277.47,758.012 1279.52,756.653 1281.58,755.294 1283.63,753.935 \n",
       "  1285.69,752.575 1287.75,751.216 1289.8,749.857 1291.86,748.498 1293.91,747.139 1295.97,745.78 1298.03,744.421 1300.08,743.062 1302.14,741.703 1304.19,740.344 \n",
       "  1306.25,738.984 1308.31,737.625 1310.36,736.266 1312.42,734.907 1314.48,733.548 1316.53,732.189 1318.59,730.83 1320.64,729.471 1322.7,728.112 1324.76,726.753 \n",
       "  1326.81,725.393 1328.87,724.034 1330.92,722.675 1332.98,721.316 1335.04,719.957 1337.09,718.598 1339.15,717.239 1341.21,715.88 1343.26,714.521 1345.32,713.162 \n",
       "  1347.37,711.802 1349.43,710.443 1351.49,709.084 1353.54,707.725 1355.6,706.366 1357.65,705.007 1359.71,703.648 1361.77,702.289 1363.82,700.93 1365.88,699.571 \n",
       "  1367.93,698.212 1369.99,696.852 1372.05,695.493 1374.1,694.134 1376.16,692.775 1378.22,691.416 1380.27,690.057 1382.33,688.698 1384.38,687.339 1386.44,685.98 \n",
       "  1388.5,684.621 1390.55,683.261 1392.61,681.902 1394.66,680.543 1396.72,679.184 1398.78,677.825 1400.83,676.466 1402.89,675.107 1404.94,673.748 1407,672.389 \n",
       "  1409.06,671.03 1411.11,669.67 1413.17,668.311 1415.23,666.952 1417.28,665.593 1419.34,664.234 1421.39,662.875 1423.45,661.516 1425.51,660.157 1427.56,658.798 \n",
       "  1429.62,657.439 1431.67,656.079 1433.73,654.72 1435.79,653.361 1437.84,652.002 1439.9,650.643 1441.96,649.284 1444.01,647.925 1446.07,646.566 1448.12,645.207 \n",
       "  1450.18,643.848 1452.24,642.488 1454.29,641.129 1456.35,639.77 1458.4,638.411 1460.46,637.052 1462.52,635.693 1464.57,634.334 1466.63,632.975 1468.68,631.616 \n",
       "  1470.74,630.257 1472.8,628.897 1474.85,627.538 1476.91,626.179 1478.97,624.82 1481.02,623.461 1483.08,622.102 1485.13,620.743 1487.19,619.384 1489.25,618.025 \n",
       "  1491.3,616.666 1493.36,615.307 1495.41,613.947 1497.47,612.588 1499.53,611.229 1501.58,609.87 1503.64,608.511 1505.69,607.152 1507.75,605.793 1509.81,604.434 \n",
       "  1511.86,603.075 1513.92,601.716 1515.98,600.356 1518.03,598.997 1520.09,597.638 1522.14,596.279 1524.2,594.92 1526.26,593.561 1528.31,592.202 1530.37,590.843 \n",
       "  1532.42,589.484 1534.48,588.125 1536.54,586.765 1538.59,585.406 1540.65,584.047 1542.71,582.688 1544.76,581.329 1546.82,579.97 1548.87,578.611 1550.93,577.252 \n",
       "  1552.99,575.893 1555.04,574.534 1557.1,573.174 1559.15,571.815 1561.21,570.456 1563.27,569.097 1565.32,567.738 1567.38,566.379 1569.43,565.02 1571.49,563.661 \n",
       "  1573.55,562.302 1575.6,560.943 1577.66,559.583 1579.72,558.224 1581.77,556.865 1583.83,555.506 1585.88,554.147 1587.94,552.788 1590,551.429 1592.05,550.07 \n",
       "  1594.11,548.711 1596.16,547.352 1598.22,545.992 1600.28,544.633 1602.33,543.274 1604.39,541.915 1606.45,540.556 1608.5,539.197 1610.56,537.838 1612.61,536.479 \n",
       "  1614.67,535.12 1616.73,533.761 1618.78,532.401 1620.84,531.042 1622.89,529.683 1624.95,528.324 1627.01,526.965 1629.06,525.606 1631.12,524.247 1633.17,522.888 \n",
       "  1635.23,521.529 1637.29,520.17 1639.34,518.811 1641.4,517.451 1643.46,516.092 1645.51,514.733 1647.57,513.374 1649.62,512.015 1651.68,510.656 1653.74,509.297 \n",
       "  1655.79,507.938 1657.85,506.579 1659.9,505.22 1661.96,503.86 1664.02,502.501 1666.07,501.142 1668.13,499.783 1670.18,498.424 1672.24,497.065 1674.3,495.706 \n",
       "  1676.35,494.347 1678.41,492.988 1680.47,491.629 1682.52,490.269 1684.58,488.91 1686.63,487.551 1688.69,486.192 1690.75,484.833 1692.8,483.474 1694.86,482.115 \n",
       "  1696.91,480.756 1698.97,479.397 1701.03,478.038 1703.08,476.678 1705.14,475.319 1707.2,473.96 1709.25,472.601 1711.31,471.242 1713.36,469.883 1715.42,468.524 \n",
       "  1717.48,467.165 1719.53,465.806 1721.59,464.447 1723.64,463.087 1725.7,461.728 1727.76,460.369 1729.81,459.01 1731.87,457.651 1733.92,456.292 1735.98,454.933 \n",
       "  1738.04,453.574 1740.09,452.215 1742.15,450.856 1744.21,449.496 1746.26,448.137 1748.32,446.778 1750.37,445.419 1752.43,444.06 1754.49,442.701 1756.54,441.342 \n",
       "  1758.6,439.983 1760.65,438.624 1762.71,437.265 1764.77,435.905 1766.82,434.546 1768.88,433.187 1770.93,431.828 1772.99,430.469 1775.05,429.11 1777.1,427.751 \n",
       "  1779.16,426.392 1781.22,425.033 1783.27,423.674 1785.33,422.315 1787.38,420.955 1789.44,419.596 1791.5,418.237 1793.55,416.878 1795.61,415.519 1797.66,414.16 \n",
       "  1799.72,412.801 1801.78,411.442 1803.83,410.083 1805.89,408.724 1807.95,407.364 1810,406.005 1812.06,404.646 1814.11,403.287 1816.17,401.928 1818.23,400.569 \n",
       "  1820.28,399.21 1822.34,397.851 1824.39,396.492 1826.45,395.133 1828.51,393.773 1830.56,392.414 1832.62,391.055 1834.67,389.696 1836.73,388.337 1838.79,386.978 \n",
       "  1840.84,385.619 1842.9,384.26 1844.96,382.901 1847.01,381.542 1849.07,380.182 1851.12,378.823 1853.18,377.464 1855.24,376.105 1857.29,374.746 1859.35,373.387 \n",
       "  1861.4,372.028 1863.46,370.669 1865.52,369.31 1867.57,367.951 1869.63,366.591 1871.68,365.232 1873.74,363.873 1875.8,362.514 1877.85,361.155 1879.91,359.796 \n",
       "  1881.97,358.437 1884.02,357.078 1886.08,355.719 1888.13,354.36 1890.19,353 1892.25,351.641 1894.3,350.282 1896.36,348.923 1898.41,347.564 1900.47,346.205 \n",
       "  1902.53,344.846 1904.58,343.487 1906.64,342.128 1908.7,340.769 1910.75,339.41 1912.81,338.05 1914.86,336.691 1916.92,335.332 1918.98,333.973 1921.03,332.614 \n",
       "  1923.09,331.255 1925.14,329.896 1927.2,328.537 1929.26,327.178 1931.31,325.819 1933.37,324.459 1935.42,323.1 1937.48,321.741 1939.54,320.382 1941.59,319.023 \n",
       "  1943.65,317.664 1945.71,316.305 1947.76,314.946 1949.82,313.587 1951.87,312.228 1953.93,310.868 1955.99,309.509 1958.04,308.15 1960.1,306.791 1962.15,305.432 \n",
       "  1964.21,304.073 1966.27,302.714 1968.32,301.355 1970.38,299.996 1972.43,298.637 1974.49,297.277 1976.55,295.918 1978.6,294.559 1980.66,293.2 1982.72,291.841 \n",
       "  1984.77,290.482 1986.83,289.123 1988.88,287.764 1990.94,286.405 1993,285.046 1995.05,283.686 1997.11,282.327 1999.16,280.968 2001.22,279.609 2003.28,278.25 \n",
       "  2005.33,276.891 2007.39,275.532 2009.45,274.173 2011.5,272.814 2013.56,271.455 2015.61,270.095 2017.67,268.736 2019.73,267.377 2021.78,266.018 2023.84,264.659 \n",
       "  2025.89,263.3 2027.95,261.941 2030.01,260.582 2032.06,259.223 2034.12,257.864 2036.17,256.504 2038.23,255.145 2040.29,253.786 2042.34,252.427 2044.4,251.068 \n",
       "  2046.46,249.709 2048.51,248.35 2050.57,246.991 2052.62,245.632 2054.68,244.273 2056.74,242.914 2058.79,241.554 2060.85,240.195 2062.9,238.836 2064.96,237.477 \n",
       "  2067.02,236.118 2069.07,234.759 2071.13,233.4 2073.18,232.041 2075.24,230.682 2077.3,229.323 2079.35,227.963 2081.41,226.604 2083.47,225.245 2085.52,223.886 \n",
       "  2087.58,222.527 2089.63,221.168 2091.69,219.809 2093.75,218.45 2095.8,217.091 2097.86,215.732 2099.91,214.372 2101.97,213.013 2104.03,211.654 2106.08,210.295 \n",
       "  2108.14,208.936 2110.2,207.577 2112.25,206.218 2114.31,204.859 2116.36,203.5 2118.42,202.141 2120.48,200.781 2122.53,199.422 2124.59,198.063 2126.64,196.704 \n",
       "  2128.7,195.345 2130.76,193.986 2132.81,192.627 2134.87,191.268 2136.92,189.909 2138.98,188.55 2141.04,187.19 2143.09,185.831 2145.15,184.472 2147.21,183.113 \n",
       "  2149.26,181.754 2151.32,180.395 2153.37,179.036 2155.43,177.677 2157.49,176.318 2159.54,174.959 2161.6,173.599 2163.65,172.24 2165.71,170.881 2167.77,169.522 \n",
       "  2169.82,168.163 2171.88,166.804 2173.93,165.445 2175.99,164.086 2178.05,162.727 2180.1,161.368 2182.16,160.008 2184.22,158.649 2186.27,157.29 2188.33,155.931 \n",
       "  2190.38,154.572 2192.44,153.213 2194.5,151.854 2196.55,150.495 2198.61,149.136 2200.66,147.777 2202.72,146.418 2204.78,145.058 2206.83,143.699 2208.89,142.34 \n",
       "  2210.95,140.981 2213,139.622 2215.06,138.263 2217.11,136.904 2219.17,135.545 2221.23,134.186 2223.28,132.827 2225.34,131.467 2227.39,130.108 2229.45,128.749 \n",
       "  2231.51,127.39 2233.56,126.031 2235.62,124.672 2237.67,123.313 2239.73,121.954 2241.79,120.595 2243.84,119.236 2245.9,117.876 2247.96,116.517 2250.01,115.158 \n",
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      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_result = run_infection(1000)\n",
    "\n",
    "plot(log10.(I_result)) # log of each element in I_result\n",
    "ylabel!(\"log(I_n)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\log(I_n) = a n + b$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Take exponentials of 10 of both sides:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$I_n = \\exp(an + b) = C e^{an}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve $I_{n+1} = \\lambda I_n$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$I_n = \\lambda^n I_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\log(\\lambda^n) = n \\log(\\lambda)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logistic Growth"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> Exponential growth is unrealistic: Assumes that there are always more people to infect &rarr; wrong (finite population)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$I_{n+1} = I_n + c I_n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each individual will be in contact with a fraction $\\alpha$ of the population. At each contact whtere will be a probability $p$ that you infect each person.\n",
    "\n",
    "So $c = p \\alpha$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Original model:\n",
    " \n",
    "$I_{n+1} = I_n + (p \\alpha N) I_n$ &rarr; good approxiamtion when almost everybody still susceptible"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "New model:\n",
    " \n",
    "Can only infect uninfected people!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$I_{n+1} = I_n + (p \\alpha S_n) I_n$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$I_{n+1} = I_n + [p \\alpha (N - I_n)] I_n = f(I_n)$\n",
    "\n",
    "\n",
    "$I_{n+1} = I_n + \\beta(I_n,S_n) I_n = f(I_n)$ &rarr; $\\beta$ is the rate of contact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "run_infection (generic function with 2 methods)"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 0.02\n",
    "α = 0.01\n",
    "N = 1000\n",
    "\n",
    "β(I,S) = p * α * (N - I)\n",
    "\n",
    "function run_infection(T=20)    \n",
    "    \n",
    "    I = zeros(T)\n",
    "    I[1] = I₀\n",
    "    \n",
    "    for n in 1:T-1\n",
    "        I[n+1] = I[n] + β(I[n], N - I[n]) * I[n]\n",
    "    end\n",
    "    \n",
    "    return I\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "20-element Vector{Float64}:\n",
       "  1.0\n",
       "  1.1998\n",
       "  1.439472095992\n",
       "  1.7269520992073721\n",
       "  2.071746046338255\n",
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   "source": [
    "I = run_infection(20)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 77,
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     "execution_count": 77,
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    "plot(I, m=:o)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 78,
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     "execution_count": 78,
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   "source": [
    "plot(I, m=:o, yscale=:log10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I = run_infection(100)\n",
    "plot(I, m=:o, yscale=:log10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 0.01\n",
    "α = 0.1\n",
    "N = 100\n",
    "\n",
    "I = run_infection(100)\n",
    "plot(I, m=:o)\n",
    "title!(\"Sigmoid shape (S-shaped)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(I, m=:o, yscale=:log10)\n",
    "title!(\"Logistic-type growth\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More realism: Heterogeneity of individuals\n",
    "\n",
    "> Idea: Instead of modelling popuylation globally, model each individual\n",
    " \n",
    "- *Patch model*: Local patches where population is **well mixed**\n",
    "- *Network model*: Links between nodes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exponential growth:\n",
    " \n",
    " $I_{n+1} = \\lambda I_n$ &rarr; growth rate changes in time"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random values\n",
    "\n",
    "For real random values go to [random.org](www.random.org)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Random numbers between 0 and 1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "rand() = 0.7259038965677891\n",
      "rand() = 0.6501870687535087\n",
      "rand() = 0.4250024512035022\n",
      "rand() = 0.8201952600359164\n",
      "rand() = 0.14422847444684228\n",
      "rand() = 0.7486822095253662\n",
      "rand() = 0.05867708511376146\n",
      "rand() = 0.7651590399926838\n",
      "rand() = 0.21867639768362301\n",
      "rand() = 0.27539760156985116\n"
     ]
    }
   ],
   "source": [
    "[rand() for i in 1:1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Random numbers from a gaussian distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "randn() = 1.8607960200711762\n",
      "randn() = 0.36272040436406894\n",
      "randn() = -2.440994129674222\n",
      "randn() = 1.0688313947433834\n",
      "randn() = -0.2612847329349688\n",
      "randn() = 1.9830022657950983\n",
      "randn() = -1.1645500710008718\n",
      "randn() = -0.6449180532234828\n",
      "randn() = -0.3731236266186792\n",
      "randn() = -0.8620642627983233\n"
     ]
    }
   ],
   "source": [
    "for i in 1:10\n",
    "    @show randn()\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Add randomness to the infection model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "run_infection (generic function with 3 methods)"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 0.02\n",
    "α = 0.01\n",
    "N = 1000\n",
    "\n",
    "β(I, S) = p * α * S\n",
    "\n",
    "function run_infection(c_average=1.1, T = 20)\n",
    "    \n",
    "    I = zeros(T)\n",
    "    I[1] = I₀\n",
    "    \n",
    "    for n in 1:T-1\n",
    "        c = c_average + 0.1*randn()\n",
    "        I[n+1] = I[n] + c * I[n]\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100-element Vector{Float64}:\n",
       " 1.1547245950261458\n",
       " 1.2324858271933266\n",
       " 1.1517988436659095\n",
       " 1.2568181577499358\n",
       " 1.060598143694681\n",
       " 1.1433696181326287\n",
       " 1.1261045717576434\n",
       " 0.9132395664543387\n",
       " 0.9421478006358643\n",
       " 1.1509613206471978\n",
       " 0.9931987379710914\n",
       " 1.1825585633431606\n",
       " 1.1291055884876235\n",
       " ⋮\n",
       " 1.0608796696100975\n",
       " 1.0217403098937177\n",
       " 1.0731176643563012\n",
       " 1.1603533011096783\n",
       " 1.0728298994546446\n",
       " 1.0671544238307153\n",
       " 1.116109557767238\n",
       " 0.9965900063495665\n",
       " 1.3235901016957947\n",
       " 1.2416965857305857\n",
       " 1.0252650189497001\n",
       " 1.155079705068042"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_average = 1.1\n",
    "cs = [c_average + 0.1*randn() for i in 1:100]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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