update

Author: tom mcandrew

outbreak_science/chap2.ipynb

@@ -933,30 +933,34 @@
     "    3. There are 2 infectors and $p=1/20$\n",
     "3. An alternative to the Reed-Frost model is Greenwood's model. For Greenwood's, the probability of infection is equal to $p$ and not $1-(1-p)^I_{t}$ as it is in the Reed-Frost Dynamical System. That is, Greenwood's model assumes exposure to two or infectors is equivalent to exposure of a single infector. In a brief 2-3 sentences, please describe when Greenwood's model may an appropriate model of the spread of a pathogen (as opposed to the Reed-Frost model). \n",
     "4. In the section `Dynamics assumed at start of outbreak' we made assumptions about $S_{0}$ and the form of $1-(1-p)^{i_{0}}$ to arrive at an approximation solution for how an epidemic grows. To arrive at a better approximation, please recompute the dynamics at the start of an outbreak using the following, more accurate, assumptions\n",
-    "$$\\begin{align}\n",
+    "\n",
+    "$\\begin{align}\n",
     "    S_{0} &= (N-1)\\\\\n",
     "    e^{x} &\\approx 1 + x + \\frac{x^{2}}{2}\\\\\n",
     "    \\log(1-x) &\\approx -x -\\frac{x^{2}}{2}\n",
-    "\\end{align}$$\n",
+    "\\end{align}$\n",
+    "\n",
     "Your final answer will be a more complicated expression than our original assumptions that led to $Np$.\n",
     "Do you feel that the more complicated mathematical expression that you found is worth the extra computation? Why or why not? \n",
     "\n",
-    "5. At the start of the an outbreak, we assumed that there was a sigle infector. How do you expect the dynamics to change if there were more than one infector? Would identifying as many infectors as possible be important to public health? Using ideas from the 'Dynamics assumed at start of outbreak' please justify your answer. \n",
+    "6. At the start of the an outbreak, we assumed that there was a sigle infector. How do you expect the dynamics to change if there were more than one infector? Would identifying as many infectors as possible be important to public health? Using ideas from the 'Dynamics assumed at start of outbreak' please justify your answer. \n",
     "\n",
-    "6.  In the section \"Relationship between Reproduction number and outbreak.\" we assumed that there was a single infector at the beginning of an outbreak and developed the expression $\\mathbb{E}(I_{t}) = \\mathcal{R}_{0}^{t}$. Please develop a more general expression for the expected number of infectors over time of there are $i^{0}$ infectors. In other words, if there are $i^{0}$ infectors than $\\mathbb{E}(I_{t}) =?$\n",
+    "7.  In the section \"Relationship between Reproduction number and outbreak.\" we assumed that there was a single infector at the beginning of an outbreak and developed the expression $\\mathbb{E}(I_{t}) = \\mathcal{R}_{0}^{t}$. Please develop a more general expression for the expected number of infectors over time of there are $i^{0}$ infectors. In other words, if there are $i^{0}$ infectors than $\\mathbb{E}(I_{t}) =?$\n",
     "\n",
-    "7.  Using your answer from (6), please plot the expected number of infectors for 5 time points, given $p=0.1,N=100,i_{0}=1$, $p=0.1,N=100,i_{0}=2$, and $p=0.1,N=100,i_{0}=0$. Review these three lines: Is it more important to correctly estimate the basic reproduction number or the initial number of infectors? If there a value of the number of infectors that is important to an outbreak? Please justify your answer.  \n",
+    "8.  Using your answer from (6), please plot the expected number of infectors for 5 time points, given $p=0.1,N=100,i_{0}=1$, $p=0.1,N=100,i_{0}=2$, and $p=0.1,N=100,i_{0}=0$. Review these three lines: Is it more important to correctly estimate the basic reproduction number or the initial number of infectors? If there a value of the number of infectors that is important to an outbreak? Please justify your answer.  \n",
     "\n",
-    "8.  Please compute the Herd Immunity Threshold for the following scenarios\n",
+    "9.  Please compute the Herd Immunity Threshold for the following scenarios\n",
     "    1. There are 5 infectors and $\\mathcal{R}_{0}=1.50$\n",
     "    2. There are 2 infectors and $\\mathcal{R}_{0}=1.25$\n",
     "    3. There are 3 infectors and $\\mathcal{R}_{0}=2.00$\n",
     "    4. There are 0 infectors and $\\mathcal{R}_{0}=2.00$\n",
     "\n",
-    "9. When we computed conditions for when an the number of expected infectors would increase (i.e. when an outbreak will take place) we used several assumptions: (1) the number of susceptibles is close to $N$, (2) the probability of infection is small. Lets investigate how the conditions for an outbreak would change if we removed assumptions (1) and (2).<br><br> If we remove assumptions (1) and (2) then we can state that our new condition for an outbreak is when the expected number of infectors at time $1$ is larger than the expected number of infectors at time zero, or\n",
-    "\\begin{align}\n",
+    "10. When we computed conditions for when an the number of expected infectors would increase (i.e. when an outbreak will take place) we used several assumptions: (1) the number of susceptibles is close to $N$, (2) the probability of infection is small. Lets investigate how the conditions for an outbreak would change if we removed assumptions (1) and (2).<br><br> If we remove assumptions (1) and (2) then we can state that our new condition for an outbreak is when the expected number of infectors at time $1$ is larger than the expected number of infectors at time zero, or\n",
+    "\n",
+    "$\\begin{align}\n",
     "    \\mathbb{E}(I_{1}) = S_{0} \\left[ 1- (1-p)^{i_{0}}\\right] > i_{0}\n",
-    "\\end{align}\n",
+    "\\end{align}$\n",
+    "\n",
     "9.1. Please continue to solve the above inequality until there is a $S_{0}$, alone, on the left of the inequality and then express this as the proportion of immune individuals. Note 1: We will assume that there are no removed at this stage. This means that s+i=n (This this to replace i_{0} on the right hand side).  Note 2: The proportion of immune is one minus the proportion of susceptible or (imm = 1-s). \n",
     "\n",
     "9.2. Assume that $N=1000$, the number of infectors at time 0 equals 200, and $p=0.002$. Will an outbreak occur? Why or why not?\n",