Ciemss scenario 2

6-month-milestone/evaluation/scenario_2/ta_3/ciemss/scenario2_q2b_SIDARTHEV_alpha.ipynb

@@ -389,10 +389,12 @@
    "id": "c65bcf10-9d12-4efe-be10-cc313e123d14",
    "metadata": {},
    "source": [
-    "## 1.d Policies that prevent infections from exceeding 1/3 of the population\n",
+    "## 2..b.i Policies that prevent infections from exceeding 1/3 of the population\n",
     "\n",
-    "Now return to the situation in b.i (constant parameters that don’t change over time). Let’s say we want to increase testing, diagnostics, and contact tracing efforts (implemented by increasing the detection parameters  and ). Assume that $\\theta \\ge 2\\epsilon$ , because a symptomatic person is more likely to be tested. What minimum constant values do these parameters need to be over the course of a 100-day simulation, to ensure that the total infected population (sum over all the infected states I, D, A, R, T) never rises above 1/3 of the total population?\n",
+    "Let’s say our goal is to ensure that the total infected population (sum over all the infected states I, D, A, R, T) never rises above 1/3 of the total population, over the course of the next 100 days. If you could choose only a single intervention (affecting only one parameter), which intervention would let us meet our goal, with minimal change to the intervention parameter? Assume that the intervention will be implemented after one month (t = day 30), and will stay constant after that, over the remaining time period (i.e. the following 70 days). What are \n",
+    "equivalent interventions of the other two intervention types, that would have the same impact on total infections?\n",
     "\n",
+    "*Note that we started implementing intervention after t=15 days. \n",
     "\n",
     "### Problem Formulation\n",
     "* **Quantity of interest**:  Total infections out of $N=60,000,000$ population\n",
@@ -405,13 +407,12 @@
     "\n",
     "\n",
     "* **Control**: $\\mathbf{u}\\in \\mathcal{U} \\subseteq \\mathbb{R}^{n_u}$\n",
-    "    * Detection parameter for Infected ($\\epsilon$) assuming that $\\theta=2\\epsilon$\n",
-    "    * Two detection parameters $\\theta, \\epsilon$, constraining $\\theta \\ge 2\\epsilon$.\n",
+    "    * Transmission parameter $\\alpha$\n",
     "    \n",
     "\n",
     "\n",
     "* **Risk-based optimization under uncertainty problem formulation**\n",
-    "    * Objective Function - Minimize the magnitude of the parameter interventions.\n",
+    "    * Objective Function - Minimize the change in the magnitude of parameter interventions from the initial value $\\alpha\\text{init}=0.57$.\n",
     "    * Constraint - Risk of normalized total infections exceeding the prescribed threshold of $1/3$, $\\mathcal{R}( M(\\mathbf{u},\\mathbf{Z}))$, is below the acceptable risk threshold, $\\mathcal{R}_\\text{threshold}$.\n",
     "\n",
     "\\begin{equation} \n",
@@ -420,22 +421,14 @@
     "\\end{split} \n",
     "\\end{equation}\n",
     "\n",
-    "#### Formulation 1: Optimize $u=\\{\\alpha, \\beta, \\gamma, \\delta\\}$\n",
+    "#### Formulation 1: Optimize for $u=\\alpha$\n",
     "\\begin{equation} \n",
     "\\begin{split} \n",
-    "u^*= \\underset{\\epsilon\\in\\mathcal{U}}{\\arg\\min}\\ & u\\\\ \\text{s.t.}\\ & \\mathbb{P}( M(u, \\mathbf{Z}) \\ge 1/3) \\le 0.05\n",
+    "u^*= \\underset{\\epsilon\\in\\mathcal{U}}{\\arg\\min}\\ & \\lvert \\alpha_\\text{init}-u \\rvert \\\\ \\text{s.t.}\\ & \\mathbb{P}( M(u, \\mathbf{Z}) \\ge 1/3) \\le 0.05\n",
     "\\end{split} \n",
     "\\end{equation}\n",
     "\n",
-    "#### Formulation 2: Optimize $\\epsilon$ and $\\theta$ constraining $\\theta \\ge 2\\epsilon$\n",
-    "\\begin{equation} \n",
-    "\\begin{split} \n",
-    "\\epsilon^\\ast, \\theta^\\ast= \\underset{\\{\\epsilon,\\theta\\}\\in\\mathcal{U}}{\\arg\\min}\\ & \\theta + \\epsilon \\\\ \\text{s.t.}\\ & \\mathbb{P}( M(\\epsilon,\\theta, \\mathbf{Z}) \\ge 1/3) \\le 0.05, \\\\\n",
-    "& \\theta \\ge 2\\epsilon\n",
-    "\\end{split} \n",
-    "\\end{equation}\n",
-    "\n",
-    "Instead of probability of exceedance, we will use superquantiles as risk measures."
+    "Instead of  probability of exceedance, we will use superquantiles as risk measures."
    ]
   },
   {
@@ -720,7 +713,7 @@
   {
    "cell_type": "code",
    "execution_count": 23,
-   "id": "0010d710",
+   "id": "89e0a5a0",
    "metadata": {},
    "outputs": [],
    "source": [

6-month-milestone/evaluation/scenario_2/ta_3/ciemss/scenario2_q2b_SIDARTHEV_beta.ipynb

@@ -389,10 +389,12 @@
    "id": "c65bcf10-9d12-4efe-be10-cc313e123d14",
    "metadata": {},
    "source": [
-    "## 1.d Policies that prevent infections from exceeding 1/3 of the population\n",
+    "## 2..b.i Policies that prevent infections from exceeding 1/3 of the population\n",
     "\n",
-    "Now return to the situation in b.i (constant parameters that don’t change over time). Let’s say we want to increase testing, diagnostics, and contact tracing efforts (implemented by increasing the detection parameters  and ). Assume that $\\theta \\ge 2\\epsilon$ , because a symptomatic person is more likely to be tested. What minimum constant values do these parameters need to be over the course of a 100-day simulation, to ensure that the total infected population (sum over all the infected states I, D, A, R, T) never rises above 1/3 of the total population?\n",
+    "Let’s say our goal is to ensure that the total infected population (sum over all the infected states I, D, A, R, T) never rises above 1/3 of the total population, over the course of the next 100 days. If you could choose only a single intervention (affecting only one parameter), which intervention would let us meet our goal, with minimal change to the intervention parameter? Assume that the intervention will be implemented after one month (t = day 30), and will stay constant after that, over the remaining time period (i.e. the following 70 days). What are \n",
+    "equivalent interventions of the other two intervention types, that would have the same impact on total infections?\n",
     "\n",
+    "*Note that we started implementing intervention after t=15 days. \n",
     "\n",
     "### Problem Formulation\n",
     "* **Quantity of interest**:  Total infections out of $N=60,000,000$ population\n",
@@ -405,13 +407,12 @@
     "\n",
     "\n",
     "* **Control**: $\\mathbf{u}\\in \\mathcal{U} \\subseteq \\mathbb{R}^{n_u}$\n",
-    "    * Detection parameter for Infected ($\\epsilon$) assuming that $\\theta=2\\epsilon$\n",
-    "    * Two detection parameters $\\theta, \\epsilon$, constraining $\\theta \\ge 2\\epsilon$.\n",
+    "    * Transmission parameter $\\beta$\n",
     "    \n",
     "\n",
     "\n",
     "* **Risk-based optimization under uncertainty problem formulation**\n",
-    "    * Objective Function - Minimize the magnitude of the parameter interventions.\n",
+    "    * Objective Function - Minimize the change in the magnitude of parameter interventions from the initial value $\\beta_\\text{init}=0.011$.\n",
     "    * Constraint - Risk of normalized total infections exceeding the prescribed threshold of $1/3$, $\\mathcal{R}( M(\\mathbf{u},\\mathbf{Z}))$, is below the acceptable risk threshold, $\\mathcal{R}_\\text{threshold}$.\n",
     "\n",
     "\\begin{equation} \n",
@@ -420,22 +421,14 @@
     "\\end{split} \n",
     "\\end{equation}\n",
     "\n",
-    "#### Formulation 1: Optimize $u=\\{\\alpha, \\beta, \\gamma, \\delta\\}$\n",
+    "#### Formulation 1: Optimize for $u=\\beta$\n",
     "\\begin{equation} \n",
     "\\begin{split} \n",
-    "u^*= \\underset{\\epsilon\\in\\mathcal{U}}{\\arg\\min}\\ & u\\\\ \\text{s.t.}\\ & \\mathbb{P}( M(u, \\mathbf{Z}) \\ge 1/3) \\le 0.05\n",
+    "u^*= \\underset{\\epsilon\\in\\mathcal{U}}{\\arg\\min}\\ & \\lvert \\beta_\\text{init}-u \\rvert \\\\ \\text{s.t.}\\ & \\mathbb{P}( M(u, \\mathbf{Z}) \\ge 1/3) \\le 0.05\n",
     "\\end{split} \n",
     "\\end{equation}\n",
     "\n",
-    "#### Formulation 2: Optimize $\\epsilon$ and $\\theta$ constraining $\\theta \\ge 2\\epsilon$\n",
-    "\\begin{equation} \n",
-    "\\begin{split} \n",
-    "\\epsilon^\\ast, \\theta^\\ast= \\underset{\\{\\epsilon,\\theta\\}\\in\\mathcal{U}}{\\arg\\min}\\ & \\theta + \\epsilon \\\\ \\text{s.t.}\\ & \\mathbb{P}( M(\\epsilon,\\theta, \\mathbf{Z}) \\ge 1/3) \\le 0.05, \\\\\n",
-    "& \\theta \\ge 2\\epsilon\n",
-    "\\end{split} \n",
-    "\\end{equation}\n",
-    "\n",
-    "Instead of probability of exceedance, we will use superquantiles as risk measures."
+    "Instead of  probability of exceedance, we will use superquantiles as risk measures."
    ]
   },
   {
@@ -720,7 +713,7 @@
   {
    "cell_type": "code",
    "execution_count": 23,
-   "id": "e6291958",
+   "id": "2f2a90ee",
    "metadata": {},
    "outputs": [
     {

6-month-milestone/evaluation/scenario_2/ta_3/ciemss/scenario2_q2b_SIDARTHEV_delta.ipynb

@@ -389,10 +389,12 @@
    "id": "c65bcf10-9d12-4efe-be10-cc313e123d14",
    "metadata": {},
    "source": [
-    "## 1.d Policies that prevent infections from exceeding 1/3 of the population\n",
+    "## 2..b.i Policies that prevent infections from exceeding 1/3 of the population\n",
     "\n",
-    "Now return to the situation in b.i (constant parameters that don’t change over time). Let’s say we want to increase testing, diagnostics, and contact tracing efforts (implemented by increasing the detection parameters  and ). Assume that $\\theta \\ge 2\\epsilon$ , because a symptomatic person is more likely to be tested. What minimum constant values do these parameters need to be over the course of a 100-day simulation, to ensure that the total infected population (sum over all the infected states I, D, A, R, T) never rises above 1/3 of the total population?\n",
+    "Let’s say our goal is to ensure that the total infected population (sum over all the infected states I, D, A, R, T) never rises above 1/3 of the total population, over the course of the next 100 days. If you could choose only a single intervention (affecting only one parameter), which intervention would let us meet our goal, with minimal change to the intervention parameter? Assume that the intervention will be implemented after one month (t = day 30), and will stay constant after that, over the remaining time period (i.e. the following 70 days). What are \n",
+    "equivalent interventions of the other two intervention types, that would have the same impact on total infections?\n",
     "\n",
+    "*Note that we started implementing intervention after t=15 days. \n",
     "\n",
     "### Problem Formulation\n",
     "* **Quantity of interest**:  Total infections out of $N=60,000,000$ population\n",
@@ -405,13 +407,12 @@
     "\n",
     "\n",
     "* **Control**: $\\mathbf{u}\\in \\mathcal{U} \\subseteq \\mathbb{R}^{n_u}$\n",
-    "    * Detection parameter for Infected ($\\epsilon$) assuming that $\\theta=2\\epsilon$\n",
-    "    * Two detection parameters $\\theta, \\epsilon$, constraining $\\theta \\ge 2\\epsilon$.\n",
+    "    * Transmission parameter $\\delta$\n",
     "    \n",
     "\n",
     "\n",
     "* **Risk-based optimization under uncertainty problem formulation**\n",
-    "    * Objective Function - Minimize the magnitude of the parameter interventions.\n",
+    "    * Objective Function - Minimize the change in the magnitude of parameter interventions from the initial value $\\delta_\\text{init}=0.011$.\n",
     "    * Constraint - Risk of normalized total infections exceeding the prescribed threshold of $1/3$, $\\mathcal{R}( M(\\mathbf{u},\\mathbf{Z}))$, is below the acceptable risk threshold, $\\mathcal{R}_\\text{threshold}$.\n",
     "\n",
     "\\begin{equation} \n",
@@ -420,22 +421,14 @@
     "\\end{split} \n",
     "\\end{equation}\n",
     "\n",
-    "#### Formulation 1: Optimize $u=\\{\\alpha, \\beta, \\gamma, \\delta\\}$\n",
+    "#### Formulation 1: Optimize for $u=\\delta$\n",
     "\\begin{equation} \n",
     "\\begin{split} \n",
-    "u^*= \\underset{\\epsilon\\in\\mathcal{U}}{\\arg\\min}\\ & u\\\\ \\text{s.t.}\\ & \\mathbb{P}( M(u, \\mathbf{Z}) \\ge 1/3) \\le 0.05\n",
+    "u^*= \\underset{\\epsilon\\in\\mathcal{U}}{\\arg\\min}\\ & \\lvert \\delta_\\text{init}-u \\rvert \\\\ \\text{s.t.}\\ & \\mathbb{P}( M(u, \\mathbf{Z}) \\ge 1/3) \\le 0.05\n",
     "\\end{split} \n",
     "\\end{equation}\n",
     "\n",
-    "#### Formulation 2: Optimize $\\epsilon$ and $\\theta$ constraining $\\theta \\ge 2\\epsilon$\n",
-    "\\begin{equation} \n",
-    "\\begin{split} \n",
-    "\\epsilon^\\ast, \\theta^\\ast= \\underset{\\{\\epsilon,\\theta\\}\\in\\mathcal{U}}{\\arg\\min}\\ & \\theta + \\epsilon \\\\ \\text{s.t.}\\ & \\mathbb{P}( M(\\epsilon,\\theta, \\mathbf{Z}) \\ge 1/3) \\le 0.05, \\\\\n",
-    "& \\theta \\ge 2\\epsilon\n",
-    "\\end{split} \n",
-    "\\end{equation}\n",
-    "\n",
-    "Instead of probability of exceedance, we will use superquantiles as risk measures."
+    "Instead of  probability of exceedance, we will use superquantiles as risk measures."
    ]
   },
   {
@@ -720,7 +713,7 @@
   {
    "cell_type": "code",
    "execution_count": 25,
-   "id": "01c8c67c",
+   "id": "f71a01d6",
    "metadata": {},
    "outputs": [
     {

6-month-milestone/evaluation/scenario_2/ta_3/ciemss/scenario2_q2b_SIDARTHEV_phiflux.ipynb

@@ -389,10 +389,12 @@
    "id": "c65bcf10-9d12-4efe-be10-cc313e123d14",
    "metadata": {},
    "source": [
-    "## 1.d Policies that prevent infections from exceeding 1/3 of the population\n",
+    "## 2..b.i Policies that prevent infections from exceeding 1/3 of the population\n",
     "\n",
-    "Now return to the situation in b.i (constant parameters that don’t change over time). Let’s say we want to increase testing, diagnostics, and contact tracing efforts (implemented by increasing the detection parameters  and ). Assume that $\\theta \\ge 2\\epsilon$ , because a symptomatic person is more likely to be tested. What minimum constant values do these parameters need to be over the course of a 100-day simulation, to ensure that the total infected population (sum over all the infected states I, D, A, R, T) never rises above 1/3 of the total population?\n",
+    "Let’s say our goal is to ensure that the total infected population (sum over all the infected states I, D, A, R, T) never rises above 1/3 of the total population, over the course of the next 100 days. If you could choose only a single intervention (affecting only one parameter), which intervention would let us meet our goal, with minimal change to the intervention parameter? Assume that the intervention will be implemented after one month (t = day 30), and will stay constant after that, over the remaining time period (i.e. the following 70 days). What are \n",
+    "equivalent interventions of the other two intervention types, that would have the same impact on total infections?\n",
     "\n",
+    "*Note that we started implementing intervention after t=15 days. \n",
     "\n",
     "### Problem Formulation\n",
     "* **Quantity of interest**:  Total infections out of $N=60,000,000$ population\n",
@@ -405,13 +407,12 @@
     "\n",
     "\n",
     "* **Control**: $\\mathbf{u}\\in \\mathcal{U} \\subseteq \\mathbb{R}^{n_u}$\n",
-    "    * Detection parameter for Infected ($\\epsilon$) assuming that $\\theta=2\\epsilon$\n",
-    "    * Two detection parameters $\\theta, \\epsilon$, constraining $\\theta \\ge 2\\epsilon$.\n",
+    "    * Transmission parameter $\\delta$\n",
     "    \n",
     "\n",
     "\n",
     "* **Risk-based optimization under uncertainty problem formulation**\n",
-    "    * Objective Function - Minimize the magnitude of the parameter interventions.\n",
+    "    * Objective Function - Minimize the magnitude of vaccination rate flux $SV_\\text{flux}$.\n",
     "    * Constraint - Risk of normalized total infections exceeding the prescribed threshold of $1/3$, $\\mathcal{R}( M(\\mathbf{u},\\mathbf{Z}))$, is below the acceptable risk threshold, $\\mathcal{R}_\\text{threshold}$.\n",
     "\n",
     "\\begin{equation} \n",
@@ -420,22 +421,14 @@
     "\\end{split} \n",
     "\\end{equation}\n",
     "\n",
-    "#### Formulation 1: Optimize $u=\\{\\alpha, \\beta, \\gamma, \\delta\\}$\n",
+    "#### Formulation 1: Optimize for $u=SV_\\text{flux}$\n",
     "\\begin{equation} \n",
     "\\begin{split} \n",
-    "u^*= \\underset{\\epsilon\\in\\mathcal{U}}{\\arg\\min}\\ & u\\\\ \\text{s.t.}\\ & \\mathbb{P}( M(u, \\mathbf{Z}) \\ge 1/3) \\le 0.05\n",
+    "u^*= \\underset{\\epsilon\\in\\mathcal{U}}{\\arg\\min}\\ & u \\\\ \\text{s.t.}\\ & \\mathbb{P}( M(u, \\mathbf{Z}) \\ge 1/3) \\le 0.05\n",
     "\\end{split} \n",
     "\\end{equation}\n",
     "\n",
-    "#### Formulation 2: Optimize $\\epsilon$ and $\\theta$ constraining $\\theta \\ge 2\\epsilon$\n",
-    "\\begin{equation} \n",
-    "\\begin{split} \n",
-    "\\epsilon^\\ast, \\theta^\\ast= \\underset{\\{\\epsilon,\\theta\\}\\in\\mathcal{U}}{\\arg\\min}\\ & \\theta + \\epsilon \\\\ \\text{s.t.}\\ & \\mathbb{P}( M(\\epsilon,\\theta, \\mathbf{Z}) \\ge 1/3) \\le 0.05, \\\\\n",
-    "& \\theta \\ge 2\\epsilon\n",
-    "\\end{split} \n",
-    "\\end{equation}\n",
-    "\n",
-    "Apart from probability of exceedance, we will explore quantiles and superquantiles as risk measures."
+    "Instead of  probability of exceedance, we will use superquantiles as risk measures."
    ]
   },
   {
@@ -729,7 +722,7 @@
   {
    "cell_type": "code",
    "execution_count": 25,
-   "id": "f4b79578",
+   "id": "7875522b",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -748,7 +741,7 @@
   {
    "cell_type": "code",
    "execution_count": 20,
-   "id": "1c28411a",
+   "id": "bc8b9444",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -854,7 +847,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "f999249d",
+   "id": "2241aebb",
    "metadata": {},
    "outputs": [],
    "source": []