SEI2HR-Econ model with quarantine and supplies scenarios

Introduction

The epidemiology compartmental model, [Wk1], presented in this notebook – SEI2HR-Econ – deals with all three rectangles in this diagram:

ImageResize[Import["https://github.com/antononcube/SystemModeling/raw/master/Projects/Coronavirus-propagation-dynamics/Diagrams/Coronavirus-propagation-simple-dynamics.jpeg"], 900]
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“SEI2HR” stands for “Susceptible, Exposed, Infected two, Hospitalized, Recovered” (populations.) “Econ” stands for “Economic”.

In this notebook we also deal with both quarantine scenarios and medical supplies scenarios. In the notebook [AA4] we deal with quarantine scenarios over a simpler model, SEI2HR.

Remark: We consider the contagious disease propagation models as instances of the more general System Dynamics (SD) models. We use SD terminology in this notebook.

The models

SEI2R

The model SEI2R is introduced and explained in the notebook [AA2]. SEI2R differs from the classical SEIR model, [Wk1, HH1], with the following elements:

  1. Two separate infected populations: one is “severely symptomatic”, the other is “normally symptomatic”
  2. The monetary equivalent of lost productivity due to infected or died people is tracked

SEI2HR

For the formulation of SEI2HR we use a system of Differential Algebraic Equations (DAE’s). The package [AAp1] allows the use of a formulation that has just Ordinary Differential Equations (ODE’s).

Here are the unique features of SEI2HR:

  • People stocks
    • There are two types of infected populations: normally symptomatic and severely symptomatic.
    • There is a hospitalized population.
    • There is a deceased from infection population.
  • Hospital beds
    • Hospital beds are a limited resource that determines the number of hospitalized people.
    • Only severely symptomatic people are hospitalized according to the available hospital beds.
    • The hospital beds stock is not assumed constant, it has its own change rate.
  • Money stocks
    • The money from lost productivity is tracked.
    • The money for hospital services is tracked.

SEI2HR-Econ

SEI2HR-Econ adds the following features to SEI2HR:

  • Medical supplies
    • Medical supplies production is part of the model.
    • Medical supplies delivery is part of the model..
    • Medical supplies accumulation at hospitals is taken into account.
    • Medical supplies demand tracking.
  • Hospitalization
    • Severely symptomatic people are hospitalized according to two limited resources: hospital beds and medical supplies.
  • Money stocks
    • Money for medical supplies production is tracked.

SEI2HR-Econ’s place a development plan

This graph shows the “big picture” of the model development plan undertaken in [AAr1] and SEI2HR (discussed in this notebook) is in that graph:

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Notebook structure

The rest of notebook has the following sequence of sections:

  • Package load section
  • SEI2HR-Econ structure in comparison of SEI2HR
  • Explanations of the equations of SEI2HR-Econ
  • Quarantine scenario modeling preparation
  • Medical supplies production and delivery scenario modeling preparation
  • Parameters and initial conditions setup
    • Populations, hospital beds, quarantine scenarios, medical supplies scenarios
  • Simulation solutions
  • Interactive interface
  • Sensitivity analysis

Load packages

The epidemiological models framework used in this notebook is implemented with the packages [AAp1-AAp4, AA3]; many of the plot functions are from the package [AAp5].

Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/Projects/Coronavirus-propagation-dynamics/WL/EpidemiologyModels.m"];
Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/Projects/Coronavirus-propagation-dynamics/WL/EpidemiologyModelModifications.m"];
Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/Projects/Coronavirus-propagation-dynamics/WL/EpidemiologyModelingVisualizationFunctions.m"];
Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/Projects/Coronavirus-propagation-dynamics/WL/EpidemiologyModelingSimulationFunctions.m"];
Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/WL/SystemDynamicsInteractiveInterfacesFunctions.m"];

SEI2HR-Econ extends SEI2HR

The model SEI2HR-Econ is an extension of the model SEI2HR, [AA4].

Here is SEI2HR:

reprTP = "AlgebraicEquation";
lsModelOpts = {"Tooltips" -> True, 
   TooltipStyle -> {Background -> Yellow, CellFrameColor -> Gray, 
     FontSize -> 20}};
modelReference = 
  SEI2HRModel[t, "InitialConditions" -> True, "RateRules" -> True, 
   "TotalPopulationRepresentation" -> reprTP];
ModelGridTableForm[modelReference, lsModelOpts]
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Here is SEI2HR-Econ:

modelSEI2HREcon = 
  SEI2HREconModel[t, "InitialConditions" -> True, "RateRules" -> True,
    "TotalPopulationRepresentation" -> reprTP];
ModelGridTableForm[modelSEI2HREcon, lsModelOpts]
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Here are the “differences” between the two models:

ModelGridTableForm@
 Merge[{modelSEI2HREcon, modelReference}, 
  If[AssociationQ[#[[1]]], KeyComplement[#], Complement @@ #] &]
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Equations explanations

In this section we provide rationale for the equations of SEI2HR-Econ.

The equations for Susceptible, Exposed, Infected, Recovered populations of SEI2R are “standard” and explanations about them are found in [WK1, HH1]. For SEI2HR those equations change because of the stocks Hospitalized Population and Hospital Beds. For SEI2HR-Econ the SEI2HR equations change because of the stocks Medical Supplies, Medical Supplies Demand, and Hospital Medical Supplies.

The equations time unit is one day. The time horizon is one year. Since we target COVID-19, [Wk2, AA1], we do not consider births.

Remark: For convenient reading the equations in this section have tooltips for the involved stocks and rates.

Verbalization description of the model

We start with one infected (normally symptomatic) person, the rest of the people are susceptible. The infected people meet other people directly or get in contact with them indirectly. (Say, susceptible people touch things touched by infected.) For each susceptible person there is a probability to get the decease. The decease has an incubation period: before becoming infected the susceptible are (merely) exposed. The infected recover after a certain average infection period or die. A certain fraction of the infected become severely symptomatic. The severely symptomatic infected are hospitalized if there are enough hospital beds and enough medical supplies. The hospitalized severely infected have different death rate than the non-hospitalized ones. The number of hospital beds might change: hospitals are extended, new hospitals are build, or there are not enough medical personnel or supplies.

The different types of populations (infected, hospitalized, recovered, etc.) have their own consumption rates of medical supplies. The medical supplies are produced with a certain rate (units per day) and delivered after a certain delay period. The hospitals have their own storage for medical supplies. Medical supplies are delivered to the hospitals only, non-hospitalized people go to the medical supplies producer to buy supplies. The hospitals have precedence for the medical supplies: if the medical supplies are not enough for everyone, the hospital needs are covered first (as much as possible.)

The medical supplies producer has a certain storage capacity (for supplies.) The medical supplies delivery vehicles have a certain – generally speaking, smaller – capacity. The hospitals have a certain capacity to store medical supplies. It is assumed that both producer and hospitals have initial stocks of medical supplies. (Following a certain normal, general preparedness protocol.)

The combined demand from all populations for medical supplies is tracked (accumulated.) The deaths from infection are tracked (accumulated.) Money for medical supplies production, money for hospital services, and money from lost productivity are tracked (accumulated.)

The equations below give mathematical interpretation of the model description above.

Code for the equations

Each equation in this section are derived with code like this:

ModelGridTableForm[modelSEI2HREcon, lsModelOpts]["Equations"][[1, 
 EquationPosition[modelSEI2HREcon, RP] + 1, 2]]

and then the output cell is edited to be “DisplayFormula” and have CellLabel value corresponding to the stock of interest.

The infected and hospitalized populations

SEI2HR has two types of infected populations: a normally symptomatic one and a severely symptomatic one. A major assumption for SEI2HR is that only the severely symptomatic people are hospitalized. (That assumption is also reflected in the diagram in the introduction.)

Each of those three populations have their own contact rates and mortality rates.

Here are the contact rates from the SEI2HR-Econ dictionary

ColumnForm@
 Cases[Normal@modelSEI2HREcon["Rates"], 
  HoldPattern[\[Beta][_] -> _], \[Infinity]]
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Here are the mortality rates from the SEI2HR-Econ dictionary

ColumnForm@
 Cases[Normal@modelSEI2HREcon["Rates"], 
  HoldPattern[\[Mu][_] -> _], \[Infinity]]
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Remark: Below with “Infected Population” we mean both stocks Infected Normally Symptomatic Population (INSP) and Infected Severely Symptomatic Population (ISSP).

Total Population

In this notebook we consider a DAE’s formulation of SEI2HR-Econ. The stock Total Population has the following (obvious) algebraic equation:

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Note that with Max we specified that the total population cannot be less than 0.

Remark: As mentioned in the introduction, the package [AAp1] allows for the use of non-algebraic formulation, without an equation for TP.

Susceptible Population

The stock Susceptible Population (SP) is decreased by (1) infections derived from stocks Infected Populations and Hospitalized Population (HP), and (2) morality cases derived with the typical mortality rate.

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Because we hospitalize the severely infected people only instead of the term

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we have the terms

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The first term is for the infections derived from the hospitalized population. The second term for the infections derived from people who are infected severely symptomatic and not hospitalized.

Births term

Note that we do not consider in this notebook births, but the births term can be included in SP’s equation:

Block[{m = SEI2HREconModel[t, "BirthsTerm" -> True]},
 ModelGridTableForm[m]["Equations"][[1, EquationPosition[m, SP] + 1, 
  2]]
 ]
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The births rate is the same as the death rate, but it can be programmatically changed. (See [AAp2].)

Exposed Population

The stock Exposed Population (EP) is increased by (1) infections derived from the stocks Infected Populations and Hospitalized Population, and (2) mortality cases derived with the typical mortality rate. EP is decreased by (1) the people who after a certain average incubation period (aincp) become ill, and (2) mortality cases derived with the typical mortality rate.

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Infected Normally Symptomatic Population

INSP is increased by a fraction of the people who have been exposed. That fraction is derived with the parameter severely symptomatic population fraction (sspf). INSP is decreased by (1) the people who recover after a certain average infection period (aip), and (2) the normally symptomatic people who die from the disease.

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Infected Severely Symptomatic Population

ISSP is increased by a fraction of the people who have been exposed. That fraction is corresponds to the parameter severely symptomatic population fraction (sspf). ISSP is decreased by (1) the people who recover after a certain average infection period (aip), (2) the hospitalized severely symptomatic people who die from the disease, and (3) the non-hospitalized severely symptomatic people who die from the disease.

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Note that we do not assume that severely symptomatic people recover faster if they are hospitalized, only that they have a different death rate.

Hospitalized Population

The amount of people that can be hospitalized is determined by the available Hospital Beds (HB) – the stock Hospitalized Population (HP) is subject to a resource limitation by the stock HB.

The equation of the stock HP can be easily understood from the following dynamics description points:

  • If the number of hospitalized people is less that the number of hospital beds we hospitalize the new ISSP people.
  • The Available Hospital Beds (AHB) are determined by the minimum of (i) the non-occupied hospital beds, and (ii) the hospital medical supplies divided by the ISSP consumption rate.
  • If the new ISSP people are more than AHB the hospital takes as many as AHB.
  • Hospitalized people have the same average infection period (aip).
  • Hospitalized (severely symptomatic) people have their own mortality rate.

Here is the HP equation:

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Note that although we know that in a given day some hospital beds are going to be freed they are not considered in the hospitalization plans for that day. Similarly, we know that new medical supplies are coming but we do not include them into AHB.

Recovered Population

The stock Recovered Population (RP) is increased by the recovered infected people and decreased by mortality cases derived with the typical mortality rate.

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Deceased Infected Population

The stock Deceased Infected Population (DIP) accumulates the deaths of the people who are infected. Note that we utilize the different death rates for HP and ISSP.

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Hospital Beds

The stock Hospital Beds (HB) can change with a rate that reflects the number of hospital beds change rate (nhbcr) per day. Generally speaking, using nhbcr we can capture scenarios, like, extending hospitals, building new hospitals, recruitment of new medical personnel, loss of medical personnel (due to infections.)

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Hospital Medical Supplies

The Hospital Medical Supplies (HMS) are decreased according to the medical supplies consumption rate (mscr) of HP and increased by a Medical Supplies (MS) delivery term (to be described next.)

The MS delivery term is build with the following assumptions / postulates:

  • Every day the hospital attempts to order MS that correspond to HB multiplied by mscr.
  • The hospital has limited capacity of MS storage, \kappa [\text{HMS}].
  • The MS producer has limited capacity for delivery, \kappa [\text{MDS}].
  • The hospital demand for MS has precedence over the demands for the non-hospitalized populations.
  • Hence, if the MS producer has less stock of MS than the demand of the hospital then MS producer’s whole amount of MS goes to the hospital.
  • The supplies are delivered with some delay period: the medical supplies delivery period (msdp).

Here is the MS delivery term:

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Here is the corresponding HMS equation:

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Medical Supplies

The equation of the Medical Supplies (MS) stock is based on the following assumptions / postulates:

  • The non-hospitalized people go to the MS producer to buy supplies. (I.e. MS delivery is to the hospital only.)
  • The MS producer vehicles have certain capacity, \kappa [\text{MSD}].
  • The MS producer has a certain storage capacity (for MS stock.)
  • Each of the populations INSP, ISSP, and HP has its own specific medical supplies consumption rate (mscr). EP, RP, and TP have the same mscr.
  • The hospital has precedence in its MS order. I.e. the demand from the hospital is satisfied first, and then the demand of the rest of the populations.

Here is the MS delivery term described in the previous section:

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Here is the MS formula with the MS delivery term replaced with “Dlvr”:

ModelGridTableForm[modelSEI2HREcon, "Tooltips" -> False][
   "Equations"][[1, EquationPosition[modelSEI2HREcon, MS] + 1, 2]] /. 
 dlvr -> Dlvr
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We can see from that equation that MS is increased by medical supplies production rate (mspr) with measuring dimension number of units per day. The production is restricted by the storage capacity, \kappa [\text{MS}]:

(*Min[mspr[HB], -MS[t] + \[Kappa][MS]]*)

MS is decreased by the MS delivery term and the demand from the non-hospitalized populations. Because the hospital has precedence, we use this term form in the equation:

(*Min[-Dlvr + MS[t], "non-hospital demand"]*)

Here is the full MS equation:

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Medical Supplies Demand

The stock Medical Supplies Demand (MSD) simply accumulates the MS demand derived from population stocks and their corresponding mscr:

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Money for Hospital Services

The stock Money for Hospital Services (MHS) simply tracks expenses for hospitalized people. The parameter hospital services cost rate (hscr) with unit money per bed per day simply multiplies HP.

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Money from Lost Productivity

The stock Money from Lost Productivity (MLP) simply tracks the work non-availability of the infected and died from infection people. The parameter lost productivity cost rate (lpcr) with unit money per person per day multiplies the total count of the infected and dead from infection.

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Quarantine scenarios

In order to model quarantine scenarios we use piecewise constant functions for the contact rates \beta [\text{ISSP}] and \beta [\text{INSP}].

Remark: Other functions can be used, like, functions derived through some statistical fitting.

Here is an example plot :

Block[{func = \[Beta]*
    Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1]},
 Legended[
  Block[{\[Beta] = 0.56, qsd = 60, ql = 8*7, qcrf = 0.25},
   ListLinePlot[Table[func, {t, 0, 365}], PlotStyle -> "Detailed"]
   ], func]]
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To model quarantine with a piecewise constant function we use the following parameters:

  • \text{qsd} for quarantine’s start
  • \text{ql} for quarantines duration
  • \text{qcrf} for the effect on the quarantine on the contact rate

Medical supplies scenarios

We consider three main scenarios for the medical supplies:

  1. Constant production rate and consistent delivery (constant delivery period)
  2. Constant production rate and disrupted delivery
  3. Disrupted production and disrupted delivery

The disruptions are simulated with simple pulse functions – we want to see how the system being modeled reacts to singular, elementary disruption.

Here is an example plot of a disruption of delivery period plot :

Block[{func = 
   dbp*Piecewise[{{1, t < dds}, {dsf, dds <= t <= dds + ddl}}, 1]},
 Legended[
  Block[{dbp = 1, dds = 70, ddl = 7, dsf = 1.8},
   ListLinePlot[Table[func, {t, 0, 365}], PlotStyle -> "Detailed"]
   ], func]]
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To model disruption of delivery with a piecewise constant function we use the following parameters:

  • \text{dbp} for the delivery base period
  • \text{dds} for delivery disruption start
  • \text{ddl} for delivery disruption duration
  • \text{dsf} for how much slower or faster the delivery period becomes

Parameters and actual simulation equations code

Here are the parameters we want to experiment with (or do calibration with):

lsFocusParams = {aincp, aip, sspf[SP], \[Beta][HP], qsd, ql, qcrf, 
   nhbcr[ISSP, INSP], nhbr[TP], mspr[HB]};

Here we set custom rates and initial conditions:

population = 10^6;
modelSEI2HREcon =
  SetRateRules[
   modelSEI2HREcon,
   <|
    TP[0] -> population,
    \[Beta][ISSP] -> 
     0.5*Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1],
    \[Beta][INSP] -> 
     0.5*Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1],
    qsd -> 60,
    ql -> 8*7,
    qcrf -> 0.25,
    \[Beta][HP] -> 0.01,
    \[Mu][ISSP] -> 0.035/aip,
    \[Mu][INSP] -> 0.01/aip,
    nhbr[TP] -> 3/1000,
    lpcr[ISSP, INSP] -> 1,
    hscr[ISSP, INSP] -> 1,
    msdp[HB] -> 
     dbp*Piecewise[{{1, t < dds}, {dsf, dds <= t <= dds + ddl}}, 1],
    dbp -> 1,
    dds -> 70,
    ddl -> 7,
    dsf -> 2
    |>
   ];

Remark: Note the piecewise functions for \beta [\text{ISSP}], \beta [\text{INSP}], and \text{msdp}[\text{HB}].

Here is the system of ODE’s we use to do parametrized simulations:

lsActualEquations = 
  ModelNDSolveEquations[modelSEI2HREcon, 
   KeyDrop[modelSEI2HREcon["RateRules"], lsFocusParams]];
ResourceFunction["GridTableForm"][List /@ lsActualEquations]
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lsActualEquations = 
  ModelNDSolveEquations[modelSEI2HREcon, modelSEI2HREcon["RateRules"]];
ResourceFunction["GridTableForm"][List /@ lsActualEquations]
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Simulation

Instead of using ParametricNDSolve as in [AA4] in this notebook we use ModelNDSolve and SetRateRules from the package [AAp4].

Different quarantine starts

Here we compute simulation solutions for a set of quarantine starts:

AbsoluteTiming[
 aVarSolutions =
   Association@
    Map[
     Function[{qsdVar},
      qsdVar -> 
       Association[
        ModelNDSolve[
          SetRateRules[
           modelSEI2HREcon, <|ql -> 56, qsd -> qsdVar|>], {t, 365}][[
         1]]]
      ],
     Range[40, 120, 20]
     ];
 ]

(*{0.366168, Null}*)

Here we plot the results for ISSP only:

SeedRandom[2532]
aVals = #[ISSP][Range[0, 365]] & /@ aVarSolutions;
ListLinePlot[
 KeyValueMap[
  Callout[Tooltip[#2, #1], #1, {If[#1 <= 70, 
      RandomInteger[{120, 200}], RandomInteger[{80, 110}]], Above}] &,
   aVals], PlotLegends -> 
  SwatchLegend[Keys[aVals], LegendLabel -> "Quarantine start"], 
 PlotRange -> All, ImageSize -> Large, 
 PlotLabel -> ISSP[t] /. modelSEI2HREcon["Stocks"]]
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Remark: We use the code in this section to do the computations in the section “Sensitivity Analysis”.

Interactive interface

Using the interface in this section we can interactively see the effects of changing parameters. (This interface is programmed without using parametrized NDSolve solutions in order to be have code that corresponds to the interface implementations in [AAr2].)

opts = {PlotRange -> All, PlotLegends -> None, 
   PlotTheme -> "Detailed", PerformanceGoal -> "Speed", 
   ImageSize -> 400};
lsPopulationKeys = {TP, SP, EP, ISSP, INSP, HP, RP, DIP, HB};
lsSuppliesKeys = {MS, MSD, HMS};
lsMoneyKeys = {MHS, MLP, MMSP};
Manipulate[
 DynamicModule[{modelLocal = modelSEI2HREcon, 
   aStocks = modelSEI2HREcon["Stocks"], aSolLocal = aParSol, 
   lsPopulationPlots, lsMoneyPlots, lsSuppliesPlots},
  
  modelLocal = 
   SetRateRules[
    modelLocal, <|aincp -> aincpM, aip -> aipM, 
     sspf[SP] -> sspfM, \[Beta][HP] -> crhpM, qsd -> qsdM, ql -> qlM, 
     qcrf -> qcrfM, nhbr[TP] -> nhbrM/1000, 
     nhbcr[ISSP, ISNP] -> nhbcrM, mspr[HB] -> msprM, 
     msdp[HB] -> msdpM|>];
  aSolLocal = Association[ModelNDSolve[modelLocal, {t, ndays}][[1]]];
  
  lsPopulationPlots =
   Quiet@ParametricSolutionsPlots[
     aStocks,
     KeyTake[aSolLocal, 
      Intersection[lsPopulationKeys, displayPopulationStocks]],
     None, ndays,
     "LogPlot" -> popLogPlotQ, "Together" -> popTogetherQ, 
     "Derivatives" -> popDerivativesQ, 
     "DerivativePrefix" -> "\[CapitalDelta]", opts, 
     Epilog -> {Gray, Dashed, 
       Line[{{qsdM, 0}, {qsdM, 1.5*population}}], 
       Line[{{qsdM + qlM, 0}, {qsdM + qlM, 1.5*population}}]}];
  
  lsSuppliesPlots =
   If[Length[
      KeyDrop[aSolLocal, Join[lsPopulationKeys, lsMoneyKeys]]] == 
     0, {},
    (*ELSE*)
    Quiet@ParametricSolutionsPlots[
      aStocks,
      KeyTake[KeyDrop[aSolLocal, Join[lsPopulationKeys, lsMoneyKeys]],
        displaySupplyStocks],
      None, ndays,
      "LogPlot" -> supplLogPlotQ, "Together" -> supplTogetherQ, 
      "Derivatives" -> supplDerivativesQ, 
      "DerivativePrefix" -> "\[CapitalDelta]", opts]
    ];
  
  lsMoneyPlots =
   Quiet@ParametricSolutionsPlots[
     aStocks,
     KeyTake[aSolLocal, Intersection[lsMoneyKeys, displayMoneyStocks]],
     None, ndays,
     "LogPlot" -> moneyLogPlotQ, "Together" -> moneyTogetherQ, 
     "Derivatives" -> moneyDerivativesQ, 
     "DerivativePrefix" -> "\[CapitalDelta]", opts];
  
  Multicolumn[Join[lsPopulationPlots, lsSuppliesPlots, lsMoneyPlots], 
   nPlotColumns, Dividers -> All, FrameStyle -> GrayLevel[0.8]],
  SaveDefinitions -> True
  ],
 {{ndays, 365, "Number of days"}, 1, 365, 1, Appearance -> {"Open"}},
 Delimiter,
 {{aincpM, 6., "Average incubation period (days)"}, 1, 60., 1, 
  Appearance -> {"Open"}},
 {{aipM, 21., "Average infectious period (days)"}, 1, 60., 1, 
  Appearance -> {"Open"}},
 {{sspfM, 0.2, "Severely symptomatic population fraction"}, 0, 1, 
  0.025, Appearance -> {"Open"}},
 {{crhpM, 0.1, "Contact rate of the hospitalized population"}, 0, 30, 
  0.1, Appearance -> {"Open"}},
 Delimiter,
 {{qsdM, 55, "Quarantine start days"}, 0, 365, 1, 
  Appearance -> {"Open"}},
 {{qlM, 8*7, "Quarantine length (in days)"}, 0, 120, 1, 
  Appearance -> {"Open"}},
 {{qcrfM, 0.25, "Quarantine contact rate fraction"}, 0, 1, 0.01, 
  Appearance -> {"Open"}},
 Delimiter,
 {{nhbrM, 2.9, "Number of hospital beds rate (per 1000 people)"}, 0, 
  100, 0.1, Appearance -> {"Open"}},
 {{nhbcrM, 0, "Number of hospital beds change rate"}, -0.5, 0.5, 
  0.001, Appearance -> {"Open"}},
 {{msprM, 200, "Medical supplies production rate"}, 0, 50000, 10, 
  Appearance -> {"Open"}},
 {{msdpM, 1.2, "Medical supplies delivery period"}, 0, 10, 0.1, 
  Appearance -> {"Open"}},
 Delimiter,
 {{displayPopulationStocks, lsPopulationKeys, 
   "Population stocks to display:"}, lsPopulationKeys, 
  ControlType -> TogglerBar},
 {{popTogetherQ, True, "Plot populations together"}, {False, True}},
 {{popDerivativesQ, False, "Plot populations derivatives"}, {False, 
   True}},
 {{popLogPlotQ, False, "LogPlot populations"}, {False, True}},
 Delimiter,
 {{displaySupplyStocks, lsSuppliesKeys, 
   "Supplies stocks to display:"}, lsSuppliesKeys, 
  ControlType -> TogglerBar},
 {{supplTogetherQ, True, "Plot supplies functions together"}, {False, 
   True}},
 {{supplDerivativesQ, False, 
   "Plot supplies functions derivatives"}, {False, True}},
 {{supplLogPlotQ, True, "LogPlot supplies functions"}, {False, 
   True}},
 Delimiter,
 {{displayMoneyStocks, lsMoneyKeys, "Money stocks to display:"}, 
  lsMoneyKeys, ControlType -> TogglerBar},
 {{moneyTogetherQ, True, "Plot money functions together"}, {False, 
   True}},
 {{moneyDerivativesQ, False, 
   "Plot money functions derivatives"}, {False, True}},
 {{moneyLogPlotQ, True, "LogPlot money functions"}, {False, True}},
 {{nPlotColumns, 1, "Number of plot columns"}, Range[5]},
 ControlPlacement -> Left, ContinuousAction -> False]
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Sensitivity analysis

When making and using this kind of dynamics models it is important to see how the solutions react to changes of different parameters. For example, we should try to find answers to questions like “What ranges of which parameters bring dramatic changes into important stocks?”

Sensitivity Analysis (SA) is used to determine how sensitive is a SD model to changes of the parameters and to changes of model’s equations, [BC1]. More specifically, parameter sensitivity, which we apply below, allows us to see the changes of stocks dynamic behaviour for different sequences (and combinations) of parameter values.

Remark: This section to mirrors to a point the section with same name in [AA4], except in this notebook we are more interested in medical supplies related SA because quarantine related SA is done in [AA4].

Remark: SA shown below should be done for other stocks and rates. In order to keep this exposition short we focus on ISSP, DIP, and HP. Also, it is interesting to think in terms of “3D parameter sensitivity plots.” We also do such plots.

Evaluations by Area under the curve

For certain stocks we might be not just interested in their evolution in time but also in their cumulative values. I.e. we are interested in the so called Area Under the Curve (AUC) metric for those stocks.

There are three ways to calculate AUC for stocks of interest:

  1. Add aggregation equations in the system of ODE’s. (Similar to the stock DIP in SEI2HR.)
    • For example, in order to compute AUC for ISSP we can add to SEI2HR the equation:
(*aucISSP'[t] = ISSP[t]*)
- More details for such equation addition are given in [AA2].
  1. Apply NIntegrate over stocks solution functions.
  2. Apply Trapezoidal rule to stock solution function values over a certain time grid.

Below we use 1 and 3.

Variation of medical supplies delivery period

Here we calculate the solutions for a certain combination of capacities and rates:

AbsoluteTiming[
 aVarSolutions =
   Association@
    Map[
     Function[{msdpVar},
      model2 = SEI2HREconModel[t];
      model2 = 
       SetRateRules[
        model2, <|\[Kappa][MS] -> 10000, \[Kappa][HMS] -> 100, 
         mspr[HB] -> 100, msdp[HB] -> msdpVar|>];
      msdpVar -> Association[ModelNDSolve[model2, {t, 365}][[1]]]
      ],
     Union[Join[Range[0.2, 1, 0.2], Range[1, 3, 0.5]]]
     ];
 ]

(*{0.231634, Null}*)

As expected more frequent delivery results in fuller utilization of the non-occupied hospital beds:

SeedRandom[23532]
focusStock = HP;
aVals = #[focusStock][Range[0, 365]] & /@ aVarSolutions;
ListLinePlot[
 KeyValueMap[
  Callout[Tooltip[#2, #1], #1, {If[#1 < 1, RandomInteger[{120, 150}], 
      RandomInteger[{160, 260}]], Above}] &, aVals], 
 PlotLegends -> 
  SwatchLegend[Keys[aVals], 
   LegendLabel -> "Medical supplies\ndelivery period"], 
 PlotRange -> All, ImageSize -> Large, 
 PlotLabel -> focusStock[t] /. modelSEI2HREcon["Stocks"]]
0xacdbpxbv6yo

Here are the corresponding AUC values:

aAUCs = TrapezoidalRule[Transpose[{Range[0, 365], #}]] & /@ aVals;
ResourceFunction["GridTableForm"][aAUCs]
1a3d3me9wh6bh
BarChart[aAUCs, ChartLabels -> Keys[aAUCs], ColorFunction -> "Pastel",
  PlotLabel -> 
  Row[{focusStock[t] /. modelSEI2HREcon["Stocks"], Spacer[5], "AUC"}]]
1ncranm3ufw0h

Variation of medical supplies production rate

In order to demonstrate the effect of medical supplies production rate (mspr) it is beneficial to eliminate the hospital beds availability restriction – we assume that we have enough hospital beds for all infected severely symptomatic people.

Here we calculate the solutions for a certain combination of capacities and rates:

AbsoluteTiming[
 aVarSolutions =
   Association@
    Map[
     Function[{msprVar},
      model2 = SEI2HREconModel[t];
      model2 = 
       SetRateRules[
        model2, <|\[Kappa][MS] -> 100000, \[Kappa][HMS] -> 
          10000, \[Kappa][MSD] -> 1000, mspr[HB] -> msprVar, 
         msdp[HB] -> 1.5, mscr[ISSP] -> 0.2, mscr[TP] -> 0.001, 
         mscr[ISSP] -> 1, nhbr[TP] -> 100/1000|>];
      msprVar -> Association[ModelNDSolve[model2, {t, 365}][[1]]]
      ],
     {20, 60, 100, 200, 300, 1000, 10000}
     ];
 ]

(*{0.156794, Null}*)

Hospitalized Population

As expected we can see that with smaller production rates we get less hospitalized people:

SeedRandom[1232]
focusStock = HP;
aVals = #[focusStock][Range[0, 365]] & /@ aVarSolutions;
ListLinePlot[
 KeyValueMap[
  Callout[Tooltip[#2, #1], #1, {RandomInteger[{180, 240}], Above}] &, 
  aVals], PlotLegends -> 
  SwatchLegend[Keys[aVals], 
   LegendLabel -> "Medical supplies\nproduction rate"], 
 PlotRange -> All, ImageSize -> Large, 
 PlotLabel -> focusStock[t] /. modelSEI2HREcon["Stocks"]]
0vxim0nbznmmi

Here are the corresponding AUC values:

aAUCs = TrapezoidalRule[Transpose[{Range[0, 365], #}]] & /@ aVals;
ResourceFunction["GridTableForm"][aAUCs]
07nfkyx6s9mjo
BarChart[aAUCs, ChartLabels -> Keys[aAUCs], ColorFunction -> "Pastel",
  PlotLabel -> 
  Row[{focusStock[t] /. modelSEI2HREcon["Stocks"], Spacer[5], "AUC"}]]
08xdwgkj6ljeh

Medical Supplies

Here we plot the availability of MS at MS producer’s storage:

SeedRandom[821]
focusStock = MS;
aVals = #[MS][Range[0, 365]] & /@ aVarSolutions;
ListLinePlot[
 KeyValueMap[
  Callout[Tooltip[#2, #1], #1, {RandomInteger[{100, 160}], Above}] &, 
  aVals], PlotLegends -> 
  SwatchLegend[Keys[aVals], 
   LegendLabel -> "Medical supplies\nproduction rate"], 
 PlotRange -> All, ImageSize -> Large, 
 PlotLabel -> focusStock[t] /. modelSEI2HREcon["Stocks"]]
05il7vfu2msga

Here are the corresponding AUC values:

aAUCs = TrapezoidalRule[Transpose[{Range[0, 365], #}]] & /@ aVals;
ResourceFunction["GridTableForm"][aAUCs]
03yccn85qjh1a
BarChart[aAUCs, 
 ChartLabels -> Map[Rotate[ToString[#], \[Pi]/6] &, Keys[aAUCs]], 
 ColorFunction -> "Pastel", 
 PlotLabel -> 
  Row[{focusStock[t] /. modelSEI2HREcon["Stocks"], Spacer[5], "AUC"}],
  ImageSize -> Medium]
0z5dthm5iuod6

Variation of delivery disruption starts

Here we compute simulation solutions for a set of delivery disruption starts using disruption length of 7 days and disruption “slowing down” factor 2:

AbsoluteTiming[
 aVarSolutions =
   Association@
    Map[
     Function[{ddsVar},
      ddsVar -> 
       Association[
        ModelNDSolve[
          SetRateRules[
           modelSEI2HREcon, <|\[Kappa][MS] -> 100000, \[Kappa][HMS] ->
              1000, mspr[HB] -> 100, ql -> 56, qsd -> 60, 
            nhbr[TP] -> 3/1000, dbp -> 1, dds -> ddsVar, ddl -> 7, 
            dsf -> 2|>], {t, 365}][[1]]]
      ],
     Append[Range[40, 120, 20], 365]
     ];
 ]

(*{0.45243, Null}*)

Note, that disruption start at day 365 means no disruption. Also, we use three hospital beds per thousand people.

Here we plot the results for HP only:

SeedRandom[009]
focusStock = HP;
aVals = #[focusStock][Range[0, 365]] & /@ aVarSolutions;
ListLinePlot[
 KeyValueMap[
  Callout[Tooltip[#2, #1], #1, {RandomInteger[{60, 140}], Bottom}] &, 
  aVals], PlotLegends -> 
  SwatchLegend[Keys[aVals], 
   LegendLabel -> "Medical supplies\ndisruption start"], 
 PlotRange -> All, ImageSize -> Large, 
 PlotLabel -> focusStock[t] /. modelSEI2HREcon["Stocks"]]
1sdi97njh54gj
1sdi97njh54gj

Here are the corresponding AUC values:

aAUCs = TrapezoidalRule[Transpose[{Range[0, 365], #}]] & /@ aVals;
ResourceFunction["GridTableForm"][aAUCs]
0s1zcqcmvxbrs
0s1zcqcmvxbrs
BarChart[aAUCs, 
 ChartLabels -> Map[Rotate[ToString[#], \[Pi]/6] &, Keys[aAUCs]], 
 ColorFunction -> "Pastel", 
 PlotLabel -> 
  Row[{focusStock[t] /. modelSEI2HREcon["Stocks"], Spacer[5], "AUC"}],
  ImageSize -> Medium]
0vq0berxbjf9v

Combined variability of delivery start and disruption

Here we calculate the solutions for a set of combinations of delivery periods and delivery disruption starts:

AbsoluteTiming[
 aVarSolutions =
   Association@
    Map[
     Function[{par},
      model2 = modelSEI2HREcon;
      model2 = 
       SetRateRules[
        model2, <|\[Kappa][MS] -> 100000, \[Kappa][HMS] -> 10000, 
         mspr[HB] -> 1000, dbp -> par[[1]], dds -> par[[2]], ddl -> 7,
          dsf -> 4, nhbr[TP] -> 3/1000|>];
      par -> Association[ModelNDSolve[model2, {t, 365}][[1]]]
      ],
     Flatten[Outer[List, {0.5, 1, 1.5}, {60, 100, 365}], 1]
     ];
 ]

(*{0.759922, Null}*)

As expected more frequent, less disrupted delivery brings fuller utilization of the non-occupied hospital beds:

SeedRandom[3233]
focusStock = HP;
aVals = #[focusStock][Range[0, 365]] & /@ aVarSolutions;
ListLinePlot[
 KeyValueMap[
  Callout[Tooltip[#2, ToString[#1]], 
    ToString[#1], {RandomInteger[{60, 160}], Left}] &, aVals], 
 PlotLegends -> 
  SwatchLegend[ToString /@ Keys[aVals], 
   LegendLabel -> 
    "Medical supplies\ndelivery period & disruption start"], 
 PlotRange -> All, ImageSize -> Large, 
 PlotLabel -> focusStock[t] /. modelSEI2HREcon["Stocks"]]
1qqm9hcj40qmx
1qqm9hcj40qmx

Here are the corresponding AUC values:

aAUCs = TrapezoidalRule[Transpose[{Range[0, 365], #}]] & /@ aVals;
ResourceFunction["GridTableForm"][aAUCs]
19kpmr551tt1c
19kpmr551tt1c
BarChart[aAUCs, 
 ChartLabels -> Map[Rotate[ToString[#], \[Pi]/6] &, Keys[aAUCs]], 
 ColorFunction -> "Pastel", 
 PlotLabel -> 
  Row[{focusStock[t] /. modelSEI2HREcon["Stocks"], Spacer[5], "AUC"}],
  ImageSize -> Medium]
1ia6ik6cr7kg5
1ia6ik6cr7kg5
SeedRandom[3233]
focusStock = DIP;
aVals = #[focusStock][Range[0, 365]] & /@ aVarSolutions;
ListLinePlot[
 KeyValueMap[
  Callout[Tooltip[#2, ToString[#1]], 
    ToString[#1], {RandomInteger[{60, 160}], Left}] &, aVals], 
 PlotLegends -> 
  SwatchLegend[ToString /@ Keys[aVals], 
   LegendLabel -> 
    "Medical supplies\ndelivery period & disruption start"], 
 PlotRange -> All, ImageSize -> Large, 
 PlotLabel -> focusStock[t] /. modelSEI2HREcon["Stocks"]]
0xwdkry34kcfq
0xwdkry34kcfq
ResourceFunction["GridTableForm"][Last /@ aVals]
1u293ap7k53o5
1u293ap7k53o5
BarChart[Last /@ aVals, 
 ChartLabels -> Map[Rotate[ToString[#], \[Pi]/6] &, Keys[aAUCs]], 
 ColorFunction -> "Pastel", 
 PlotLabel -> "Deceased Population at day 365", ImageSize -> Medium]
0m5va3qv461nx
0m5va3qv461nx

References

Articles

[Wk1] Wikipedia entry, “Compartmental models in epidemiology”.

[Wl2] Wikipedia entry, “Coronavirus disease 2019”.

[HH1] Herbert W. Hethcote (2000). “The Mathematics of Infectious Diseases”. SIAM Review. 42 (4): 599–653. Bibcode:2000SIAMR..42..599H. doi:10.1137/s0036144500371907.

[BC1] Lucia Breierova, Mark Choudhari, An Introduction to Sensitivity Analysis, (1996), Massachusetts Institute of Technology.

[AA1] Anton Antonov, “Coronavirus propagation modeling considerations”, (2020), SystemModeling at GitHub.

[AA2] Anton Antonov, “Basic experiments workflow for simple epidemiological models”, (2020), SystemModeling at GitHub.

[AA3] Anton Antonov, “Scaling of Epidemiology Models with Multi-site Compartments”, (2020), SystemModeling at GitHub.

[AA4] Anton Antonov, “SEI2HR model with quarantine scenarios”, (2020), SystemModeling at GitHub.

Repositories, packages

[WRI1] Wolfram Research, Inc., “Epidemic Data for Novel Coronavirus COVID-19”, WolframCloud.

[AAr1] Anton Antonov, Coronavirus propagation dynamics project, (2020), SystemModeling at GitHub.

[AAr2] Anton Antonov, “Epidemiology Compartmental Modeling Monad in R”, (2020), ECMon-R at GitHub.

[AAp1] Anton Antonov, “Epidemiology models Mathematica package”, (2020), SystemModeling at GitHub.

[AAp2] Anton Antonov, “Epidemiology models modifications Mathematica package”, (2020), SystemModeling at GitHub.

[AAp3] Anton Antonov, “Epidemiology modeling visualization functions Mathematica package”, (2020), SystemModeling at GitHub.

[AAp4] Anton Antonov, “Epidemiology modeling simulation functions Mathematica package”, (2020), SystemModeling at GitHub.

[AAp5] Anton Antonov, “System dynamics interactive interfaces functions Mathematica package”, (2020), SystemsModeling at GitHub.

SEI2HR model with quarantine scenarios

Introduction

The epidemiology compartmental model, [Wk1], presented in this notebook – SEI2HR – deals with the left-most and middle rectangles in this diagram:

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0y1f7ckbmwf5i

“SEI2HR” stands for “Susceptible, Exposed, Infected two, Hospitalized, Recovered” (populations.)

In this notebook we also deal with quarantine scenarios.

Remark: We consider the contagious disease propagation models as instances of the more general System Dynamics (SD) models. We use SD terminology in this notebook.

The models

SEI2R

The model SEI2R are introduced and explained in the notebook [AA2]. SEI2R differs from the classical SEIR model , [Wk1, HH1], with the following elements:

  1. Two separate infected populations: one is “severely symptomatic”, the other is “normally symptomatic”
  2. The monetary equivalent of lost productivity due to infected or died people is tracked.

SEI2HR

For the formulation of SEI2HR we use a system of Differential Algebraic Equations (DAE’s). The package [AAp1] allows the use of a formulation that has just Ordinary Differential Equations (ODE’s).

Here are the unique features of SEI2HR:

  • People stocks
    • Two types of infected populations: normally symptomatic and severely symptomatic
    • Hospitalized population
    • Deceased from infection population
  • Hospital beds
    • Hospital beds are a limited resource that determines the number of hospitalized people
    • Only severely symptomatic people are hospitalized according to the available hospital beds
    • The hospital beds stock is not assumed constant, it has its own change rate.
  • Money stocks
    • The money from lost productivity are tracked
    • The money for hospital services are tracked

SEI2HR’s place a development plan

This graph shows the “big picture” of the model development plan undertaken in [AAr1] and SEI2HR (discussed in this notebook) is in that graph:

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1cg8j8gglqwi1

Notebook structure

The rest of notebook has the following sequence of sections:

  • Package load section
  • SEI2HR structure in comparison of SEI2R
  • Explanations of the equations of SEI2HR
  • Quarantine scenario modeling preparation
  • Parameters and initial conditions setup
    • Populations, hospital beds, quarantine scenarios
  • Parametric simulation solution
  • Interactive interface
  • Sensitivity analysis

Load packages

The epidemiological models framework used in this notebook is implemented with the packages [AAp1, AAp2, AA3]; many of the plot functions are from the package [AAp4].

SEI2HR extends SEI2R

The model SEI2HR is an extension of the model SEI2R, [AA2].

Here is SEI2R:

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0een3ni1vr18e

Here is SEI2HR:

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0dsi0svozimcl

Here are the “differences” between the two models:

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0h6b6x7di5e05

Equations explanations

In this section we provide rationale of the model equations of SEI2HR.

The equations for Susceptible, Exposed, Infected, Recovered populations of SEI2R are “standard” and explanations about them are found in [WK1, HH1]. For SEI2HR those equations change because of the stocks Hospitalized Population and Hospital Beds.

The equations time unit is one day. The time horizon is one year. Since we target COVID-19, [Wk2, AA1], we do not consider births.

Remark: For convenient reading the equations in this section have tooltips for the involved stocks and rates.

Verbalization description of the model

We start with one infected (normally symptomatic) person, the rest of the people are susceptible. The infected people meet other people directly or get in contact with them indirectly. (Say, susceptible people touch things touched by infected.) For each susceptible person there is a probability to get the decease. The decease has an incubation period: before becoming infected the susceptible are (merely) exposed. The infected recover after a certain average infection period or die. A certain fraction of the infected become severely symptomatic. If there are enough hospital beds the severely symptomatic infected are hospitalized. The hospitalized severely infected have different death rate than the non-hospitalized ones. The number of hospital beds might change: hospitals are extended, new hospitals are build, or there are not enough medical personnel or supplies. The deaths from infection are tracked (accumulated.) Money for hospital services and money from lost productivity are tracked (accumulated.)

The equations below give mathematical interpretation of the model description above.

Code for the equations

Each equation in this section are derived with code like this:

and then the output cell is edited to be “DisplayFormula” and have CellLabel value corresponding to the stock of interest.

The infected and hospitalized populations

SEI2HR has two types of infected populations: a normally symptomatic one and a severely symptomatic one. A major assumption for SEI2HR is that only the severely symptomatic people are hospitalized. (That assumption is also reflected in the diagram in the introduction.)

Each of those three populations have their own contact rates and mortality rates.

Here are the contact rates from the SEI2HR dictionary

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0ooy1eklv8ies

Here are the mortality rates from the SEI2HR dictionary

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0kskzm0nsvjo3

Remark: Below with “Infected Population” we mean both stocks Infected Normally Symptomatic Population (INSP) and Infected Severely Symptomatic Population (ISSP).

Total Population

In this notebook we consider a DAE’s formulation of SEI2HR. The stock Total Population has the following (obvious) algebraic equation:

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0peb290e1lx2d

Note that with Max we specified that the total population cannot be less than 0.

Remark: As mentioned in the introduction, the package [AAp1] allows for the use of non-algebraic formulation, without an equation for TP.

Susceptible Population

The stock Susceptible Population (SP) is decreased by (1) infections derived from stocks Infected Populations and Hospitalized Population (HP), and (2) morality cases derived with the typical mortality rate.

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0h1zwgod2xs4g

Because we hospitalize the severely infected people only instead of the term

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09xmdu8xq7e0p

we have the terms

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0dkc6td55qv7p

The first term is for the infections derived from the hospitalized population. The second term for the infections derived from people who are infected severely symptomatic and not hospitalized.

Births term

Note that we do not consider in this notebook births, but the births term can be included in SP’s equation:

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0ny25blzp2rub

The births rate is the same as the death rate, but it can be programmatically changed. (See [AAp2].)

Exposed Population

The stock Exposed Population (EP) is increased by (1) infections derived from the stocks Infected Populations and Hospitalized Population, and (2) mortality cases derived with the typical mortality rate. EP is decreased by (1) the people who after a certain average incubation period (aincp) become ill, and (2) mortality cases derived with the typical mortality rate.

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0mrj0gt13ppsn

Infected Normally Symptomatic Population

INSP is increased by a fraction of the people who have been exposed. That fraction is derived with the parameter severely symptomatic population fraction (sspf). INSP is decreased by (1) the people who recover after a certain average infection period (aip), and (2) the normally symptomatic people who die from the disease.

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0hy96yb7pg654

Infected Severely Symptomatic Population

ISSP is increased by a fraction of the people who have been exposed. That fraction is corresponds to the parameter severely symptomatic population fraction (sspf). ISSP is decreased by (1) the people who recover after a certain average infection period (aip), (2) the hospitalized severely symptomatic people who die from the disease, and (3) the non-hospitalized severely symptomatic people who die from the disease.

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19hml5ravi0a8

Note that we do not assume that severely symptomatic people recover faster if they are hospitalized, only that they have a different death rate.

Hospitalized Population

The amount of people that can be hospitalized is determined by the available Hospital Beds (HB) – the stock Hospitalized Population (HP) is subject to a resource limitation by the stock HB.

The equation of the stock HP can be easily understood from the following dynamics description points:

  • If the number of hospitalized people is less that the number of hospital beds we hospitalize the new ISSP people.
  • If the new ISSP people are more than the Available Hospital Beds (AHB) we take as many as AHB.
  • Hospitalized people have the same average infection period (aip).
  • Hospitalized (severely symptomatic) people have their own mortality rate.
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0ippnyp2hf90k

Note that although we know that in a given day some hospital beds are going to be freed they are not considered in the hospitalization plans for that day.

Recovered Population

The stock Recovered Population (RP) is increased by the recovered infected people and decreased by mortality cases derived with the typical mortality rate.

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042iy3oy8o454

Deceased Infected Population

The stock Deceased Infected Population (DIP) accumulates the deaths of the people who are infected. Note that we utilize the different death rates for HP and ISSP.

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1glekhgj1v0e7

Hospital Beds

The stock Hospital Beds (HB) can change with a rate that reflects the number of hospital beds change rate (nhbcr) per day. Generally speaking, using nhbcr we can capture scenarios, like, extending hospitals, building new hospitals, recruitment of new medical personnel, loss of medical personnel (due to infections.)

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1is45mamhsayd

Money for Hospital Services

The stock Money for Hospital Services (MHS) simply tracks expenses for hospitalized people. The parameter hospital services cost rate (hscr) with unit money per bed per day simply multiplies HP.

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0evzoim1tjwsj

Money from Lost Productivity

The stock Money from Lost Productivity (MLP) simply tracks the work non-availability of the infected and died from infection people. The parameter lost productivity cost rate (lpcr) with unit money per person per day multiplies the total count of the infected and dead from infection.

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1133o7bm975sr

Quarantine scenarios

In order to model quarantine scenarios we use piecewise constant functions for the contact rates \beta [\text{ISSP}] and \beta [\text{INSP}].

Remark: Other functions can be used, like, functions derived through some statistical fitting.

Here is an example plot :

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0gtvprtffih1o

To model quarantine with a piecewise constant function we use the following parameters:

  • \text{qsd} for quarantine’s start
  • \text{ql} for quarantines duration
  • \text{qcrf} for the effect on the quarantine on the contact rate

Parameters and actual simulation equations code

Here are the parameters we want to experiment with (or do calibration with):

Here we set custom rates and initial conditions:

Remark: Note the piecewise functions for \beta [\text{ISSP}] and \beta [\text{INSP}].

Here is the system of ODE’s we use to do parametrized simulations:

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0i35a50x9mdtq

Simulation

Solutions

Straightforward simulation for one year using ParametricNDSolve :

Example evaluation

Here are the parameters of a stock solution:

Here we replace the parameters with concrete rate values (kept in the model object):

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0m3ccwuzjtg7d

Here is an example evaluation of a solution using the parameter values above:

Interactive interface

Using the interface in this section we can interactively see the effects of changing the focus parameters.

opts = {PlotRange -> All, PlotLegends -> None, 
   PlotTheme -> "Detailed", PerformanceGoal -> "Speed", 
   ImageSize -> 400};
lsPopulationKeys = {TP, SP, EP, ISSP, INSP, HP, RP, DIP, HB};
lsEconKeys = {MHS, MLP};
Manipulate[
 DynamicModule[{lsPopulationPlots, lsEconPlots, lsRestPlots},
  
  lsPopulationPlots =
   ParametricSolutionsPlots[
    modelSEI2HR["Stocks"],
    KeyTake[aSol, Intersection[lsPopulationKeys, displayStocks]],
    {aincp, aip, spf, crhp, qsd, ql, qcrf, nhbcr, nhbr/1000}, ndays,
    "LogPlot" -> popLogPlotQ, "Together" -> popTogetherQ, 
    "Derivatives" -> popDerivativesQ, 
    "DerivativePrefix" -> "\[CapitalDelta]", opts];
  
  lsEconPlots =
   ParametricSolutionsPlots[
    modelSEI2HR["Stocks"],
    KeyTake[aSol, Intersection[lsEconKeys, displayStocks]],
    {aincp, aip, spf, crhp, qsd, ql, qcrf, nhbcr, nhbr/1000}, ndays,
    "LogPlot" -> econLogPlotQ, "Together" -> econTogetherQ, 
    "Derivatives" -> econDerivativesQ, 
    "DerivativePrefix" -> "\[CapitalDelta]", opts];
  
  lsRestPlots =
   If[Length[KeyDrop[aSol, Join[lsPopulationKeys, lsEconKeys]]] == 
     0, {},
    (*ELSE*)
    ParametricSolutionsPlots[
     modelSEI2HR["Stocks"],
     KeyTake[KeyDrop[aSol, Join[lsPopulationKeys, lsEconKeys]], 
      displayStocks],
     {aincp, aip, spf, crhp, qsd, ql, qcrf, nhbcr, nhbr/1000}, ndays,
     "LogPlot" -> econLogPlotQ, "Together" -> econTogetherQ, 
     "Derivatives" -> econDerivativesQ, 
     "DerivativePrefix" -> "\[CapitalDelta]", opts]
    ];
  
  Multicolumn[Join[lsPopulationPlots, lsEconPlots, lsRestPlots], 
   nPlotColumns, Dividers -> All, FrameStyle -> GrayLevel[0.8]]
  ],
 {{displayStocks, Join[lsPopulationKeys, lsEconKeys], 
   "Stocks to display:"}, Join[lsPopulationKeys, lsEconKeys], 
  ControlType -> TogglerBar},
 {{aincp, 6., "Average incubation period (days)"}, 1, 60., 1, 
  Appearance -> {"Open"}},
 {{aip, 21., "Average infectious period (days)"}, 1, 60., 1, 
  Appearance -> {"Open"}},
 {{spf, 0.2, "Severely symptomatic population fraction"}, 0, 1, 0.025,
   Appearance -> {"Open"}},
 {{qsd, 65, "Quarantine start days"}, 0, 365, 0.01, 
  Appearance -> {"Open"}},
 {{ql, 8*7, "Quarantine length (in days)"}, 0, 120, 1, 
  Appearance -> {"Open"}},
 {{qcrf, 0.25, "Quarantine contact rate fraction"}, 0, 1, 0.01, 
  Appearance -> {"Open"}},
 {{crhp, 0.1, "Contact rate of the hospitalized population"}, 0, 30, 
  0.1, Appearance -> {"Open"}},
 {{nhbcr, 0, "Number of hospital beds change rate"}, -0.5, 0.5, 0.001,
   Appearance -> {"Open"}},
 {{nhbr, 2.9, "Number of hospital beds rate (per 1000 people)"}, 0, 
  100, 0.1, Appearance -> {"Open"}},
 {{ndays, 365, "Number of days"}, 1, 365, 1, Appearance -> {"Open"}},
 {{popTogetherQ, True, "Plot populations together"}, {False, True}},
 {{popDerivativesQ, False, "Plot populations derivatives"}, {False, 
   True}},
 {{popLogPlotQ, False, "LogPlot populations"}, {False, True}},
 {{econTogetherQ, True, "Plot economics functions together"}, {False, 
   True}},
 {{econDerivativesQ, False, 
   "Plot economics functions derivatives"}, {False, True}},
 {{econLogPlotQ, True, "LogPlot economics functions"}, {False, 
   True}},
 {{nPlotColumns, 1, "Number of plot columns"}, Range[5]},
 ControlPlacement -> Left, ContinuousAction -> False]
1su0ilr7pjpgz
1su0ilr7pjpgz

Sensitivity analysis

When making and using this kind of dynamics models it is important to see how the solutions react to changes of different parameters. For example, we should try to find answers to questions like “What ranges of which parameters bring dramatic changes into important stocks?”

Sensitivity analysis is used to determine how sensitive is a SD model to changes of the parameters and to changes of model’s equations, [BC1]. More specifically, parameter sensitivity, which we apply below, allows us to see the changes of stocks dynamic behaviour for different sequences (and combinations) of parameter values.

Remark: The sensitivity analysis shown below should be done for other stocks and rates. In order to keep this exposition short we focus on ISSP, DIP, and HP.

It is interesting to think in terms of “3D parameter sensitivity plots.” We also do such plots.

Evaluations by Area under the curve

For certain stocks we might be not just interested in their evolution in time but also in their cumulative values. I.e. we are interested in the so called Area Under the Curve (AUC) metric for those stocks.

There are three ways to calculate AUC for stocks of interest:

  1. Add aggregation equations in the system of ODE’s. (Similar to the stock DIP in SEI2HR.)
    • For example, in order to compute AUC for ISSP we can add to SEI2HR the equation:
(*aucISSP'[t] = ISSP[t]*)
- More details for such equation addition are given in [AA2].
  1. Apply NIntegrate over stocks solution functions.
  2. Apply Trapezoidal rule to stock solution function values over a certain time grid.

Below we use 1 and 3.

Remark: The AUC measure for a stock is indicated with the prefix “∫”. For example AUC for ISSP is marked with “∫ISSP”.

Ranges

Below we use the following sets of quarantine starts and quarantine durations.

Note that putting the quarantine start to be at day 365 means “no quarantine.”

Number of infected people

Quarantine starts sensitivity

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13ewmnrwebn37

Note that the plots and tabulated differences with “no quarantine” indicate that there is a very narrow range to choose an effective quarantine start.

Quarantine duration sensitivity

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0jgocyy3n73g1

Number of deceased people

Quarantine starts sensitivity

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0upvp860txpwb

Number of hospitalized people

Quarantine starts sensitivity

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0jtq60dibh8ns

Infected Severely Symptomatic Population stock integral with respect to quarantine start and length

In this section the 3D plot of AUC of ISSP is calculated using Trapezoidal rule.

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1pm82rs4erlsy

Deceased Infected Population stock with respect to quarantine start and length

We can see from SEI2HR’s equations that DIP is already an AUC type of value. We can just plot the DIP values at the time horizon (one year.)

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1mtkajep4247q

Hospitalized Population stock integral with respect to quarantine start and length

In this section the 3D plot of AUC of HP is calculated using Trapezoidal rule.

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16mnmktwxtv14

References

Articles

[Wk1] Wikipedia entry, “Compartmental models in epidemiology”.

[Wl2] Wikipedia entry, “Coronavirus disease 2019”.

[HH1] Herbert W. Hethcote (2000). “The Mathematics of Infectious Diseases”. SIAM Review. 42 (4): 599–653. Bibcode:2000SIAMR..42..599H. doi:10.1137/s0036144500371907.

[BC1] Lucia Breierova, Mark Choudhari, An Introduction to Sensitivity Analysis, (1996), Massachusetts Institute of Technology.

[AA1] Anton Antonov, “Coronavirus propagation modeling considerations”, (2020), SystemModeling at GitHub.

[AA2] Anton Antonov, “Basic experiments workflow for simple epidemiological models”, (2020), SystemModeling at GitHub.

[AA3] Anton Antonov, “Scaling of Epidemiology Models with Multi-site Compartments”, (2020), SystemModeling at GitHub.

Repositories, packages

[WRI1] Wolfram Research, Inc., “Epidemic Data for Novel Coronavirus COVID-19”, WolframCloud.

[AAr1] Anton Antonov, Coronavirus propagation dynamics project, (2020), SystemModeling at GitHub.

[AAp1] Anton Antonov, “Epidemiology models Mathematica package”, (2020), SystemsModeling at GitHub.

[AAp2] Anton Antonov, “Epidemiology models modifications Mathematica package”, (2020), SystemsModeling at GitHub.

[AAp3] Anton Antonov, “Epidemiology modeling visualization functions Mathematica package”, (2020), SystemsModeling at GitHub.

[AAp4] Anton Antonov, “System dynamics interactive interfaces functions Mathematica package”, (2020), SystemsModeling at GitHub.

Project management files

[AAo1] Anton Antonov, WirVsVirus-Hackathon-work-plan.org, (2020), SystemsModeling at GitHub.

[AAo2] Anton Antonov, WirVsVirus-hackathon-Geo-spatial-temporal-model-mind-map, (2020), SystemsModeling at GitHub.

WirVsVirus 2020 hackathon participation

Introduction

Last weekend – 2020-03-20 ÷ 2020-03-22 – I participated in the (Germany-centric) hackathon WirVsVirus. (I friend of mine who lives in Germany asked me to team up and sign up.)

Our idea proposal was accepted, listed in the dedicated overview table (see item 806). The title of our hackathon project is:

“Geo-spatial-temporal Economic Model for COVID-19 Propagation and Management in Germany”

Nearly a dozen of people enlisted to help. (We communicated through Slack.)

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Multiple people helped with the discussion of ideas, directions where to find data, with actual data gathering, and related documented analysis. Of course, just discussing the proposed solutions was already a great help!

What was accomplished

Work plans

The following mind-map reflects pretty well what was planned and done:

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There is also a related org-mode file with the work plan.

Data

I obtained Germany city data with Mathematica’s build-in functions and used it to heuristically derive a traveling patterns graph, [AA1].

Here is the data:

Here is Geo-histogram of that data:

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We considered a fair amount of other data. But because of the time limitations of the hackathon we had to use only the one above.

Single-site models

During the development phase I used the model SEI2R, but since we wanted to have a “geo-spatial-temporal epidemiological economics model” I productized the implementation of SEI2HR-Econ, [AAp1].

Here are the stocks, rates, and equations of SEI2HR-Econ:

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Multi-site SEI2R (SEI2HR-Econ) over a hexagonal grid graph

I managed to follow through with a large part of the work plan for the hackathon and make multi-site scaled model that “follows the money”, [AA1]. Here is a diagram that shows the travelling patterns graph and solutions at one of the nodes:

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Here is (a snapshot of) an interactive interface for studying and investigating the solution results:

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For more details see the notebook [AA1]. Different parameters can be set in the “Parameters” section. Especially of interest are the quarantine related parameters: start, duration, effect on contact rates and traffic patterns.

I also put in that notebook code for exporting simulations results and programmed visualization routines in R, [AA2]. (In order other members of team to be able to explore the results.)

References

[DP1] 47_wirtschaftliche Auswirkung_Geo-spatial-temp-econ-modell, DevPost.

[WRI1] Wolfram Research, Inc., Germany city data records, (2020), SystemModeling at GitHub.

[AA1] Anton Antonov, “WirVsVirus hackathon multi-site SEI2R over a hexagonal grid graph”, (2020), SystemModeling at GitHub.

[AA2] Anton Antonov, “WirVsVirus-Hackathon in R”, (2020), SystemModeling at GitHub.

[AAp1] Anton Antonov, “Epidemiology models Mathematica package”, (2020), SystemsModeling at GitHub.

Scaling of epidemiology models with multi-site compartments

Version 1.0

Introduction

In this notebook we describe and exemplify an algorithm that allows the specification and execution geo-spatial-temporal simulations of infectious disease spreads. (Concrete implementations and examples are given.)

The assumptions of the typical compartmental epidemiological models do not apply for countries or cities that are non-uniformly populated. (For example, China, USA, Russia.) There is a need to derive epidemiological models that take into account the non-uniform distribution of populations and related traveling patterns within the area of interest.

Here is a visual aid (made with a random graph over the 30 largest cities of China):

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1acjs15vamk0b

In this notebook we show how to extend core, single-site epidemiological models into larger models for making spatial-temporal simulations. In the explanations and examples we use SEI2R, [AA2, AAp1], as a core epidemiological model, but other models can be adopted if they adhere to the model data structure of the package “EpidemiologyModels.m”, [AAp1].

From our experiments with we believe that the proposed multi-site extension algorithm gives a modeling ingredient that is hard emulate by other means within single-site models.

Definitions

Single-site: A geographical location (city, neighbourhood, campus) for which the assumptions of the classical compartmental epidemiological models hold.

Single site epidemiological model: A compartmental epidemiological model for a single site. Such model has a system of Ordinary Differential Equations (ODE’s) and site dependent initial conditions.

Multi-site area: An area comprised of multiple single sites with known traveling patterns between them. The area has a directed graph G with nodes that correspond to the sites and a positive matrix \text{\textit{tpm}}(G) for the traveling patterns between the sites.

Problem definition: Given (i) a single site epidemiological model M, (ii) a graph G connecting multiple sites, and (iii) a traveling patterns matrix \text{\textit{tpm}}(G) between the nodes of G derive an epidemiological model S(M,\text{\textit{tpm}}(G)) that simulates more adequately viral decease propagation over G.

Multi-Site Epidemiological Model Extension Algorithm (MSEMEA): An algorithm that derives from a given single site epidemiological model and multi-site area an epidemiological model that can be used to simulate the geo-spatial-temporal epidemics and infectious disease spreads. (The description of MSEMEA is the main message of this notebook.)

Load packages

The epidemiological models framework used in this notebook is implemented with the packages [AAp1, AAp2, AA3]; the interactive plots functions are from the package [AAp4].

Notebook structure

The section “General algorithm description” gives rationale and conceptual steps of MSEMEA.

The next two sections of the notebook follow the procedure outline using the SEI2R model as M, a simple graph with two nodes as G, and both constant and time-dependent matrices for \text{\textit{tpm}}(G).

The section “Constant traveling patterns over a grid graph” presents an important test case with a grid graph that we use to test and build confidence in MSEMEA. The sub-section “Observations” is especially of interest.

The section “Time-dependent traveling patterns over a random graph” presents a nearly “real life” application of MSEMEA using a random graph and a time dependent travelling patterns matrix.

The section “Money from lost productivity” shows how to track the money losses across the sites.

The last section “Future plans” outlines envisioned (immediate) extensions work presented in this notebook.

General algorithm description

In this section we describe a modeling approach that uses different mathematical modeling approaches for (i) the multi-site travelling patterns and (ii) the single-site disease spread interactions, and then (iii) unifies them into a common model.

Splitting and scaling

The traveling between large, densely populated cities is a very different process of the usual people mingling in those cities. The usual large, dense city mingling is assumed and used in the typical compartmental epidemiological models. It seems it is a good idea to split the two processes and derive a common model.

Assume that all journeys finish within a day. We can model the people arriving (flying in) into a city as births, and people departing a city as deaths.

Let as take a simple model like SIR or SEIR and write the equation for every site we consider. This means for every site we have the same ODE’s with site-dependent initial conditions.

Consider the traveling patterns matrix K, which is a contingency matrix derived from source-destination traveling records. (Or the adjacency matrix of a travelling patterns graph.) The matrix entry of K(i,j) tells us how many people traveled from site i to site j. We systematically change the ODE’s of the sites in following way.

Assume that site a had only travelers coming from site b and going to site b. Assume that the Total Population (TP) sizes for sites a and b are N_a and N_b respectively. Assume that only people from the Susceptible Population (SP) traveled. If the adopted single-site model is SIR, [Wk1], we take the SP equation of site a

\text{SP}_a'(t)=-\frac{\beta  \text{IP}_a(t) \text{SP}_a(t)}{N_a}-\text{SP}_a(t) \mu

and change into the equation

\text{SP}_a'(t)=-\frac{\beta  \text{IP}_a(t) \text{SP}_a(t)}{N_a}-\text{SP}_a(t) \mu -\frac{K(a,b)\text{SP}_a(t)}{N_a}+\frac{K(b,a)\text{SP}_b(t)}{N_b},

assuming that

\frac{K(a,b)\text{SP}_a(t)}{N_a}\leq N_a ,\frac{K(b,a)\text{SP}_b(t)}{N_b}\leq N_b.

Remark: In the package [AAp3] the transformations above are done with the more general and robust formula:

\min \left(\frac{K(i,j)\text{SP}_i(t)}{\text{TP}_i(t)},\text{TP}_i(t)\right).

The transformed systems of ODE’s of the sites are joined into one “big” system of ODE’s, appropriate initial conditions are set, and the “big” ODE system is solved. (The sections below show concrete examples.)

Steps of MSEMEA

MSEMEA derives a compartmental model that combines (i) a graph representation of multi-site traveling patterns with (ii) a single-site compartmental epidemiological model.

Here is a visual aid for the algorithm steps below:

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0i0g3m8u08bj4
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05c2sz8hd3ryj
  1. Get a single-site epidemiological compartmental model data structure, M.
    1. The model data structure has stocks and rates dictionaries, equations, initial conditions, and prescribed rate values; see [AA2, AAp1].
  2. Derive the site-to-site traveling patterns matrix K for the sites in the graph G.
  3. For each of node i of G make a copy of the model M and denote with M[i].
    1. In general, the models M[i], i\in G have different initial conditions.
    2. The models can also have different death rates, contact rates, etc.
  4. Combine the models M[i], i\in G into the scaled model S.
    1. Change the equations of M[i], i\in G to reflect the traveling patterns matrix K.
    2. Join the systems of ODE’s of M[i], i\in G into one system of ODE’s.
  5. Set appropriate or desired initial conditions for each of the populations in S.
  6. Solve the ODE’s of S.
  7. Visualize the results.

Precaution

Care should be taken when specifying the initial conditions of MSEMEA’s system of ODE’s (sites’ populations) and the traveling patterns matrix. For example, the simulations can “blow up” if the traveling patterns matrix values are too large. As it was indicated above, the package [AAp3] puts some safe-guards, but in our experiments with random graphs and random traveling patterns matrices occasionally we still get “wild” results.

Analogy with Large scale air-pollution modeling

There is a strong analogy between MSEMEA and Eulerian models of Large Scale Air-Pollution Modeling (LSAPM), [AA3, ZZ1].

The mathematical models of LSAPM have a “chemistry part” and an “advection-diffusion part.” It is hard to treat such mathematical model directly – different kinds of splittings are used. If we consider 2D LSAPM then we can say that we cover the modeling area with steer tank reactors, then with the chemistry component we simulate the species chemical reactions in those steer tanks, and with the advection-diffusion component we change species concentrations in the steer tanks (according to some wind patterns.)

Similarly, with MSEMEA we separated the travel of population compartments from the “standard” epidemiological modeling interaction between the population compartments.

Similarly to the so called “rotational test” used in LSAPM to evaluate numerical schemes, we derive and study the results of “grid graph tests” for MSEMEA.

Single site epidemiological model

Here is the SEI2R model from the package [AAp1]:

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18y99y846b10m

Here we endow the SEI2R model with a (prominent) ID:

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0alzg909zg4h0

Thus we demonstrated that we can do Step 3 of MSEMEA.

Below we use ID’s that correspond to the nodes of graphs (and are integers.)

Scaling the single-site SIR model over a small complete graph

Constant travel matrices

Assume we have two sites and the following graph and matrix describe the traveling patterns between them.

Here is the graph:

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0vgm31o9drq4f

And here is the traveling patterns matrix:

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0lbp0xgso2tgt

Note that there are much more travelers from 1 to 2 than from 2 to 1.

Here we obtain the core, single-site model (as shown in the section above):

Make the multi-site compartments model with SEI2R and the two-node travel matrix using the function ToSiteCompartmentsModel of [AAp2]:

Show the unique stocks in the multi-site model:

From the symbolic form of the multi-model equations derive the specific equations with the adopted rate values:

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0mjliik7acoyd

Show the initial conditions:

Show the total number of equations:

Solve the system of ODE’s of the extended model:

Display the solutions for each site separately:

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1o9362wmczxo6

From the plots above we see that both sites start with total populations of 100000 people. Because more travelers go from 1 to 2 we see that the exposed, infected, and recovered populations are larger at 2.

Time dependent travel matrices

Instead of using constant traveling patterns matrices we can use matrices with time functions as entries. It is instructive to repeat the computations above using this matrix:

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1gsoh03lixm6y

Here are the corresponding number of traveling people functions:

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0qh7kbtwxyatf

Here we scale the SIR model, solve the obtained system of ODE’s, and plot the solutions:

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0trv1vnslv1rm

Note that the oscillatory nature of the temporal functions in the travelling patterns matrix are reflected in the simulation results.

Constant traveling patterns over a grid graph

In this section we do the model extension and simulation over a regular grid graph with a constant traveling patterns matrix.

Here we create a grid graph with directed edges:

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0l2m5npcrlnvw

Note that:

  • There is one directed edge between any two edge-connected nodes
  • All horizontal edges point in one direction
  • All vertical edges point in one direction
  • The edges are directed from nodes with smaller indexes to nodes with larger indexes.

Here we make a constant traveling matrix and summarize it:

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0d4ocoa6gibfj

Here we scale the SEI2R model with the grid graph constant traveling matrix:

Change the initial conditions in the following way:

  • Pick initial population size per site (same for all sites)
  • Make a constant populations vector
  • At all sites except the first one put the infected populations to zero; the first site has one severely symptomatic person
  • Set the susceptible populations to be consistent with the total and infected populations.

Solve the system of ODE’s of the scaled model:

Randomly sample the graph sites and display the solutions separately for each site in the sample:

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030qbpok8qfmc

Display solutions of the first and last site:

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0mcxwp8l1vqlb

As expected from the graph structure, we can see in the first site plot that its total population is decreasing – nobody is traveling to the first site. Similarly, we can see in the last site plot that its total population is increasing – nobody leaves the last site.

Graph evolution visualizations

We can visualize the spatial-temporal evolution of model’s populations using sequences of graphs. The graphs in the sequences are copies of the multi-site graph each copy having its nodes colored according to the populations in the solutions steps.

Here is a sub-sequence for the total populations:

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070ld135tkf7y

Here is a sub-sequence for the sum of the infected populations:

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0rv8vap59g8vk

Here is a sub-sequence for the recovered population:

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0dnvpy20nafpz

Here is an animation of the sum of the infected populations:

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1hfd1mqh0iwk7

Curve shapes of the globally-aggregated solutions

Let us plot for each graph vertex the sum of the solutions of the two types of infected populations. Here is a sample of those graphs:

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0jwgipeg8uznn

We can see from the plot above that at the grid vertexes we have typical SEIR curve shapes for the corresponding infected populations.

Let us evaluate the solutions for the infected populations for over all graph vertexes and sum them. Here is the corresponding “globally-aggregated” plot:

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03nzzmh7yydeu

We can see that the globally aggregated plot has a very different shape than the individual vertex plots. The globally aggregated plot has more symmetric look; the individual vertex plots have much steeper gradients on their left sides.

We can conjecture that a multi-site model made by MSEMEA would capture better real life situations than any single-site model. For example, by applying MSEMEA we might be more successful in our calibration attempts for the Hubei data shown (and calibrated upon) in [AA2.].

Interactive interface

With this interactive interface we see the evolution of all populations across the graph:

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0fv3dbwmah3sh

Observations

Obviously the simulations over the described grid graph, related constant traveling patterns matrix, and constant populations have the purpose to build confidence in conceptual design of MSEMEA and its implementation.

The following observations agree with our expectations for MSEMEA’s results over the “grid graph test”.

  1. The populations plots at each site resemble the typical plots of SEI2R.
  2. The total population at the first site linearly decreases.
  3. The total population at the last site linearly increases.
  4. The plots of the total populations clearly have gradually increasing gradients from the low index value nodes to the high index value nodes.
  5. For the infected populations there is a clear wave that propagates diagonally from the low index value nodes to the high index value nodes.
    1. In the direction of the general “graph flow.“
  6. The front of the infected populations wave is much steeper (gives “higher contrast”) than the tail.
    1. This should be expected from the single-site SEI2R plots.
  7. For the recovered populations there is a clear “saturation wave” pattern that indicates that the recovered populations change from 0 to values close to the corresponding final total populations.
  8. The globally aggregated solutions might have fairly different shapes than the single-site solutions. Hence, we think that MSEMEA gives a modeling ingredient that is hard to replicate or emulate by other means in single-site models.

Time-dependent traveling patterns over a random graph

In this section we apply the model extension and simulation over a random graph with random time-dependent traveling patterns matrix.

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1elzu67nqndly

Remark: The computations in this section work with larger random graphs; we use a small graph for more legible presentation of the workflow and results. Also, the computations should clearly demonstrate the ability to do simulations with real life data.

Derive a traveling patterns matrix with entries that are random functions:

Here is a fragment of the matrix:

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17ym9q0uehfbt

Summarize and plot the matrix at t=1:

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0bw9rrd64615r

Here we scale the SEI2R model with the random traveling matrix:

Change the initial conditions in the following way:

  • Pick maximum population size per site
  • Derive random populations for the sites
  • At all sites except the first one put the infected populations to zero; the first site has one severely symptomatic person
  • Set the susceptible populations to be consistent with the total and infected populations.

Here solve the obtained system of ODE’s:

Here we plot the solutions:

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1pzmli04rbchr

Graph evolution visualizations

As in the previous section we can visualize the spatial-temporal evolution of model’s populations using sequences of graphs.

Here is a globally normalized sequence:

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1v8xq8jzm6ll5

Here is a locally normalized (“by vertex”) sequence:

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029w2jtxsyaen

Money from lost productivity

The model SEI2R from [AAp1] has the stock “Money from Lost Productivity” shown as \text{MLP}(t) in the equations:

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1sareh7ovkmtt

Here are MLP plots from the two-node graph model:

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0cm5n3xxlewns

Here we plot the sum of the accumulated money losses:

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0n6gz7j3qlq07

Here is the corresponding “daily loss” (derivative):

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0jrk2ktzeeled

Future plans

There are multiple ways to extend the presented algorithm, MSEMEA. Here are a few most immediate ones:

  1. Investigate and describe the conditions under which MSEMEA performs well, and under which it “blows up”
  2. Apply MSEMEA together with single site models that have large economics parts
  3. Do real data simulations related to the spread of COVID-19.

References

Articles, books

[Wk1] Wikipedia entry, “Compartmental models in epidemiology”.

[HH1] Herbert W. Hethcote (2000). “The Mathematics of Infectious Diseases”. SIAM Review. 42 (4): 599–653. Bibcode:2000SIAMR..42..599H. doi:10.1137/s0036144500371907.

[AA1] Anton Antonov, “Coronavirus propagation modeling considerations”, (2020), SystemModeling at GitHub.

[AA2] Anton Antonov, “Basic experiments workflow for simple epidemiological models”, (2020), SystemModeling at GitHub.

[AA3] Anton Antonov, “Air pollution modeling with gridMathematica”, (2006), Wolfram Technology Conference.

[ZZ1] Zahari Zlatev, Computer Treatment of Large Air Pollution Models. 1995. Kluwer.

Repositories, packages

[WRI1] Wolfram Research, Inc., “Epidemic Data for Novel Coronavirus COVID-19”, WolframCloud.

[AAr1] Anton Antonov, Coronavirus propagation dynamics project, (2020), SystemModeling at GitHub.

[AAp1] Anton Antonov, “Epidemiology models Mathematica package”, (2020), SystemsModeling at GitHub.

[AAp2] Anton Antonov, “Epidemiology models modifications Mathematica package”, (2020), SystemsModeling at GitHub.

[AAp3] Anton Antonov, “Epidemiology modeling visualization functions Mathematica package”, (2020), SystemsModeling at GitHub.

[AAp4] Anton Antonov, “System dynamics interactive interfaces functions Mathematica package”, (2020), SystemsModeling at GitHub.

UML diagrams creation and generation

This post is to show how to create and generate Unified Modeling Language (UML) diagrams in Mathematica. It is related to programming in Mathematica using Object-Oriented Design Patterns.

Although this post is not about prediction per se it is going to be referenced and used in future posts about predictive algorithms and challenges.

Package functions

This command imports the package UMLDiagramGeneration.m :

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/Misc/UMLDiagramGeneration.m"]

The package provides the functions UMLClassNode and UMLClassGraph.

The function UMLClassNode has the signature

UMLClassNode[classSymbol, opts]

UMLClassNode creates a Grid object with a class name and its methods for the specified class symbol. The option “Abstact” can be used to specify abstract class names and methods. The option “EntityColumn” can be used to turn on and off the explanations column.

The function UMLClassGraph that has the signature:

UMLClassGraph[symbols, abstractMethodsPerSymbol, symbolAssociations, symbolAggregations, opts]

UMLClassGraph creates an UML graph diagram for the specified symbols (representing classes) and their relationships. It takes as options the options of UMLClassNode and Graph.

UML diagrams creation

Let us visualize a simple relationship between buildings, people, books, and a client program.

UMLClassGraph[{Library \[DirectedEdge] Building,
Museum \[DirectedEdge] Building,
Member \[DirectedEdge] Person}, {}, {Library <-> Member,
Museum \[DirectedEdge] Member, Client \[DirectedEdge] Building,
Client \[DirectedEdge] Person}, {Library \[DirectedEdge] Book},
"Abstract" -> {Building, Person},
"EntityColumn" -> False, VertexLabelStyle -> "Text",
ImageSize -> Large, GraphLayout -> "LayeredDigraphEmbedding"]

UML-diagram-for-Inheritance-Composition-Association

In the diagram above the classes Person and Building are abstract (that is why are in italic). Member inherits Person, Library and Museum inherit Building. Library can contain (many) Book objects and it is associated with Member. Client associates with Building and Person.

UML diagram generation

The main package function UMLClassGraph is capable of generating UML diagrams over Design Patterns code written in the style exemplified and described in my WTC 2015 talk Object-Oriented Design Patterns.

Let us look into a simple UML generation example for the design pattern Template Method.

Here is the Mathematica code for that design pattern:

 Clear[AbstractClass, ConcreteOne, ConcreteTwo];
 
 CLASSHEAD = AbstractClass;
 AbstractClass[d_]["Data"[]] := d;
 AbstractClass[d_]["PrimitiveOperation1"[]] := d[[1]];
 AbstractClass[d_]["PrimitiveOperation2"[]] := d[[2]];
 AbstractClass[d_]["TemplateMethod"[]] :=
 CLASSHEAD[d]["PrimitiveOperation1"[]] + CLASSHEAD[d]["PrimitiveOperation2"[]]
 
 ConcreteOne[d_][s_] := Block[{CLASSHEAD = ConcreteOne}, AbstractClass[d][s]]
 ConcreteOne[d_]["PrimitiveOperation1"[]] := d[[1]];
 ConcreteOne[d_]["PrimitiveOperation2"[]] := d[[1]]*d[[2]];
 
 ConcreteTwo[d_][s_] := Block[{CLASSHEAD = ConcreteTwo}, AbstractClass[d][s]]
 ConcreteTwo[d_]["PrimitiveOperation1"[]] := d[[1]];
 ConcreteTwo[d_]["PrimitiveOperation2"[]] := d[[3]]^d[[2]];

This command generates an UML diagram over the code above:

UMLClassGraph[{AbstractClass, ConcreteOne,
ConcreteTwo}, {AbstractClass -> {"PrimitiveOperation1",
"PrimitiveOperation2"}}, "Abstract" -> {AbstractClass},
VertexLabelStyle -> "Subsubsection"]

UML-diagram-generated-over-TemplateMethod-code

Here is a diagram generated over a Mathematica implementation of Decorator:

UML-diagram-for-Decorator

And here is a diagram for a concrete implementation of Interpreter for Boolean expressions:

UML-diagram-for-Interpreter-of-BooleanExpr

(Interpreter is my favorite Design Pattern and I have made several Mathematica implementations that facilitate and extend its application. See these blog posts of mine: “Functional parsers” category in MathematicaForPrediction at WordPress).