Message Boards

GROUPS:

Staff Picks Biological Sciences Mathematics Medical Sciences Operations Research Mathematica Modeling

Anton Antonov

SEI2HR-Econ model with quarantine and supplies scenarios

Anton Antonov, Accendo Data LLC

Posted 2 years ago

4746 Views

2 Replies

3 Total Likes

Follow this post

MODERATOR NOTE: coronavirus resources & updates: https://wolfr.am/coronavirus

SEI2HR-Econ model with quarantine and supplies scenarios

Version 0.9

Anton Antonov
MathematicaForPrediction at WordPress
SystemModeling at GitHub
March, April 2020

Introduction

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

In[]:=

ImageResize[Import["https://github.com/antononcube/SystemModeling/raw/master/Projects/Coronavirus-propagation-dynamics/Diagrams/Coronavirus-propagation-simple-dynamics.jpeg"],900]

Out[]=

“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:

Out[]=

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].

In[]:=

Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/Projects/Coronavirus-propagation-dynamics/WL/EpidemiologyModels.m"];Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/Projects/Coronavirus-propagation-dynamics/WL/EpidemiologyModelModifications.m"];Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/Projects/Coronavirus-propagation-dynamics/WL/EpidemiologyModelingVisualizationFunctions.m"];Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/Projects/Coronavirus-propagation-dynamics/WL/EpidemiologyModelingSimulationFunctions.m"];Import["https://raw.githubusercontent.com/antononcube/SystemModeling/master/WL/SystemDynamicsInteractiveInterfacesFunctions.m"];

SEI2HR-Econ extends SEI2HR

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

Here is SEI2HR:

In[]:=

reprTP="AlgebraicEquation";lsModelOpts={"Tooltips"True,TooltipStyle{BackgroundYellow,CellFrameColorGray,FontSize20}};modelReference=SEI2HRModel[t,"InitialConditions"True,"RateRules"True,"TotalPopulationRepresentation"reprTP];ModelGridTableForm[modelReference,lsModelOpts]

Out[]=

Stocks

#	Symbol	Description
1	TP[t]	Total Population
2	SP[t]	Susceptible Population
3	EP[t]	Exposed Population
4	INSP[t]	Infected Normally Symptomatic Population
5	ISSP[t]	Infected Severely Symptomatic Population
6	RP[t]	Recovered Population
7	MLP[t]	Money of Lost Productivity
8	HP[t]	Hospitalized Population
9	DIP[t]	Deceased Infected Population
10	HB[t]	Hospital Beds
11	MHS[t]	Money for Hospital Services

,Rates

#	Symbol	Description
1	μ[TP]	Population death rate
2	μ[INSP]	Infected Normally Symptomatic Population death rate
3	μ[ISSP]	Infected Severely Symptomatic Population death rate
4	sspf[SP]	Severely Symptomatic Population Fraction
5	β[INSP]	Contact rate for the normally symptomatic population
6	β[ISSP]	Contact rate for the severely symptomatic population
7	aip	Average infectious period
8	aincp	Average incubation period
9	lpcr[ISSP,INSP]	Lost productivity cost rate (per person per day)
10	μ[HP]	Hospitalized Population death rate
11	β[HP]	Contact rate for the hospitalized population
12	nhbr[TP]	Number of hospital beds rate
13	hscr[ISSP,INSP]	Hospital services cost rate (per bed per day)
14	nhbcr[ISSP,INSP]	Number of hospital beds change rate (per day)

,Equations

Equation

TP[t]

Max[0,

EP[t]

INSP[t]

ISSP[t]

RP[t]

SP[t]

]

′

[t]-

HP[t]

SP[t]

β[HP]

TP[t]

INSP[t]

SP[t]

β[INSP]

TP[t]

Max[0,-

HP[t]

ISSP[t]

]

SP[t]

β[ISSP]

TP[t]

SP[t]

μ[TP]

′

[t]

HP[t]

SP[t]

β[HP]

TP[t]

INSP[t]

SP[t]

β[INSP]

TP[t]

Max[0,-

HP[t]

ISSP[t]

]

SP[t]

β[ISSP]

TP[t]

EP[t]

aincp

μ[TP]

′

INSP

[t]-

INSP[t]

aip

EP[t]

(1-

sspf[SP]

)

aincp

INSP[t]

μ[INSP]

′

ISSP

[t]-

ISSP[t]

aip

EP[t]

sspf[SP]

aincp

HP[t]

μ[HP]

-(-

HP[t]

ISSP[t]

)

μ[ISSP]

′

[t]

Min HB[t] - HP[t] , EP[t] sspf[SP] aincp 	HP[t] < HB[t]
0	True

HP[t]

aip

HP[t]

μ[HP]

′

[t]

INSP[t]

ISSP[t]

aip

RP[t]

μ[TP]

′

DIP

[t]

HP[t]

μ[HP]

INSP[t]

μ[INSP]

+(-

HP[t]

ISSP[t]

)

μ[ISSP]

′

[t]

HB[t]

nhbcr[ISSP,INSP]

′

MHS

[t]

HP[t]

hscr[ISSP,INSP]

′

MLP

[t]

lpcr[ISSP,INSP]

(

INSP[t]

ISSP[t]

HP[t]

μ[HP]

INSP[t]

μ[INSP]

+(-

HP[t]

ISSP[t]

)

μ[ISSP]

)

,RateRules

#	Symbol	Value
1	μ[TP]	1 45625
2	μ[ISSP]	0.035 aip
3	μ[INSP]	0.01 aip
4	β[ISSP]	0.15
5	β[INSP]	0.15
6	aip	26
7	aincp	6
8	sspf[SP]	0.2
9	lpcr[ISSP,INSP]	1
10	μ[HP]	0.25 μ[ISSP]
11	β[HP]	0.1 β[ISSP]
12	nhbr[TP]	0.0029
13	nhbcr[ISSP,INSP]	0
14	hscr[ISSP,INSP]	1

,InitialConditions

#	Equation
1	SP[0]99998
2	EP[0]0
3	ISSP[0]1
4	INSP[0]1
5	RP[0]0
6	MLP[0]0
7	TP[0]100000
8	HP[0]0
9	DIP[0]0
10	HB[0] nhbr[TP] TP[0]
11	MHS[0]0



Here is SEI2HR-Econ:

In[]:=

modelSEI2HREcon=SEI2HREconModel[t,"InitialConditions"True,"RateRules"True,"TotalPopulationRepresentation"reprTP];ModelGridTableForm[modelSEI2HREcon,lsModelOpts]

Out[]=

Here are the “differences” between the two models:

In[]:=

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

Out[]=

Stocks

#	Symbol	Description
1	MS[t]	Medical Supplies
2	MSD[t]	Medical Supplies Demand
3	MMSP[t]	Money for Medical Supplies Production
4	HMS[t]	Hospital Medical Supplies

,Rates

#	Symbol	Description
1	hpmscr[ISSP,INSP]	Hospitalized population medical supplies consumption rate (per day)
2	upmscr[ISSP,INSP]	Un-hospitalized population medical supplies consumption rate (units per day)
3	mspr[HB]	Medical supplies production rate (units per pay)
4	mspcr[HB]	Medical supplies production cost rate (per unit)
5	msdr[HB]	Medical supplies delivery rate (delay factor)
6	msdp[HB]	Medical supplies delivery period (number of days)
7	mscr[TP]	Medical supplies consumption rate (units per day per person)
8	mscr[INSP]	Medical supplies consumption rate (units per day per person)
9	mscr[ISSP]	Medical supplies consumption rate (units per day per person)
10	mscr[HP]	Medical supplies consumption rate (units per day per person)
11	κ[HMS]	Capacity to store Hospital Medical Supplies
12	κ[MS]	Capacity to store produced Medical Supplies
13	κ[MSD]	Capacity to transport produced Medical Supplies

,Equations

Equation

′

HMS

[t]-Min[

HMS[t]

,HP[t]

mscr[ISSP]

Min[

MS[t]

,HB[t]

mscr[HP]

HMS[t]

κ[HMS]

κ[MSD]

]

msdp[HB]

′

[t]-

HP[t]

aip

MinHB[t]-HP[t], EP[t]sspf[SP] aincp , HMS[t] mscr[ISSP] 	HP[t]<HB[t]
0	True

-HP[t]μ[HP]

′

MMSP

[t]

MSD[t]

mspcr[HB]

′

[t]-MinINSP[t]

mscr[INSP]

+(-HP[t]+ISSP[t])

mscr[ISSP]

+(EP[t]+RP[t]+SP[t])

mscr[TP]

MS[t]

Min[

MS[t]

,HB[t]

mscr[HP]

HMS[t]

κ[HMS]

κ[MSD]

]

msdp[HB]

+Min[

mspr[HB]

MS[t]

κ[MS]

Min[

MS[t]

,HB[t]

mscr[HP]

HMS[t]

κ[HMS]

κ[MSD]

]

msdp[HB]

′

MSD

[t]HP[t]

mscr[HP]

+INSP[t]

mscr[INSP]

+ISSP[t]

mscr[ISSP]

+(EP[t]+RP[t]+SP[t])

mscr[TP]

,RateRules

#	Symbol	Value
1	hpmscr[ISSP,INSP]	4
2	upmscr[ISSP,INSP]	2
3	mspcr[HB]	1
4	msdp[HB]	1.5
5	mspr[HB]	100nhbr[TP]TP[0]
6	mscr[TP]	0.02
7	mscr[INSP]	0.7
8	mscr[ISSP]	3
9	mscr[HP]	4
10	κ[HMS]	2HB[0] mscr[ISSP]
11	κ[MS]	2 mscr[TP] TP[0]
12	κ[MSD]	HB[0] 10

,InitialConditions

#	Equation
1	HMS[0] κ[HMS]
2	MMSP[0]0
3	MS[0] κ[MS]
4	MSD[0]0



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:

In[]:=

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

In[]:=

ColumnForm@Cases[Normal@modelSEI2HREcon["Rates"],HoldPattern[β[_]_],∞]

Out[]=

β[INSP]Contact rate for the normally symptomatic population

β[ISSP]Contact rate for the severely symptomatic population

β[HP]Contact rate for the hospitalized population

Here are the mortality rates from the SEI2HR-Econ dictionary

In[]:=

ColumnForm@Cases[Normal@modelSEI2HREcon["Rates"],HoldPattern[μ[_]_],∞]

Out[]=

μ[TP]Population death rate

μ[INSP]Infected Normally Symptomatic Population death rate

μ[ISSP]Infected Severely Symptomatic Population death rate

μ[HP]Hospitalized Population death rate

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:

(TP)

TP[t]

Max[0,

EP[t]

INSP[t]

ISSP[t]

RP[t]

SP[t]

]

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

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.

(SP)

′

[t]-

HP[t]

SP[t]

β[HP]

TP[t]

INSP[t]

SP[t]

β[INSP]

TP[t]

Max[0,-

HP[t]

ISSP[t]

]

SP[t]

β[ISSP]

TP[t]

SP[t]

μ[TP]

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

ISSP[t]SP[t]β[ISSP]

TP[t]

we have the terms

HP[t]

SP[t]

β[HP]

TP[t]

Max[0,-

HP[t]

ISSP[t]

]

SP[t]

β[ISSP]

TP[t]

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:

In[]:=

Block[{m=SEI2HREconModel[t,"BirthsTerm"True]},ModelGridTableForm[m]["Equations"]〚1,EquationPosition[m,SP]+1,2〛]

Out[]=

′

[t]-

SP[t]

μ[TP]

HP[t]

SP[t]

β[HP]

TP[0]

INSP[t]

SP[t]

β[INSP]

TP[0]

Max[0,-

HP[t]

ISSP[t]

]

SP[t]

β[ISSP]

TP[0]

μ[TP]

TP[0]

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.

(EP)

′

[t]

HP[t]

SP[t]

β[HP]

TP[t]

INSP[t]

SP[t]

β[INSP]

TP[t]

Max[0,-

HP[t]

ISSP[t]

]

SP[t]

β[ISSP]

TP[t]

EP[t]

aincp

μ[TP]

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.

(INSP)

′

INSP

[t]-

INSP[t]

aip

EP[t]

(1-

sspf[SP]

)

aincp

INSP[t]

μ[INSP]

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.

(ISSP)

′

ISSP

[t]-

ISSP[t]

aip

EP[t]

sspf[SP]

aincp

HP[t]

μ[HP]

-(-

HP[t]

ISSP[t]

)

μ[ISSP]

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:

(HP)

′

[t]

Min

HB[t]