In this document we give usage examples for the functions of the package, “SystemDynamicsModelGraph.m”, [AAp1]. The package provides functions for making dependency graphs for the stocks in System Dynamics (SD) models. The primary motivation for creating the functions in this package is to have the ability to introspect, proofread, and verify the (typical) ODE models made in SD.
A more detailed explanation is:
For a given SD system S of Ordinary Differential Equations (ODEs) we make Mathematica graph objects that represent the interaction of variable dependent functions in S.
Those graph objects give alternative (and hopefully convenient) way of visualizing the model of S.
The following commands load the packages [AAp1, AAp2, AAp3]:
The organizers and I did a poll for what people want to hear. After discussing the results of the 15 votes from that poll we decided the presentation to be a methodological one instead of a know-how one.
Approximately 70% of the presentation was based on an R-programmed software monad for epidemiology compartmental models, ECMMon-R, [AAr2]. For the rest were used frameworks, simulations, and graphics made with Mathematica, [AAr1], and Wolfram System Modeler .
The presentation was given online (because of COVID-19) using Zoom. 190 people registered. Nearly 70 showed up (and maybe 60 stayed throughout.)
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:
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
Remark: Below with “Infected Population” we mean both stocks Infected Normally Symptomatic Population (INSP) and Infected Severely Symptomatic Population (ISSP).
In this notebook we consider a DAE’s formulation of SEI2HR-Econ. The stock Total Population has the following (obvious) algebraic equation:
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.
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.
Because we hospitalize the severely infected people only instead of the term
we have the terms
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.
Note that we do not consider in this notebook births, but the births term can be included in SP’s equation:
The births rate is the same as the death rate, but it can be programmatically changed. (See [AAp2].)
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.
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.
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.
Note that we do not assume that severely symptomatic people recover faster if they are hospitalized, only that they have a different death rate.
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:
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.
The stock Recovered Population (RP) is increased by the recovered infected people and decreased by mortality cases derived with the typical mortality rate.
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.
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.)
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, .
The MS producer has limited capacity for delivery, .
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:
Here is the corresponding HMS equation:
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, .
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:
Here is the MS formula with the MS delivery term replaced with “Dlvr”:
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, :
(*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:
Medical Supplies Demand
The stock Medical Supplies Demand (MSD) simply accumulates the MS demand derived from population stocks and their corresponding mscr:
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.
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.
In order to model quarantine scenarios we use piecewise constant functions for the contact rates and .
Remark: Other functions can be used, like, functions derived through some statistical fitting.
Remark: We use the code in this section to do the computations in the section “Sensitivity Analysis”.
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].)
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:
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].
Apply NIntegrate over stocks solution functions.
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:
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:
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
The following mind-map reflects pretty well what was planned and done:
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:
Here is (a snapshot of) an interactive interface for studying and investigating the solution results:
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.)
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):
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.
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 with nodes that correspond to the sites and a positive matrix for the traveling patterns between the sites.
Problem definition: Given (i) a single site epidemiological model , (ii) a graph connecting multiple sites, and (iii) a traveling patterns matrix between the nodes of derive an epidemiological model that simulates more adequately viral decease propagation over .
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.)
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].
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 , a simple graph with two nodes as , and both constant and time-dependent matrices for .
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 , which is a contingency matrix derived from source-destination traveling records. (Or the adjacency matrix of a travelling patterns graph.) The matrix entry of tells us how many people traveled from site to site . We systematically change the ODE’s of the sites in following way.
Assume that site had only travelers coming from site and going to site . Assume that the Total Population (TP) sizes for sites and are and 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
and change into the equation
Remark: In the package [AAp3] the transformations above are done with the more general and robust formula:
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:
Get a single-site epidemiological compartmental model data structure, .
The model data structure has stocks and rates dictionaries, equations, initial conditions, and prescribed rate values; see [AA2, AAp1].
Derive the site-to-site traveling patterns matrix for the sites in the graph .
For each of node of make a copy of the model and denote with .
In general, the models have different initial conditions.
The models can also have different death rates, contact rates, etc.
Combine the models into the scaled model .
Change the equations of , to reflect the traveling patterns matrix .
Join the systems of ODE’s of , into one system of ODE’s.
Set appropriate or desired initial conditions for each of the populations in .
Solve the ODE’s of .
Visualize the results.
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.
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.
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.].
With this interactive interface we see the evolution of all populations across the graph:
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”.
The populations plots at each site resemble the typical plots of SEI2R.
The total population at the first site linearly decreases.
The total population at the last site linearly increases.
The plots of the total populations clearly have gradually increasing gradients from the low index value nodes to the high index value nodes.
For the infected populations there is a clear wave that propagates diagonally from the low index value nodes to the high index value nodes.
In the direction of the general “graph flow.“
The front of the infected populations wave is much steeper (gives “higher contrast”) than the tail.
This should be expected from the single-site SEI2R plots.
For the recovered populations there is a clear “saturation wave” pattern that indicates that the recovered populations change from to values close to the corresponding final total populations.
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.
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:
It is important to calibrate these kind of models with real data, or at least to give a serious attempt to such a calibration. If the calibration is “too hard” or “impossible” that would indicate that the model is not that adequate. (If adequate at all.)
The calibration efforts can be (semi-)automated using special model-to-data goodness of fit functions and a minimization procedure. (See for example, [AA2].)
In this section we just attempt to calibrate SEI2R over real data taken from [WRI1] using a specialized interactive interface.
Here is COVID-19 data taken from [WRI1] for the Chinese province Hubei:
But we have to use a fraction of that population in order to produce good fits. That can be justified with the conjecture that the citizens of Hubei are spread out and it is mostly one city (Wuhan) where the outbreak is.
The real data have to be padded with a certain number of 0’s related to the infectious and incubation periods in order to make good fits. Such padding is easy to justify: if we observe recovered people that means that they passed through the incubation and infectious periods.
Calibration interactive interface
In this interface we put the Infected Severely Symptomatic Population (ISSP) to zero. That way it is easier to compare the real data with the simulated results (and pick parameter values that give close fits.) Also note that since SEI2R is simple in this interface the system is always solved.