Pandemic Dynamics with Social Effects:
Rapid Model Prototyping with Fuzzy Logic
Tsan Sheng Ng’,
* Department of Industrial and Systems Engineering, National University of Singapore,
Faculty of Engineering, 10 Kent Ridge Crescent, Singapore 119260
isentsa@nus.edusg
Shao Wei Lam**
** Department of Decision Sciences, National University of Singapore
NUS Business School, 10 Kent Ridge Crescent, Singapore 119260
lamshaowei@gnail.com
Mong Soon Sim*,
* DSO National Laboratories, 20 Science Park Drive, Singapore 118230
smongsoo@dso.org.sg
Abstract
The human behavior aspect of pandemic prevention and mitigation involve uncertainties
manifested as a range of responses, from the extreme to the indifferent. Relationships
between variables influencing human behavior are usually described qualitatively, and as
such do not suffice for stock and flow models. These uncertainties can slow down the
modelling process considerably, thus limiting the effectiveness of a model-based
approach in time-critical studies such as an impending pandemic outbreak. Our proposed.
approach utilizes fuzzy modelling concepts integrated within the system dynamics
modelling framework to create a rapid model prototyping process of developing a
pandemic dynamics model. This can facilitate quantitative analysis for policy making in
pandemic mitigation interventions. We use the recent H1N1 pandemic in Singapore as a
case example to demonstrate the practical usefulness of our approach.
Keywords: pandemic dynamics, social effects, system dynamics, fuzzy logic
Introduction
In the past decade, we have witnessed the impact of pandemic on our modem society
(Keogh-Brown and Smith, 2008). As an illustration, the 2003 Severe Acute Respiratory
Syndrome (SARS) outbreak affected a number of countries and the total economic loss
can amount to at least a few US$ billion. Apart from economic impact, the pandemic can
trigger other risks pertaining to the operation of critical infrastructures. If the effect of
pandemic is not checked, it will propagate to a national crisis of a much larger scale,
where human resources are removed from all sectors including critical infrastructure and
essential services. When this happens, the daily functioning of a society may be severely
affected.
To counter the spread of pandemic in a community, there are various strategies employed.
by the govemment (Ferguson e al 2005). Usually a combination of prophylaxis and.
social distancing measures is proved effective to contain the spread of disease (Ooi et al
2005). During the 2003 SARS outbreak in Singapore, the govemment adopted two main
strategies (Le. early detection and isolation of all cases and quarantine of all close
contacts of symptomatic cases) and it helped to break the chain of transmission. In
addition, Singapore's national plan for pandemic response makes reference to mitigating
the effect of the first pandemic wave through securing the co-operation of the general
public (Singapore Ministry of Home Affairs, 009). Thi This is to be achieved by impressing
the need for each individual to have a sense of collective responsibility in detecting and.
preventing the spread of flu. The public will be educated and expected to practice
improved personal hygiene and adopt socially responsible behavior.
In general, the pandemic response strategy in Singapore is founded on the key
observation that pandemic dynamics is significantly dependent on the rich interplay
between the dynamics of pathogen transmission and the structure and behaviors of social
responses. Given the high population density in Singapore (7,022/kn’) (Singapore
Department of Statistics, 2009), which is perhaps one of the most densely populated
country in the world (United Nations Department of Economic and Social Affairs, 2009),
there has to be a more significant consideration of social effects in understanding
pandemic dynamics.
In the literature, various attempts have been made to use large-scale population dynamics
models to evaluate the socio-economic impacts of a pandemic and the effectiveness of
various mitigation strategies. Ewers and Dauelsberg (2007) integrated an industrial
system model to evaluate the impact of an outbreak on labor, sales and economic
performance. Lant et al. (2008) used a hierarchical system dynamics modelling approach.
to simulate the execution of the pandemic preparedness plan in a public university. Other
than influenza, Ritchie and Galvan (1999) studied the effects of strategies such
fumigation, implementation of larvacide programs and education in a dengue fever
outbreak in Mexico. These studies however, remain as open-loop analysis. In a small
country such as Singapore, it is postulated that closed-loop effects can become significant
and thus should be appropriately accounted for in a system model.
In this study, we will propose an approach to investigate how societal responses may be
incorporated as a feedback in a pandemic model. Considering the nature of pandemic
transmission, such an approach has to inevitably account for the difficulty in obtaining
quality data for the modeling of pandemic dynamics in traditional statistical pandemic
models. The rapidity of pandemic transmission is of more significant concem to
Singapore considering its high population density. In fact, we lack the luxury of time to
collect quality data for intricate statistical modeling of pandemic dynamics especially for
a model that considers social effects. Hence, a rapid prototyping approach based on
fuzzy-modeling integrated within system dynamics modeling framework would be
proposed to deal with the time-compression impact on data availability for modeling
pandemic dynamics with social effects.
The structure of this paper is as follows. The basic Susceptible-Exposed-Infectious-
Recovered (SEIR) model most commonly used for influenza modelling is first discussed.
In order to incorporate social effects and considering the need to rapidly develop useful
pandemic dynamics model under the dual constraints of data insufficiency and time
pressures, the fSEIR model, incorporating fuzzy modelling concepts with a system
dynamics model, is proposed. A simple case study is described to demonstrate the
improvement of the fSEIR model for the predicting of the evolution of pandemic over
time using real data from the Singapore govemment. The study is still ongoing. This
paper will focus on the Influenza A (H1N1) virus and the model described in this paper is
a preliminary one.
Pandemic Dynamics Models
SEIR model
A compartmental model (Ma and Li, 2007) is typically used for studying pandemics. Of
such models, the Susceptible-Exposed-Infectious- Recovered (SEIR) model is most
commonly used for influenza pandemic modelling. The SEIR model is essentially a set of
ordinary differential equations which are solved to derive the dynamic behaviour of the
system over time. One of the advantages of the SEIR model is that it has few variables
which can be determined relatively quickly by experts. State variables include the stock
of susceptible and infectious population, and rate-related parameters include the
probability of viral transmission and the “infectivity” of the cases over time. In order to
consider biological, social and environmental effects on infectiousness, infectivity can be
decomposed into these 3 effects respectively. For example, it can be decomposed into a
product of biological infectiousness and contact rates, thus allowing the assessment of
interventions aimed to mitigate the social effects on infectiousness.
We first conducted a preliminary investigation on using the basic SEIR model for
predicting the evolution of HIN1 pandemic in Singapore. In the SEIR model, the driver
of the pandemic is the rate of infection within the susceptible population. This rate of
infection is determined by the probability of transmission, the size of the susceptible
population (“susceptible” in the model) and the number of infectious people in the system.
The probability of transmission (“Beta” in the model) can incorporate more complex real-
world considerations, such as the impact of human behaviours, to produce a more
representative model. After infection, the susceptible population moves into the exposed
stock where the disease will undergo an incubation period before manifesting symptoms.
When this happens, the person joins the infectious stock, where he has a chance to infect
other members of the susceptible population in the system, before he recovers and joins
the recovered stock. The SEIR model can be summarized in the Stock and Flow model
shown in Figure 1.
bela ~~ Pe cases
\t+ +
Susceptible { — | Exposed b Infectious t —=> Recovered
latent =, Recovery
Period (2 Period
days) (4 days)
Figure 1: A SEIR model using stock and flow diagram
We developed a basic SEIR system dynamics model for the H1N1 pandemic cases in
Singapore. Parameters of the virus gathered from official WHO sources as shown in
Figure 2. It is clear that the “Number of infectious” produced from the model and the
“Reported number of infected people” from the MOH press Teleases differ greatly. The
SEIR model alone is unable to replicate past data satisfactorily’. Since the pandemic
occurred while the pandemic preparedness plan was in effect, we hypothesise that the
pandemic preparations have had an effect on the probability of transmission. This
provides some evidence that the SEIR model may not be adequate when social effects are
significant but not considered.
RMSE-306
Figure 2: An example of poor replication of the SEIR model
fSEIR - Fuzzy-Based Rapid Model Prototyping
The premise of our work is to develop a more representative model of the system by
integrating the influence of human behaviour to the probability of transmission over the
progression of the pandemic. The dynamic hypothesis is shown in Figure 3. In the actual
system, the population behaviour responds to the spread of the pandemic. In particular,
1 Ttis possible that with more data training, the SEIR model can replicate the actual number of infected
people with better accuracy. However, our study focus aims to look at incorporating other societal factor
that is not taken into consideration in SEIR model.
the rise in the stock of infectious people sets up three balancing loops which act to reduce
the beta value and thus reduce the number of susceptible getting infected. DORSCON
which impacts the temperature regime compliance among the population is a specific
policy measure in Singapore. It is the acronym for “Disease Outbreak Response System’
and it lists out the pandemic responses that Singapore will take as a nation when
threatened by a pandemic flu or infectious agent. The different levels of DORSCON are
green, yellow, orange, red and black. The different policy levers within each threat level
can have varying structural impact on the probability of transmission. Furthermore, the
frequency of temperature regime, social distancing and better hygiene practices rises as
the rise in the number of infectious people within the population is telegraphed by the
media attention paid to it. The process is not instantaneous, as delays exist in the system
due to perception and reporting delays. Some mitigation measures are easier to adopt
while others take longer to come into effect.
+
Temperature
: * DORSCON
45) regime
ee +
q ty iW
Social 4. oo Media \. WHO
Distancing attention alert level
B v +
| Hygiene
Practice
“Ww
Beta — : t) Import
“ : —__ / cases
V+
Qt
¥ Exposed +> Infectious + Recovered
Susceptible
Latent _/ Recovery _-
Period Period
Figure 3: Integrating societal response in SEIR model
In the modelling process, implementing changing aspects of human behaviour in
response to stimuli poses a quantitative dilemma. There is no universally accepted way of
quantifying these relationships. However, we do have an idea of what the policy makers’
mental model of the system is from the pandemic preparedness plan. In view of this, a
fuzzy-logic based SD modelling approach (Ng et al, 2009) is proposed to mitigate the
difficulties of handling the issues of incomplete structural information of the system
Fuzzy numbers (Tanaka and Niimura 1996, Bojadziev and Bojadziev, 2007) provide the
most natural interface for modellers to incorporate linguistic expert knowledge into a
quantitative model. Fuzzy logic has been used extensively in modelling qualitative
variables such as those that may arise from the modelling of social response during
pandemics. During the early stages of model conception, knowledge sharing sessions
with those who are familiar with the system may take the form of more qualitative
descriptions on how the system works and the variables that are involved Thus,
implementing a framework to formulate SEIR models with support for linguistic
variables using fuzzy logic will allow for a quicker modelling process, particularly when.
social effects are considered. We term the fuzzy pandemic dynamics model as the fSEIR.
We translate the limited information that has been gathered from public domain resources
into a form amenable to simulation using the fuzzy logic tool (Bojadziev and Bojadziev,
2007). This is represented in the block diagram shown in Figure 4, with the inputs and
outputs of each of the various fuzzy logic blocks displayed.
Temperature Regime
DORSCON Compliance among
population %of
population
Probability of
Transmission of
Infection
Number of Social Distancing/ Self No. of
advertisements \/ Quarantine contacts/day
% of population % of
Number of practicinglmproved population
H1N1 cases Hygiene
reported
Figure 4: Factor effects on probability of transmission
Each of the blocks in Figure 4 represents a single fuzzy logic controller that consists of a
tules base and membership functions for each of the inputs and outputs. Most of the
blocks are self explanatory. The rules base in the logic structure shown in Figure 4 is a
collection of IF-THEN rules which capture how the variables are related to each other
qualitatively. It attempts to capture the mental model of the system as is held by the
domain experts. Relationships between the variables are not the only aspect which can be
qualitative. The variables themselves can be described in qualitative tems. The
membership functions which represent these variables map the range of values the
variables can take into levels such as HIGH, LOW, MANY or FEW etc. Each variable
has its own unique membership functions, with as many levels as appropriate.
The rules and membership functions can be gleaned from sources such as reports, expert
knowledge, intuition, or data mining. The last piece of the puzzle is the fuzzy inference
method which translates the qualitative descriptions used in the construction of the fuzzy
logic controller into quantitative output which can is used in simulation. The Mamdani-
Sugeno method (Tanaka and Niimura 1996) has been used in this model.
The fuzzy logic blocks are incorporated into the structure of the SEIR model is shown
below in Figure 5.
SS
Figure 5. Structure Diagram of the fSEIR Pandemic Model
Preliminary Results
The incorporation of AI tools into the model produces a better match for the initial stages
of the pandemic spread (see Figure 6). Compared to the original SETR model, the model
output is not seen to diverge wildly as time passes, lending credibility to the results of the
model. Using AI tools has allowed us to make use of what limited quantitative and
qualitative information is available and still be able to construct a model that mimics the
behavior of the actual system.
4500
RMSE:79
4000 =
3500
Figure 6: Integrating societal response in SEIR model
Without access to domain experts, the only sources available to us are journal articles and.
govemment releases. Despite this limitation, sufficient knowledge can still be leamt
about a system from these sources to allow the formulation of an fSEIR model. In
situations where rapid prototyping is required, or data is unavailable or in a qualitative
form, the fSEIR model can be used to quickly bridge the gap between a causal loop
diagram and a stock and flow model useful for simulations. With better information about
the system available, the model can be further refined and improved. For example, data
on the number of infected residents over a longer time period will provide for better
calibration of the model parameters. The Fuzzy Logic controllers can also be improved
by incorporating empirical data. Specifically, the fuzzy logic rules and linguistic
variables and memberships can be better calibrated through surveys on public reaction to
DORSCON and media attention, as well as SME’s domain knowledge on DORSCON
determination policies. The structure of the human behaviour system can be further
improved with the input of domain experts as well.
Comparing the infection rates over time for the SEIR and fSEIR models (Figure 7), the
peak of the pandemic has been observed to be delayed by about 1 month and lowered by
approximately half. Using this result instead of the pure SEIR model can mean, for
example, that policy makers can allocate lesser beds for pandemic purposes, which
allows for lesser disruptions to nonal hospital operations and the related economic costs.
Stocks of anti-virals and prophylaxis that have to be maintained can be suitable adjusted
as well. The time gained can be used to gauge how much time is available before the
pandemic peaks, and how long the measures have to be continued before they are
rescinded.
SEIRSD Model
SSEIRSD Model
oO 20 40 60 80 100 «120 «140 160 180 200
Figure 7: Comparison of infection rates over time for SEIRSD and sSEIRSD models
The predictions can be improved by estimating statistical prediction or confidence
intervals for pertinent measures through simulation. The pandemic peak value and time
involved, the number of infectious people expected can all fall within a certain range that
reflects the uncertainty inherent in the system. Using this information, policy makers can
make better and more optimal decisions for the well being of society.
Conclusion and Future Direction
In this paper we developed the dynamic systems model of a pandemic attack with
inclusion of the feedback effects of social response on the disease infectivity. This closes
the pandemic-socio-behavioral loop and in particular is of relevance to a closely knit
society such as Singapore. Such a model serves as a valuable decision-support to
healthcare policy-makers looking to evaluate the dynamic impacts of various pandemic
mitigation instruments. Furthermore, our proposed fSEIR model and approach enables
the rapid prototyping of system models for effective deployment under time-critical
situations. Ongoing work includes model calibration with medical domain experts,
disaggregation of the susceptible into different behavior groups, and optimizing budget
allocations in pandemic preparedness accounting for social effects.
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