Spatial modeling of an epidemic using differential equations with parameters Zixin He Professor Ivan T. Ivanov Vanier College Abstract In this paper, the classical SIR model has been expanded to accommodate the modeling of additional characteristics of an outbreak. A new population subgroup has been added to account for the effects of a latent period on the infectious cycle. New parameters have also been introduced to model for an imperfect post-recovery immunity acquisition, temporary immunity period and global immunization efforts. This new model was then expanded for computation on a population grid. Cells are linked through a proximity effect algorithm and the effect of air travel through major transportation hubs is predicted. Incorporating these long distance connections allows for an accurate depiction of the spread of most infectious diseases if left uncontrolled. When compared the current Zika epidemic, the model proves to be reliable with the correct set of parameters.
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Spatial modeling of an epidemic using
differential equations with parameters
Zixin He
Professor Ivan T. Ivanov
Vanier College
Abstract
In this paper, the classical SIR model has been expanded to accommodate the modeling
of additional characteristics of an outbreak. A new population subgroup has been added
to account for the effects of a latent period on the infectious cycle. New parameters have
also been introduced to model for an imperfect post-recovery immunity acquisition,
temporary immunity period and global immunization efforts. This new model was then
expanded for computation on a population grid. Cells are linked through a proximity effect
algorithm and the effect of air travel through major transportation hubs is predicted.
Incorporating these long distance connections allows for an accurate depiction of the
spread of most infectious diseases if left uncontrolled. When compared the current Zika
epidemic, the model proves to be reliable with the correct set of parameters.
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Table of Contents Abstract ........................................................................................................................... 1
The connectivity factor calculates the ratio of daily passenger traffic compared to the
population at the start of the computation and uses it to determine the fraction of infected
that gets added to the pool of infected possibly travelling between the modeled cities. This
pool of infected individuals is then redistributed according to relative air passenger volume
compared to the total air passenger volume. These individuals would then be added to
the density of infected for the city’s corresponding cell. To ease calculations, the current
model only considers cities with a recorded annual passenger volume above 50 million in
2015.
Figure 1 – Color map of the proximity values for computing the effect of neighboring cells
of the center cell. This is a grid of 11 values by 11 values.
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Table 1 - List of cities modeled with their geographical coordinate and their annual air
passenger volume in 2015. Taken from Wikipedia since all other reputable and organised
sources require a large access fee.
City Name Coordinates Volume London 51.5074° N, 0.1278° W 153,487,957 New York 40.7128° N, 74.0059° W 126,651,526 Tokyo 35.6895° N, 139.6917° E 110,070,000 Atlanta 33.7490° N, 84.3880° W 101,491,106 Paris 48.8566° N, 2.3522° E 99,761,998 Shanghai 31.2304° N, 121.4737° E 99,189,000 Chicago 41.8781° N, 87.6298° W 99,170,835 Beijing 39.9042° N, 116.4074° E 96,130,390 Los Angeles 34.0522° N, 118.2437° W 95,793,140 Istanbul 41.0082° N, 28.9784° E 89,435,167 Dubai 25.2048° N, 55.2708° E 88,935,250 Bangkok 13.7563° N, 100.5018° E 83,206,293 Dallas 32.7767° N, 96.7970° W 78,671,661 Miami 25.7617° N, 80.1918° W 77,557,288 Moscow 55.7558° N, 37.6173° E 75,285,463 Seoul 37.5665° N, 126.9780° E 72,445,198 San Francisco 37.7749° N, 122.4194° W 71,014,239 Sao Paulo 23.5505° S, 46.6333° W 68,715,919 Hong Kong 22.3964° N, 114.1095° E 68,488,000 Washington 38.9072° N, 77.0369° W 66,669,853 Frankfurt 50.1109° N, 8.6821° E 63,697,127 Amsterdam 52.3702° N, 4.8952° E 58,284,864 Jakarta 6.1745° S, 106.8227° E 57,000,000 Singapore 1.3521° N, 103.8198° E 55,448,964 Guangzhou 23.1291° N, 113.2644° E 55,208,000 Denver 39.7392° N, 104.9903° W 54,014,502 Houston 29.7604° N, 95.3698° W 53,414,359 Kuala Lumpur 3.1390° N, 101.6869° E 51,678,211
Solving the System Using Euler’s Method
Given the strong interdependencies and the large number of variables within the system,
it would be a waste of time to try and obtain an analytical answer. The relationships
between neighboring cells only increases the challenge to obtaining exact and analytic
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solutions to the system. The most obvious way is to solve the initial value problem using
a numerical method. Time allowing, higher order methods can be used. For the moment,
Euler’s method proves sufficient in solving the system.
Euler’s method is a numerical process that incrementally computes the rate of change
from a previous step and adds an increment of it to the values in the previous step to
obtain the values of the current step. Since each step is an approximation based on the
previous step, this method inherently leads to runaway errors as the calculations progress
forward. Luckily, as computerized systems can be programmed to do very rapidly the
repetitive calculations, they allow for a large number of minuscule steps, minimizing the
error introduced at each step of calculation.
The general idea of Euler’s method is to discretize the integrand in an integral. Instead of
looking for a continuous function, it computes a series of points that allow for an estimation
for the shape of the function. However, the faster a function changes, the faster the
approximation becomes off. This method may also jump over discontinuities such that
special arrangements are required for discontinuous functions. The general form of
Euler’s method is as follows:
𝑓𝑓(𝑡𝑡𝑛𝑛+1) = 𝑓𝑓(𝑡𝑡𝑛𝑛) + 𝑓𝑓′(𝑡𝑡𝑛𝑛) × 𝑑𝑑𝑡𝑡 (13)
The 𝑃𝑃 is the step number and the 𝑑𝑑𝑡𝑡 is the width of each step. Given a duration 𝐷𝐷 and a
𝑃𝑃 number of steps to compute, 𝑑𝑑𝑡𝑡 is determined as follows:
𝑑𝑑𝑡𝑡 = 𝐷𝐷𝑛𝑛. (14)
When applied to the current SEIR system, it results the following set of equations for a
The recent outbreak of the Zika virus can serve as an example to demonstrate the
effectiveness of the current modeling method.
Background
The Zika virus has been first observed in Uganda back in 1947. Since then multiple
outbreaks have been reported in Africa, South-East Asia and Polynesia. The most recent
outbreak started in Brazil in late 2015. It has since spread to 43 countries and cases are
still multiplying. The virus is of the genus Flavivirus, making it similar to the more well-
known dengue, West Nile and yellow fever viruses.
Infection by the Zika virus produces a mild fever and followed by back pain. Some report
a maculopapular rash developing mostly on the face, neck, trunk, and upper arms. It
occasionally causes diarrhea and abdominal discomfort. There are no recoded cases of
death due to the Zika virus. Immunization against the disease after infection is unclear,
but there are observations that the virus may be able to survive in the male testes for
more than two months after the original infection.
Until recently, this virus is little known as it has never been behind a large outbreak. The
most important known threat from Zika has only been established recently with a strong
increase in the number of births with microcephaly in Brazil. There is strong evidence of
a link between the Zika virus and microcephaly in fetuses whose mother have contracted
the Zika virus at some point during the pregnancy. The mechanism leading to
microcephaly is still unknown and there are no vaccines nor treatments specifically
targeting the Zika virus.
Observed Trends of the Epidemic
Up to this point, data sources are unsure about how many people are infected by the Zika
virus due to the mild and often absent symptoms of an infection. The most predominant
sources of information are the Centers for Disease Control and Prevention in the US and
the World Health Organization. The following two figures show the spread of the Zika
virus one year after what is believed to be the start of the outbreak.
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Figure 3 – States in the US with reported cases of Zika. May 2016 data from the CDC.
As of May 11, 2016, there are a total of 507 laboratory-confirmed, travel acquired cases
in the continental US (Centers for Disease Control and Prevention, 2016).
Figure 4 - All Countries and Territories with Active Zika Virus Transmission as of May 12,
2016. From the CDC (Centers for Disease Control and Prevention, 2016).
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The Pan American Health Organization also has a data set on reported and estimated
cases of Zika in central and south American countries. While the cases of new infections
seem to have largely slowed down, the data is constituted of fragments that may not
depict the whole picture. For example, Brazil only has data on confirmed cases
discontinuously up to February. There are no estimated numbers for Brazil, which makes
this graph less reliable especially at more recent time periods. However, recent
containment efforts have indeed slowed the spread of the Zika virus.
Figure 5 – Epicurve from the Pan American Health Organization (Pan American Health
Organization / World Health Organization, 2016).
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Predicted Trends by the SEIR Model
The available data can be compared to the data generated by the SEIR model elaborated
in this paper. The initial values for the simulation are as follows.
System constants Value Description
𝛼𝛼 0.00002943 Daily natural population growth constant
𝛽𝛽 0.6 Effective contact rate
𝛾𝛾 0.1 Inverse of infectious period in days
𝛿𝛿 0 Ratio for immunization after infection
𝜖𝜖 0.5 Inverse of latent period in days
𝜅𝜅 0 Inverse of immunized period in days
𝜇𝜇 0 Death rate from epidemic disease
Given the uncertain information about immunization after an infection, the ratio of
immunization post-recovery is presumed to be null. The average latent period for the
disease is estimated to be 2 days and the infectious period, 10 days. The initial infected
population was set to 50 originated within Brazil.
Figure 6 – Progression of the Simulated Epidemic after One Year. The scale is logarithmic.
The simulation did predict a few cases on the continental US and Europe. The Central
Americas were also affected in good measure. The infections did not spread to null
population areas of the Amazon, which was also expected.
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Comparison
Overall, the simulation is mostly on point. A closer look at the numbers of infected in each
country show that the model is relatively accurate. The number of infected in Brazil one
year after the outbreak was estimated to be 120 thousand individuals. The simulation
model shows an infected population in roughly the same range. The cases in the United
States also corresponds to the predictions of the simulation. The cases dotting the East
and West Coasts of the US is represented accurately in the simulation at roughly the
same time frame. The simulation shows a few hundred cases scattered across Europe
which matches with predictions and observed trends (European Centre for Disease
Prevention and Control, 2016).
However, two issues put the accuracy of the model into question. The first is that the Zika
virus is mostly transmitted by mosquitoes carrying the virus and human-to-human
transmission is limited. The simulation model might not depict accurately the underlying
endemic nature of the outbreak. The second issue is that the air travel model does not
respect the travel preferences of passengers from different continents nor the seasonal
travel peaks. A spike in reported cases in the US during the school and holiday breaks
was also not represented. Fortunately, these issues are minor events within the simulated
time frame and does not affect the overall outcome of the predictions.
Conclusion
The modeling of an epidemic spread allows healthcare providers to distinguish more at-
risk regions and develop strategies to limit the effect of an outbreak by allocating
appropriate resources before the outbreak spreads. While accurate, the current model is
insufficient for accurate prediction of the spread of an outbreak through air travel.
Including additional cities will improve the representation of the efficacy of air travel in
transporting patients in different population dense locations of the world. Incorporating
data on the most traveled routes will eliminate the errors induced by uniformly distributing
infected individuals across the network of high air traffic cells.
As for the original question about the likelihood of a widespread pandemic affect the
globe, the simulation certainly shows the inherent capability of air travel to spread
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diseases to many areas of the world rapidly. In the end, it all depends on how the disease
affect individuals and how it transmits. As common sense would likely tell you, the easier
the disease spreads, the harder it is to contain it. A long latent period or an infectious
period presenting little symptoms will allow the outbreak to spread very rapidly before
authorities can react. In general, as past examples have already demonstrated, air travel
and increased mobility greatly increases the risk of spreading diseases around the globe.
In the end, it seems that living in an isolated island may not be such a bad idea. The
isolation greatly reduces the risk of exposure to an epidemic outbreak. However, there
are only a limited number of isolated islands on the globe. With the difficulty of being self-
sustaining and the rising sea levels, getting sick from a pandemic disease will be the least
of your troubles if you do resettle to a small island in the Pacific.
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