Spatial Explicit Model to Visualize the Spread of
Epidemic Disease in a Network
Mohan Timilsina1 ,Raphael Duboz2 and Hideaki Takeda3
1Computer Science and Information Management,
Asian Institute of Technology,
Pathumthani, Thailand
http://www.ait.ac.th
2
CIRAD - UPR Agirs,
Montpellier, France
http://www.cirad.fr
3
National Institute of Informatics,
2-1-2 Hitotsubashi, Chiyoda,
Tokyo, Japan
http://www.nii.ac.jp

Abstract. The development of the visualization tool of any infectious spreading disease requires an elegant graphical user interface which can show theresult of simulation at every time steps. The goal of this work is to provide
an effective way to visualize epidemic diseases, e.g., flu, cholera, influenza,etc. in a network for the decision makers to determine the level of epidemic
threats in any vulnerable zones. The Gravity and Susceptible, Infected andRecovered (SIR) epidemic models are used to draw a network on the map
and simulate the disease status. Thus the results obtained are visualized
to give a clear picture of spreading behavior of a disease. This visualization tool provides an understanding for the epidemical analysis of diseases
through graphs and network statistics.

Key words: Susceptible Infected Recovered, gravity model, visualization
tools, network

1 Introduction
The epidemic of any infectious disease can be better understood by decision makers ifthey are provided with a proper tool to visualize this. If we consider an example for anew outbreak of infectious disease in a particular geographical region it is always crucial for a decision maker to come out with epidemic scenarios. In fact, visualization of
the vulnerable zone plays an important role for understanding populations from serious
epidemic hazards. As a result of simulation of diseases in a network, early visualizationof such threats can be obtained. Hence, a well-developed decision support system withsimulations to identify the vulnerable zone has become a common area of interest in
this field. The growing interest in simulation modeling techniquea has been used in
the context of visualizing epidemic diseases. A review of some existing epidemic disease visualization tools using different technologies follows. GLEAMviz [1] visualization
software has been made to visualize epidemic levels on a worldwide scale. This projectuses the multi scale mobility model from the airport listed in International Air Transport Associations (IATA) database for the disease transmission route. Geographical
Information Systems (GIS) and electronic disease surveillance databases [2] are extensively used to visualize the impact of epidemics of geographical regions. Furthermore,social network analysis [3] is also used for epidemic simulations for disease transmission
in human beings. Although the above mentioned technologies provide epidemic visualization they need huge data sets about diseases and are not flexible to be applied indifferent emerging contagious diseases because they rely on epidemic network data sets
[4]. Hence a decision support system that can provide policy makers with the ability
to create epidemic scenarios in absence of disease routes for an emerging contagious
disease seems to be essential. To address this problem an epidemic visualization systemwith simulation is proposed. The purpose of this tool is to visualize epidemic disease
networks through simulation results. The specific goals are listed as follows:• To use gravity model for epidemic disease network formation;• To use SIR for epidemic modeling; and• To visualize epidemic scenarios on a map.
The epidemic disease network is formed by using the gravity model. This model provides
the rate of disease transmission across distance. The strength of disease transmission isdecreased with the increase in distance Zipf in the 1940s [5] has provided a theoreticalmotivation for movement between cities 1 and 2 being governed by a P1dP2 relationship,
where P is the respective city population and d is separation distance. The strength ofthe movement is affected by the size of the population and their separations. Hence, in
our tool we used distance as an important factor to construct the network. The closer
the nodes the denser is the network. The visualization of epidemic diseases for deci-sion support system using Susceptible(S), Infected(I) and Recovered(R) (SIR) epidemic
model is one of the main components of this tool. It is a good modeling tool for many
infectious disease like small pox[6], influenza and measles, etc.; where every population
can be categorized into three different stages[7].It has three stages and in a population which are prone to disease are in Susceptible stage. Those who carry the disease
and transmit the disease to others are in Infected stage and those who are immune orrecovered from the disease and those removed from the population are in Recovered
stage. In the tool the three stages of the model are depicted in the nodes which inthis case are cities on the map. This tool is intended for general purpose epidemic disease visualization to decision makers at different disease transmission rates, epidemic
distances,simulation timesteps and infection probabilities. The results thus obtained
are visualized on a map. The system uses two dimensional coordinates of cities in the
map to draw a network underlying the principle of gravity model[8].The system is very
flexible to change the network as per the necessity of decision makers, whereas previously developed technologies requires data as a fixed source for the network formation.
Section 2 introduces network topology. Section 3 is about the SIR epidemic model,
Section 4 provides the system description, Section 5 describes the implementation of
the system, Section 6 describes simulation and analysis, Section 7 is about user study
and evaluation, Section 8 discusses about the limitations of the system and Section 9
draws conclusions.

2 Network Topology
The strength of interconnection of two nodes is determined by how closely they arelocated. The affiliation between the cities is the function of the distance and thus the
network is built upon physical proximity between them. The force of epidemics decays
with the increase in distance [9].The spread of any infectious disease can be depictedby human travel pattern between two cities. An analogy with the physics law, it isexpected that disease transmission increases with decrease in geographical distance[10]. The generic model for the gravity is described as:
Fi,j = G

Mi Mj
d2i,j

where G is the gravitational constant: M is the mass of the bodies: i,j and d is the
distance between them. In the context of this work the force of an epidemic is taken
as the link formation between the two nodes, i.e., two nodes will construct the link ifthey fall into the range of the epidemic distance. This means connectivity completelydepends upon closeness of the cities. Fig. 1 visualizes the network of cities with two
different epidemic radius and its degree distribution.

(a)

(b)

Fig. 1. Example of epidemic network . (a) Networks with 70 units epidemic radius. (b)Networks with 100 units epidemic radius.

3 Epidemic Model
The core of the system is based upon epidemiological states of the cities. Hence theconnection among different cities is a vital part of network formation. The transportation network is also an important role for the disease carrier[11].Every connected city
is susceptible to the disease. Those cities which are already infected will either stay
infected or will change the status to be recovered. The spreading process of disease canbe understood as: at each time step the susceptible nodes becomes infected at a rate ofβ; the infected nodes will be recovered at a rate of γ. The transmission of the diseaseβγ

in the nodes is S −→ I −→ R. SIR model is represented by the differential equationas [12][13]dsdidr= −βis,= βis − γi,
= γi
dt
dt
dt
Each state is fraction of population at a particular timestamp. The population is oneof the stages at a particular time. This model is particularly meant for fast spreading
disease and for a constant population. Fig. 2 visualizes the mode of spread of diseasestages.

Fig. 2. SIR epidemic model.
The overall composition of the cities in a map will be of three different kinds of categories: susceptible cities, infected cities and recovered or removed cities.

4 System Description
4.1 Methodology
The system is developed by integrating the Geographical Information System (GIS)
shape files for reading the map, graph theory for analyzing the network and SIR epidemic model for the simulation. The input of the system is epidemic radius, diseasetransmission rate , simulation timestep and infection period. The computation of the
network data is done by the open source boost library4 and for the visualization open
source 2D Cairo graphics5 is used. The system first reads the GPS position of the cities
and locates on a map. Then after getting the input for the epidemic radius it draws
the network on top of it. The user can then select any city on a map as an epidemic
outbreak and provide the transmission probability[14] and simulation timestep and cansee the epidemic visualization on a map.

4.2 Simulation ModelIn the SIR epidemic model the cities on the map are classified according to their states:
susceptible(S), infected(I) and recovered(R).The epidemic spreads through the links between the cities. The spreading process of disease can be understood as: At time (t=0)
the node infects its neighboring susceptible nodes with the random probability(p) lessthan the transmission probability(tp). At time (t=1) the neighbor node gets infection
and at time (t=2) the initial node which has transmitted the disease passes the infectious period and becomes immune or recovered in the system. The overall scenario canbe viewed in the figure below:
45

http://www.boost.org/
http://www.cairographics.org/

At time = 0
p< tp

At time = 1
SAt time = 2
I

I

I

I

R

S

S

S

(a)

(b)

(c)

Fig. 3. S,I and R are the Susceptible, Infected and Recovered nodes of the disease network. (a) Infected nodes transmits the disease. (b) Susceptible nodes receive infection.
(c) Recovered nodes recover from infection.

Pseudocode for the infection and recovered statusprogram Infective and Recovered nodes (Output){Get the transmission probablity,infectious period from a user
For every nodes which is infected state:=1};
varnodes: 0..maxNodes;
statusVector;
timeVector;
transmissionProbablity,randomProbablity: Real;
begin
nodes := 0;
statusVector:= 0;
timeVector:= 0;

repeatrandomProbablity:= rand() / (RAND_MAX+1))*-1;
if(randomProbablity<transmissionProbablity){
statusVector:=1; //infects the randomly selected outdegree nodes
timeVector :=timeVector+1;}if(timeVector>infectiousPeriod &&statusVector==1){
statusVector:=2; //Nodes will be recovered or removed}
nodes:= nodes+1;
until nodes = maxNodes;
end.
The model works undertaking two major widely used components in epidemiology
called as infectious period to produce the symptoms of disease in the nodes and transmission probability to transfer the disease from infected to the susceptible nodes. Thus
the epidemic visualization system, that enhanced by this model, will have sufficient
information to simulate the disease and create epidemic scenario in a disease network.

5 Implementation

The proposed system is made which allows the user to enter the epidemic radius,transmission probability, infectious period of the disease and simulation time steps.
The network is then drawn by calculating the Euclidean distance between the pairsof the cities. From,the whole cities user can select any of them as an outbreak of theepidemic and run the simulations and the result is dynamically visualized on the map
for every timestep. At the same time the user can also see the graph density, degree
distribution and epidemic curves in the interface. Figure below is the User interface ofthe prototype.

Fig. 4. Screenshot of the epidemic disease visualization of the proposed system.

6 Simulations and Analysis

(a)

(b)

(c)

Fig. 5. SIR Simulation result for 60 unit epidemic radius

In figure(a), Susceptible curve decreases rapidly with the increase in transmission
probability, with the increase in simulation timesteps the curve starts to decrease andafter some period the curve flattens out. In figure(b), Infected curve increases rapidly
with the increase in transmission probability and reaches the peak of the infection, as
the number of simulation timeteps increases the infection curve also falls down rapidly.
It starts to decrease and finally the curve reaches to minimum 0 and flattens out where
no infection is seen. Similarly in figure(c) the recovery node is only seen after the
infectious period of the outbreak in this case the infectious period of the disease is fourdays. It is observed from the figure(c) that the recovery curve also increases rapidly
with the increase in transmission probability and after some period the curve reaches
to the peak and becomes constant.

7 User Study and Evaluation
Users of this prototype system are only few members including the developer and wehave not verified results yet. However we think this prototype will be able to create
the epidemic scenarios for decision makers without explicitly requiring the network
construction data. Apart from the user study the evaluation of this prototype has beendone with the existing visualization tool Epigrass6 and Gleamviz7 . The results of theevaluation follows:
1. Ability to visualize SIR epidemic disease: The developed prototype fully
supports this feature by asking the user to set up their parameters in the tools.
GleamViz and Epigrass can also support this feature moreover they can be modeled for all types of epidemic disease.
2. Ability to visualize the disease in a local scale: The developed prototype andEpigrass fully supports this feature by prompting the user to load their specific GIS
shape files and to run the simulations whereas GleamViz provides the global levelepidemic visualization.
3. Ability to create epidemic scenarios: All of these tools support this feature.
4. Ability to create the disease network with the Euclidean distance between the nodes: This is an interesting feature of the prototype where the network is drawn underlying straight line distance between the two nodes. Due to thisreason the prototype does not require the network construction data else it only
needs the epidemic radius from the user. Whereas Gleamviz requires multi scale
mobility model of the International Air Transport Association (IATA) databasefor the network formation and Epigrass uses the transportation and passenger flow
data for the network formation. To the best of our knowledge in the absence ofthe network data in GleamViz and Epigrass both cannot simulate and visualize theepidemic disease.
7.1 Comparision of the prototype with the existing visualization tools
Table 1 provides a comparision of the prototype with the GLEAMViz and EpiGrassvisualization tools.

67

http://sourceforge.net/projects/epigrass/
http://www.gleamviz.org/

Table 1.
Criteria

Prototype

GLEAMviz

EpiGrass

Visualization of spatial data Yes

Yes

Yes

Visualization of models

Yes

Yes

Yes

Visualization of simulated Yes

Yes

Yes

results
Types of epidemic disease

Contagious

Every kind of epi- Every kind of

acute infectious demic disease

epidemic diseasediseaseData for disease network No network data IATA databases

Transportation

formationandrequirement

passenger

flow data
Software modularity and Yes

Client server based Yes

extensibility

online tool

The strength of the prototype is to draw a network for the disease simulation in theabsence of the network data which is not present in either of the two systems. The
prototype calculates the Euclidean distance between the nodes within the epidemic
radius and draws the network on the map. Currently the prototype is only applicablefor the visualization of acute infectious disease which has a very short exposed period.
Table 1 compares the important feature of the popular epidemic visualization tools
with the prototype.

8 Limitations
The prototype is developed using SIR epidemic model. This model also has shortcomings because it does not consider the immigrants and emmigrants of the population
and assumes the total population as constant [15]. In the prototype we have cities andassume the population is well mixed and the same in all parts of the country which iscompletely impossible in a real situation. In the developed prototype the latent period
for acquisition of infection and the start of infectiousness has not been addressed due
to which the chronic infectious disease cannot be modeled. The diseases with latent
periods that can be modeled using the extensible SIR model with latent period called
SEIR epidemic model cannot be used by this prototype. However the prototype can be
useful for the decision makers to observe and create the scenarios for the acute infectious disease having a very short exposed period like influenza, chicken pox, distemper.

etc. In the current version of the system we can only address the SIR model but inour upcoming version of the system we will also be able to provide the flexiblility tochange the infectious disease model like SEIR, MSIR, SIS and SIQR to visualize theepidemic spread of the disease.

9 Conclusion and Future Work
This research proposes the visualization of the epidemic disease using a simple epidemic
and gravitational model. To accomplish this a prototype is developed without explicitly
requiring predefined network construction data. The epidemic spread as a result of
simulation is visualized in the map. The usage of graph theory has become very useful
to visualize the epidemic network, degree distribution and finding the central nodes
in the network. The developed prototype can visualize the states of every node at
the given time stamp. This tool is very flexible for visualizing SIR modeled infectious
disease for a given transmission rate and epidemic radius. Currently, this prototype
cannot address all the epidemic diseases. In the current context we also lack the real
epidemic data to test with our system generated results. Due to this reason we cannot
ensure that the prototype can be very accurate with its simulated results. Howeverin the upcoming version the prototype will come out with tested real data and also
be capable of visualization for other existing epidemic models. Therefore the proposed
visualization approach is enhanced in the further development of the tool.

References
1. Broeck, W.V.d., Gioannini, C., Goncalves, B., Quaggiotto, M.,Colizza, V., : The
GLEaMviz computational tool, a publicly available software to explore realistic
epidemic spreading scenarios at the global scale J. BMC. Infectious.Diseases. 11,37(2011)
2. Burns, J., Hatt, C., Brooks, C., Keefauver, E., Wells, E.V., Shuchman, R., Wilson,
M.L.,: Visualization and Simulation of Disease Outbreaks: Spatially-Explicit Applications Using Disease Surveillance Data. In: 26th Annual Esri International User
Conference, Redlands,CA (2006)
3. Bisong, H., Jianhua, G.,: Simulation of Epidemic Spread in Social Network. In:
Management and Service Science, International Conference , Wuhan, China (2009)
4. Chen, H., Zeng, D.,: AI for Global Disease Surveillance J. Intelligent Systems, IEEE.
24, no.6, 66-82(2009)
5. Zipf GK.,: The P1 P2/D Hypothesis: On the Intercity Movement of Persons. Am
Sociol Rev 11: 677-686 (1946)

6. Ferguson, N. M., Keeling, M. J., Edmunds, W. J., Gani, R., Grenfell, B. T., Anderson, R. M., Leach, S.,: Planning for smallpox outbreaks. Nature, 425(6959), 681-685(2003).
7. Antulov, N., LančIć, A., ŠTefančIć, H., ŠIkić, M.,: FastSIR algorithm: A fast algorithm for the simulation of the epidemic spread in large networks by using thesusceptible-infected-recovered compartment model, Inf. Sci. 239 226-240(August
2013)
8. Eggo, M.R., Cauchemez, S., Ferguson, N.M.,: Spatial dynamics of the 1918 influenza
pandemic in England, Wales and the United States J. J.R.Soc.Interface, 8, 233243(2011)
9. Barthélemy, M.,: Spatial networks, Physics Reports, 499:1101,(2011)
10. Sarzynska, M., Udiani, O., Zhang, N.,: A study of gravity-linked metapopulation
models for the spatial spread of dengue fever, CORD Conference Proceedings (August 2013)
11. Zhang, Y., Zhang, Yo., Liu, Z.,: The role of different transportation in the spreadingof new pandemic influenza in mainland China. In: Geoinformatics, International
Conference , Beijing, China (2011)
12. Newman, M. E.: Spread of epidemic disease on networks. Physical review E, 66(1),016128 (2002).
13. Hethcote, H. W. :The mathematics of infectious diseases. SIAM review, 42(4),599-653 (2000)
14. Meyers, L. A., Pourbohloul, B., Newman, M. E., Skowronski, D. M., Brunham, R.
C.,: Network theory and SARS: predicting outbreak diversity. Journal of theoreticalbiology, 232(1), 71-81 (2005).
15. Ahmed, E., Agiza, H. N.,:On modeling epidemics including latency, incubationand variable susceptibility. Physica A: Statistical Mechanics and its Applications,253(1), 347-352 (1998).