Computing the SIR Model: Fractal dimensionleads to vaccination strategies

Ben O’Donnell, Brandon McPhail, Duy Tran, Patrick Carlisle

September 13, 2004

Abstract

We created a suite of computer programs to model epidemics andanalyzed the resultant data for various fractal patterns. In particular,we were interested in exploring the relations between the infectiousnessof a disease and the fractal nature of its survivor sets or infectionboundary. We hypothesized a successful vaccination strategy basedon the measured fractal dimension of an outbreak.

1 Introduction

Our computer simulations performed a stochastic implementation of theSusceptible-Infected-Recovered or SIR model. This epidemiological modelassumes that, given a contagious illness, a population can be partitionedinto individuals that are susceptible to the illness, infected by the illness,and recovered from (and no longer susceptible to) the illness.1 Perhaps thesimplest SIR model is the Kermack-McKendrick model[2], which assumes aclosed population of identical individuals in which the number of suscepti-ble, infected, and recovered is respectively governed by the three couplednonlinear differential equations

dS

dt= −βSI

dI

dt= βSI − γI

dR

dt= γI

1This is to be contrasted with the Susceptible-Infected-Susceptible or SIS model, inwhich individuals may return to a susceptible state after contracting the illness.

1

where

S(t) := Number of individuals at time t susceptible to the illness

I(t) := Number of individuals at time t infected with the illness

R(t) := Number of individuals at time t no longer susceptible

The number of susceptibles decreases2 according to the number of availablesusceptibles and the number of infecteds, governed by the rate β at whichthe disease is transmitted.3 The number of infecteds increases at this samerate minus the number of infecteds that recover at rate γ from the illness.Note that since the total number of individuals is conserved,

dS

dt+

dI

dt+

dR

dt= 0

from which it follows that the number of recovered individuals increases ata rate

dR

dt= −dS

dt− dI

dt= −(−βSI) − (βSI − γI)= γI.

As simple as these equations are, Kermack and McKendrick successfullyapplied this model to describe the progression of the bubonic plague in In-dia in 1905-1906. Nevertheless, the model is often extended to account forcharacteristics specific to a particular disease.

We extend the Kermack-McKendrick model to allow for multiple stagesof infection and a new parameter δ governing the rate of death due to theillness. In addition, variation is accounted for over a 2-dimensional space.Given n ordered stages of infection, we associate with each stage i the threeparameters

βi, γi, δi ∈ [0, 1].βi is the rate at which individuals in the i-th stage of the illness cause neigh-boring susceptibles to enter the first infected stage. γi is the rate at whichindividuals in infection stage i pass into stage i + 1. If i is the n-th stage,then γi governs the rate at which these individuals recover from the illness.δi is the death rate associated with i-th stage of infection. Susceptible andrecovered states have no β, γ or δ associated with them since susceptibles

2For our purposes, we assume that β and γ always have values between 0 and 1.3For brevity, at time t, we refer to cells in S(t) as susceptibles, cells in I(t) as infecteds,

and cells in R(t) as recovereds.

2

and recovereds do not infect other individuals, nor do they die from anyother causes. Given an illness with n stages of infection, and ignoring for amoment this 2-dimensional space, we can now reformulate the equations ofKermack-McKendrick as

dS

dt= −

n∑

i=1

βiIiS

dI1dt

=n∑

i=1

βiIiS − (γ1 + δ1)I1dI2dt

= γ1I1 − (γ2 + δ2)I2...

dIndt

= γn−1In−1 − (γn + δn)IndRSdt

= γnIn

dRDdt

=n∑

i=1

δiIi

where

RS(t) := Total number of survivors of the illness at time t

RD(t) := Total number of deaths due to the illness at time t.

1.1 The Conflagration Model

We consider first an SIR model that has been simplified to mimic a forest fire.Susceptibles may be imagined as green, healthy trees. We allow only a singlestage of infection and imagine infected trees as “on fire” or “burning”. Tofurther simplify, we assume that burning trees will always die and thereforeset γ := 0. Our model then consists of two parameters, β and δ, with whichwe may determine the course of a forest fire. If our trees are particularlysensitive to fire, δ will be large. If we wish to model a forest of densely-packed trees, we may account for this by likewise choosing a large β. If,however, the area has just received heavy rainfall, lower values for β andperhaps also δ could account for this. For our purposes, we will set δ := 1.0such that a tree burns up almost immediately in time dt.

1.2 The 3-Stage Model

In addition to the conflagration model, we also ran simulations of an illnesswith 3 stages of infection, with parameters perhaps consistent with a disease

3

such as smallpox. We model an illness which displays no immediate symp-toms upon contraction and set δ := 0.0. Ignoring again the 2-dimensionaldistribution, we might express this model as

dS

dt= −(β1I1 + β2I2 + β3I3)S

dI1dt

= (β1I1 + β2I2 + β3I3)S − γ1I1dI2dt

= γ1I1 − (γ2 + δ2)I2dI3dt

= γ2I2 − (γ3 + δ3)I3dRSdt

= γ3I3

dRDdt

= δ2I2 + δ3I3.

2 Methods

We programmed a custom suite of software, written in the ANSI C language,to perform a stochastic implementation of the SIR model described above.Populations are discretely divided into a 2-dimensional lattice of individuals(or cells), where β, γ, and δ describe the probability of a single cell changingstate in a single time step. Each cell in this lattice may transmit the infectionto its nearest neighbors. We refer to this implementation as a discrete-digitalSIR model.

2.1 The Model

An initial distribution of cells at time t0 leads to the spread of the illness.4

At every time step t, the state of each cell in the lattice is updated as follows.If the cell in question is at infected stage i, then a weighted coin is flipped forevery susceptible neighbor to determine with probability βi if the susceptibleneighbor should become infected (initially at stage 1). After all infectedcells in the lattice have “exposed” their susceptible neighbors to the illness,a weighted coin is flipped for each to determine with probability δi if theyshould die of the illness. If the cell does not die, then another weighted coin

4The simulation software additionally allows for arbitrary distributions to be introducedat any time t during the course of the simulation. Such distributions were “recorded” toallow, for example, a series of delayed outbreaks. For our simulations, however, all cellsare introduced at time t0.

4

is flipped to determine with probability γi if the cell should proceed to thenext stage of infection, or, if none exists, to the survivor state (recovered).Note that cells that contract the illness at time t must wait until time t + dtbefore they expose their neighbors and allow the illness to run its course.When, after some number of steps, no infected cells remain, the simulationhalts.

For our simulations, the nearest neighbors, those which may be exposedto the illness, are the 4 adjacent cells above, below, right, and left of theinfected cell. In addition, outbreaks were not allowed to spread beyond theedge of the lattice, which we treat topologically as a plane (and not, forexample, a torus).5

After each simulation of our model (after the illness had run its course),we performed a statistical measure of the fractal dimension of the set ofremaining susceptible cells. Additional data, such as the total number ofsteps in the simulation and the number of remaining cells of each state werealso recorded.

For some simulations of our conflagration model, we additionally recordedthe fractal dimension of the infection boundary, that is, those infected cellswith susceptible neighbors, at each step of the simulation.

2.2 Batching Simulations

Computations of the simulations were “embarrassingly parallel,” with severalthousand runs of the model (each with a different random seed) performedindependently across several machines. Jobs were managed with configura-tion files and scripts to chop the job up across the parameter fields into tasksfor each processor, and resultant data were then collected only after the lastprocess completed.6

2.3 Box-Counting

Fractal dimension of a set F was calculated statistically using a box-countingmethod. Overlaying the simulation lattice with boxes of different sizes 1/s,7

the program counts the number of boxes ns which contain at least one cell inF . Next, the program finds a best fit line for all the data points (log s, log ns)

5The simulation software actually allows for different neighborhoods and lattice-boundary conditions; however, for these simulations, we chose a single configuration.

6Approximately 1015 ops were required for each job, distribute across 6 to 10 processors.7This ensures that s represents the number of boxes that fit on one side of the whole

lattice, rather than the size of each box

5

0 20 40 60 80 100

1.58

01.

582

1.58

41.

586

1.58

81.

590

SeedPct

Fra

ctD

im

Figure 1: Scatter plot of random seeds used to generate a Sierpinski gasketversus calculated fractal dimension of that gasket using our box countingcode. The seeds are displayed as percentages of the total space of possiblerandom seeds, rather than as their integer values.

using a least-squares approximation. The slope of this line is considered tobe the fractal dimension of the data set F .

The value of ns for large box sizes tends to be noisy, and so it is neces-sary to disregard a few data points when calculating the best fit line. Afterattempting to use a method that would dynamically determine which pointswere the noisy ones, we found it was more effective to simply disregard thefirst 5 data points (i.e. the 5 data points with the smallest values of s). Theaccuracy of these two methods was determined by using a non-deterministicmethod of generating a Sierpinski gasket, and by testing each of the twoversions for 1000 gaskets generated by unique random seeds. The version ofthe code which was finally used with the SIR model simulations performedadmirably, and as is demonstrated in Fig. 1, the calculated fractal dimen-sion was always accurate to at least four significant digits with respect to theanalytically calculated fractal dimension of a Sierpinski gasket, which is:

log 3

log 2≈ 1.5849

The error in the box counting calculation is too small to be visible on a graph

6

Figure 2: Initial state for iterated cross distribution

of this resolution.

2.4 The Simulations

In our simulations, we explored the interaction of the fractal dimension ofthe survivor set and infection boundary with the total number of resultingdeaths. We hypothesized a successful vaccination strategy that distributesvaccine to areas where the infection boundary has highest fractal dimension.Fig. 2, for example, illustrates a sort of “worst-case scenario” in which thefractal dimension of the outbreak is very high and from which we wouldexpect higher death counts than if the outbreak had been distribution withina tight cluster.

To test our vaccination strategy, we also ran two batches of simulations inwhich an outbreak initially has a “rough” side and a “smooth” side. In thefirst batch, vaccine (recovered cells) was distributed around the smooth sideof the outbreak (see Fig. 3) . In the second batch, vaccine was distributedaround the rough side (see Fig. 4). We hypothesized that the rough-sidevaccination strategy would result in fewer deaths on average and thus proveto be the better of the two. In both simulations, the number of initiallyinfected cells is equal.

7

Figure 3: Initial state for the smooth-side vaccination strategy

Figure 4: Initial state for the rough-side vaccination strategy

8

3 Results

All of our data was graphed and analyzed using the GNU R package forstatistical computing.

3.1 Conflagration Model

3.1.1 Fractal Dimension vs. Transmission Rate

Primarily we were concerned with the relations between the fractal dimensionof the survivor sets and the rate of transmission of infection. Using a simpleconflagration model, we performed 100 runs8 for each rate of transmissionβ ∈ {0, 0.005, 0.010, . . .1.0}, with fixed γ := 0, δ := 1, on a square lattice of16.7 million cells (4092 by 4092). Fig. 5 shows a scatter plot of the rate oftransmission plotted against the resulting fractal dimension of the survivorset. To see how the duration of the outbreak is affected by our choice of β,Fig. 6 shows a scatter plot of the rate of transmission plotted against thenumber of steps required for that simulation.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

RatePct

Fra

ctD

im

Figure 5: Scatter plot of infection rates used to run a conflagration simulationversus calculated fractal dimension of the survivor set.

8For each run, a different seed was chosen to seed the pseudo-random number generator.

9

0.0 0.2 0.4 0.6 0.8 1.0

020

0040

0060

0080

0010

000

1200

0

RatePct

Ste

ps

Figure 6: Scatter plot of infection rates used to run an conflagration simula-tion versus the number of time steps taken for that simulation to complete.The horizontal line is y = 2050.

3.1.2 Fractal Dimension of Infection Boundary

To examine the relation between the fractal dimension of the infection bound-ary and the rate of infection, we again used our conflagration model, this timewith a lattice size of 1024 by 1024. We performed 10 runs9 for each rate oftransmission β ∈ {0, 0.01, 0.02, . . .1.0}, again with fixed γ := 0, δ := 1. Un-like before, where we calculated the fractal dimension of the survivor set, inthese runs we calculated the fractal dimension of the infection-boundary atevery time step in the simulation. Fig. 7 plots the progression for a run witha transmission rate β = 0.51 while Fig. 8 plots the progression for a run witha transmission rate β = 0.77. In both plots the data points are connected bylines to help clarify the order in which they occur.

3.2 Vaccination Strategy (3-Stage Model)

To test our vaccination strategy, we chose a 3-stage SIR model with

β2 := 0.4 β3 := 0.1

9Again, for each the pseudo-random number generator was initialized with a differentseed.

10

0 500 1000 1500

0.0

0.2

0.4

0.6

0.8

Step

Fire

Dim

Figure 7: Scatter plot, connected by lines, of the fractal dimension of theinfection boundary at every time step of a simulation run with transmissionrate β = 0.51.

γ1 := 0.2 γ3 := 0.9

δ1 := 0.0 δ2 := 0.4 δ3 := 0.1

and

β1 ∈ {0.05, 0.05625, 0.0625, . . .0.1}γ2 ∈ {0.4, 0.42, 0.44, . . .0.7}

For each of the two strategies (vaccinate smooth-side, vaccinate rough-side),we performed 100 runs on a 1024 by 1024 lattice for each combination of β1and γ2.

Looking at the scatter plot in Fig. 9, it is apparent that vaccinating thesmooth side results in a consistently higher death count, and box plots inFigs. 10 and 11 confirm this. A one-sample t-test of the difference in deathsyields a p-value of 2.2× 10−16, allowing us to reject the null hypothesis thatthese two distributions do not differ in the resulting death count.

11

0 200 400 600 800 1000

0.0

0.2

0.4

0.6

0.8

1.0

Step

Fire

Dim

Figure 8: Scatter plot, connected by lines, of the fractal dimension of theinfection boundary at every time step of a simulation run with transmissionrate β = 0.77.

4 Conclusion

4.1 Survivor Set

The fractal dimension of the survivor set is especially interesting if viewedfrom the perspective of percolation theory. Essentially, the sharp downturn atβ = 0.5 represents a critical value, where it becomes possible for the infectionto reach the border of the simulation lattice. At lower values, where thisborder is never reached, the fractal dimension of 2 implies that the survivorset spans the entire plane. This makes sense if we change our perspectiveslightly. Imagine that as we increase the size of the simulation lattice, the fewrecovered cells clustered in the center of the lattice fill a smaller percentageof the overall cells. As our scope increases, the recovered cells at the centeroccupy a smaller and smaller portion of the plane, vanishing to zero, atwhich point the survivor set does indeed span the plane. This critical pointmarks a fascinating transition between what might be termed relative sizingof the survivor set and absolute. In the previous example, where the fractaldimension of the survivor set is 2 (or arbitrarily close to 2), the numberof recovereds remains the same as our scope increases, and thus accounts

12

0.4 0.45 0.5 0.55 0.6 0.65 0.7

5e+

0555

0000

6e+

0565

0000

7e+

05

0

0.01

0.02

0.03

0.04

0.05

Gamma002

Bet

a001

Dea

ths

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

**************************

************************

************************

************************

************************

************************

************************

************************

************************

Figure 9: Scatter plot of the death toll associated with each pair of (β1, γ2).The larger, blue points represent vaccination of the smooth side, while thesmaller, red points represent vaccination of the rough side.

for a diminishing percentage of the total number of cells. However, oncethe fractal dimension takes a steep downturn from 2, and we begin to seeinfections which run into the edge of our simulation lattice, we can see thatif there were eight neighboring lattices of the same size to encompass andextend this one, the infection would likely spread all the way across themas well. Thus, as the size of our lattice increases, so does the size of ourrecovered set, thus maintaining a somewhat stable portion of the overallnumber of cells, even as the size of the lattice increases indefinitely.

4.2 Infection Boundary

The fractal dimension of the infection boundary is related to it’s “jagged-ness”. A higher fractal dimension means that the curve of the infectionboundary is much more convoluted and is taking up a greater portion ofthe available space. A smaller fractal dimension means that the boundaryis smoother, less convoluted, and more closely approximates a straight line.The results visible in Figs. 7 and 8 both display the same tendency: whenthe infection boundary is jagged, it tends to smooth itself out but once itbecomes smooth, it tends to return to being jagged. If an infection bound-

13

0.05 0.05625 0.06875 0.075 0.08125 0.09375 0.1

020

0040

0060

0080

00

Rate of Infection (Stage 1)

Dea

ths

(Sm

ooth

) −

Dea

ths

(Rou

gh)

Figure 10: Box plot of the difference between death counts in the rough andsmooth data sets as they vary over β1. The positive values indicate a higherdeath count for the smooth vaccination strategy.

14

0.4 0.44 0.48 0.52 0.56 0.6 0.64 0.68

020

0040

0060

0080

00

Rate of Progression (Stage 2)

Dea

ths

(Sm

ooth

) −

Dea

ths

(Rou

gh)

Figure 11: Box plot of the difference between death counts in the rough andsmooth data sets as they vary over γ2. The positive values indicate a higherdeath count for the smooth vaccination strategy.

15

ary becomes too jagged, the various edges run into each other and, mergingtogether, become smoother again. The amount of variability in the infectionboundary’s fractal dimension is most likely related to the number of timesteps in the simulation. Those simulations which run for a long time (Fig. 6;around β = 0.5) tend to have extremely variable fractal dimensions on the in-fection boundary, since it keeps spreading in a very non-regular fashion, often“backtracking” in some sense before smoothing and slowing again. As therates of infection grow, it is less likely for an infection boundary to becomejagged at all, since the jaggedness can only occur if some (but not all) por-tions of the fire boundary continue to infect some (but not all) neighboringcells.

4.3 Smooth or Rough

Our hypothesis that a vaccination strategy of placing vaccine along theboundary of the outbreak with the higher fractal dimension should mini-mize the number of resulting deaths could not be proved. However, due tothe increased coverage of the rough side of the initial outbreak, more vac-cine (roughly 3000 more cells) was required to encompass the rough side tothe same thickness with vaccinated cells than was required to encompass thesmooth side. We cannot confirm or disprove our hypothesis without control-ling for this bias in the initial number of vaccinated individuals.10

References

[1] R. Crandall, G. Oxman, M. Helman, and A. Sullivan. Intuitive physicsleads to epidemic models: Focus on smallpox. Reed College EpidemiologyJournal, 2004.

[2] W.O. Kermack and A.G. McKendrick. A Contribution to the Mathemat-ical Theory of Epidemics. Proceedings of the Royal Society of LondonSeries A, (115):700–721, 1927.

10Note that simply removing the 3274 vaccinated cells from the death count for therough vaccination strategy is not enough, since the vaccination arc also contains another7639 susceptible cells which we would assume to be less likely to contract the infectionsince many of their vaccinated neighbors stand as a “buffer” against attack.

16