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Physica A
journal homepage: www.elsevier.com/locate/physa
Mobility matrix evolution for an SIS epidemic patch model
Johnathan Sanders a, Benjamin Noble a, Robert A. Van Gorder b,∗, Cortney Riggs a
a Department of Electrical Engineering and Computer Science, University of Central Florida, Orlando, FL 32816, USAb Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
a r t i c l e i n f o
Article history:Received 19 April 2012Available online 20 July 2012
Intercommunity disease spread can be modeled using a collection of discrete community‘‘patches’’ with continuous population flow between them. In a susceptible–infected–susceptible (SIS) model residents of a community may either be classified as susceptible orinfected. Infected individualsmay heal and become susceptible again but are not permittedto die or become immune. The spread of disease can be controlled by modifying the rateand direction of resident movement across patch boundaries. In this work we use geneticalgorithms to evolve optimal connections between patch boundaries such that the totalnumber of infected individuals is minimized.
In recent years there has been a great deal of interest in the study of epidemiological events. The interest has beenbrought about by a variety of factors including new domains such as the Internet where a computer virus epidemic can haveunpredictable effects and be nearly as damaging as its real world counterpart. Computer domains are especially attractive aswe can gather information quickly about the spread, persistence, and eventual death of infections. Theoretical solutions canbe tested on infected computer networks before being deployed in real world epidemics. This has provoked the design of aflurry of mathematical models to describe the behavior of multiple types of infections in multiple population distributions.This work focuses on one specific type of epidemic which is applicable both to the spread of computer viruses and realworld diseases. In a susceptible–infected–susceptible (SIS) epidemic, infected individuals do not gain immunity from aninfection after their recovery. This is analogous to computer individuals which have no natural immune system. It is worthmentioning that software updates do not fall under the category of natural immunity as they essentially immunize the entirepopulation of computers in a single shot. In other words software updates are not an individual computer’s response to itsown infection. This begs the question then why an epidemic model for computers is needed at all. After all, if an update cansimply be downloaded onto every computer then the transmission of an infection from computer to computer is negatedand there is nothing to model. The answer is cost. There are transmission costs associated with downloading updates. Everytime a computer node checks an individual transmission for a virus it slows down the propagation of that transmission.Additionally there are deployment costs such as the update download time and resource usage. The virus, the update, andthe constant need to filter network traffic causes network communication to grind to a halt. We suggest two alternativesolutions. Rather than deploying an update to every computer node choose one of the following: either strategically deployvirus removal software to nodes where it will be most effective, or reroute network traffic along paths such that the spreadof infection is minimized. Once network traffic is restored to normal capacity, updates can be released to prevent futureinfection.
∗ Corresponding author.E-mail address: [email protected] (R.A. Van Gorder).
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The main contributions of this work are as follows:
1. Investigate the feasibility of using aGA tominimize disease spread in an epidemic patchmodel (described in the literaturereview);
2. Compare infection minimization while modifying only network routing with infection minimization while modifyingonly the deployment of infection recovery resources;
3. Compare the GA’s solutions to manual observations made in previous work.
In doing so we hope to lay some groundwork in the use of mobility models to minimize infection spread in both computerand human populations.
The rest of the paper is organized as follows. In Section 2 we provide a brief literature review on epidemiological models.Section 3 gives a mathematical analysis of the SIS model and restates it as an optimal control theory problem. In Section 4we present the experimental setup as well as our metrics for evaluation. Section 5 will be reserved for experimental resultsand a brief discussion.
2. Literature review
Epidemic models are composed of two components, a disease state model and amobility model. The disease state modeldescribes how the infection affects the population over time while the mobility model determines movement and contactbetween individuals. The epidemic model then tells us in what way and to what extent a disease is affecting an entirepopulation of individuals. In this section we give a brief overview of some popular disease state and mobility models foundin the literature.
2.1. Disease state models
Most disease state models are mathematical models that involve equations for the total amount of different categoriesof individuals at specific locations. The individuals can represent people, packets of information on a computer network,disease treatment supplies, or even the virus itself. Most models are generally systems of ordinary differential equations.
The firstmajor type of disease statemodel is the SIRmodel. This represents individualswho start in a susceptible state andthen move to an infectious state which can cause others they come in contact with to move from susceptible to infectious.After becoming infectious the next phase is to become recovered. In this state the individual is then immune to the virus andwill no longer be able to move into an infectious state. This model is commonly used to represent smallpox, measles, andother diseases in which recovered individuals become immune [1]. In the simplest case of this model there are 3 differentialequations, onewhich represents the number of susceptible people, S, onewhich represents the number of infectious people,I , and one which represents the number of recovered people, R:
∂S∂t
=−βISN
, (2.1)
∂ I∂t
=βISN
− γ I, (2.2)
∂R∂t
= γ I, (2.3)
which are taken from Ref. [1]. Here β is the infection rate, γ is the recovery rate, and N is the constant total population size.This assumes that an individual can come into contact with every other individual in the population. This is a reasonableassumption for small populations. In large populations it is more likely that individuals will have a smaller area of influenceand will have some other individuals with which there is no potential for contact. Additional factors can be added such asbirths, deaths, exposed but not yet infectious individuals, passively immune infants, and a variable population size.
The second major disease state model type is SIS. In the SIS model individuals initially start off susceptible and thenbecome infectious upon contactwith the disease. Then after recovering, the infectious individuals become susceptible again.This model can be used to represent certain types of influenza and other diseases where an individual can become infectedafter having recovered from a previous infection. A more common use for this model however is to represent a computervirus epidemic. Oftenwhen computer viruses are found they can be cleaned from a computer but the computer still remainssusceptible to getting the virus again. The most basic version of the model is very similar to the previous model:
∂S∂t
=−βISN
+ γ I, (2.4)
∂ I∂t
=βISN
− γ I, (2.5)
also appearing in Ref. [1] where again S is the total number of susceptible individuals, I is the total number of infectiousindividuals, and N is the total population. The same additions such as births, deaths, exposed individuals, passively immune
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infants, population diffusion, and a variable population size can be included in this model as in the last model; however theequations would be updated differently. One such model is given in Ref. [2] where a diffusion term is added. In Ref. [3] arepresentation for demographics is introduced as well as a seasonally variable infection rate.
The previous twomodels described are deterministic. An individual in a particular state can onlymove into the next state.One disease state model is given in Ref. [4] which is not deterministic. Outbreaks are stochastically generated which changethe problem into optimizing the flow of treatment supplies from treatment centers. All disease state models basically fallinto one of these three categories with some variation based on the specific application.
2.2. Mobility models
As stated in the previous section, assuming that all individuals have an equal likelihood of making contact is only a goodassumption for small populations. The most common way to account for this in an epidemic model is to assume randommixing in small populations and then add a diffusion term to the differential equations to account for movement from onesmall population to another. Some of themodels focus only onwhat happenswithin a small population and assume randommixing while others use a type of mobility model which is represented by the diffusion term. In this section we will discussa variety of epidemic models which use mobility.
Onemethod formodelingmobility is to have a connectivitymatrix which represents direct connections between smallerpopulations. One such model is given in Ref. [2]. Each node in the model is a small subset of the population and has a set ofdifferential equations with a connectivity matrix as the diffusion term to represent its current population. The connectivitymatrix is essentially an adjacencymatrix with a one at every place where there is a connection between two subpopulationsand a zero at every place where there is no connection. This diffusion term then adjusts the number of susceptible andinfected individuals based on the number of incoming and outgoing connections.
A mobility matrix model is different than a connectivity matrix model in that the mobility matrix represents the numberof individuals that move from one subpopulation to another as opposed to just the connection between subpopulations. Themodel presented in Ref. [5] is a model that uses a mobility matrix.
In Ref. [4] instead of introducing a diffusion term which represents individuals a diffusion term is introduced whichrepresents the movement of supplies to treat infected individuals. These supplies are limited to a maximum capacity atsupply centers and trucks with limited capacity are used to transport the supplies to the individuals. The goal of Ref. [4] wasto find an optimal logistics network that chooses the appropriate time and amount of resources to send to stochasticallygenerated areas of demand.
A constraint that can be added to a network mobility model is the constraint of being a scale free network. A node’sdegree is the number of connections it has to other nodes. The degree distribution P(k) is the probability that a node in thenetwork has degree k. Scale free networks have probability distributions that follow
P(k) ≈ ck−λ
where c is a constant used to normalize P(k), k is the degree mentioned previously, and λ is a power constant that variesfrom network to network. Scale free networks follow what is known as the power law. In other words a few high degreenodes (known as hubs) connect many lower degree nodes together. These lower degree nodes may themselves be smallerhubs connecting even lower degree nodes and so on. There are benefits and drawbacks to scale free networks. Imagine thatone node in the network becomes disconnected. The chances are relatively low that node is a hub (there are very few) andthe network can maintain connectivity. For this reason the Internet has been modeled as a scale free network. It is worthnoting that if a hub does become disconnected the network connectivity falls apart. Scale free networks and how they relateto epidemiology are discussed in more detail in Ref. [6].
Hence, there is much interest in SIS models on networks. For some advanced treatments of the SIS models on networkstructures, see the modern Refs. [7–13] and references therein.
There are also other models that do not use networks. The model presented in Ref. [14] uses a reaction–diffusion modelto represent the motion of individuals between subpopulations.
3. Optimal control problem for SIS/SIR epidemic patch models
We may rephrase the problem of finding an optimal mobility matrix in the language of optimal control problems. Ingeneral, the continuous time control problem for the SIS patch model takes the form
minLi,j∈0,1
V (S(t), I(t), α) , (3.1)
subject toS(t) = F(S(t), I(t); L, β, γ , dS),
I(t) = G(S(t), I(t); L, β, γ , dI),
S(0) = S0, (3.2)I(0) = I0,0 ≤ Lij ≤ 1,
J. Sanders et al. / Physica A 391 (2012) 6256–6267 6259
where V ∈ R is an objective function, S(t) = (S1(t), . . . , Sn(t)) ∈ Rn is the time-dependent vector of susceptible individualsin each of the n patches, I(t) = (I1(t), . . . , In(t)) ∈ Rn is the time-dependent vector of infected individuals in each of the npatches, L ∈ Rn2 is a real-valued vector containing the n2 components, β ∈ Rn is the vector giving patch dependent ratesof disease transmission, γ ∈ Rn is the vector giving patch dependent rates of disease recovery, dS is the rate of diffusion ofthe susceptible population, dI is the rate of diffusion of the infected population, α ∈ Ra denotes a vector of other constantswhich may appear in the objective function (we shall say more on this later), and both S0 ∈ Rn and I0 ∈ Rn are constantvectors holding the initial populations.
The function V should be somemeasure of the understandability of the disease persistence in the population. It might bean aggregate measure of lost productivity over time due to infections, it might be a measure of health-care expendables (inthe case of human or animal disease) or costs associated with repairing computers (in the case of computer viruses), or itmight simply be a measure of the number of infected individuals either over time or at a specific point in time. The systemof 2n differential equations (3.2) describes the time evolution of the state variables (S1(t), . . . , Sn(t)) and (I1(t), . . . , In(t)).In a model without births or deaths,
whereN is the total population size. The functions F : Rn2+4n+1→ Rn andG : Rn2+4n+1
→ Rn are assumed to be continuousin all variables. Assumptions on differentiability can be helpful for some applications, but for sake of generality we refrainfrommaking further restrictions on F and G for the time being. Assuming all model parameters are fixed, our goal is to selectthe value of the vector L which minimizes the objective function V subject to the constraints (3.2).
We may also define a discrete time control problem for the SIS patch model. For such a model, the only modificationneeded to (3.1)–(3.2) will be to the constraints on the state variables, which now read
S(t + 1) = F(S(t), I(t); L, β, γ , dS),
I(t + 1) = G(S(t), I(t); L, β, γ , dI),
S(0) = S0 (3.4)I(0) = I0,0 ≤ Lij ≤ 1.
Meanwhile, SIR models are also of interest in those cases where immunity to the epidemic can be developed. Thecontinuous time optimal control problem (3.1) is then held subject to the constraints
S(t) = F(S(t), I(t); L, β, γ , dS),
I(t) = G(S(t), I(t); L, β, γ , dI),
R(t) = H(I(t); L, β, γ , dI),
S(0) = S0, (3.5)I(0) = I0,R(0) = 0n,
0 ≤ Lij ≤ 1,
whereas the discrete time problem is held subject to the constraints
S(t + 1) = F(S(t), I(t); L, β, γ , dS),
I(t + 1) = G(S(t), I(t); L, β, γ , dI),
R(t + 1) = H(I(t); L, γ ),
S(0) = S0, (3.6)I(0) = I0,R(0) = 0n,
0 ≤ Lij ≤ 1.
The function H : Rn2+2n→ Rn is assumed to be continuous in all variables. Here, 0n denotes the vector of zeros in Rn.
3.1. Possible forms of the objective function V
Up to this point, we have not specified a particular form for the objective function. Possible simple objective functionscould be
V = maxt∈[0,t∗]
ni=1
Ii(t), (3.7)
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which is the maximum value of the sum of the infected populations over the temporal interval t ∈ [0, t∗],
V =
t∗
0
n
i=1
Ii(t)
dt (continuous time model),
t∗t=0
ni=1
Ii(t) (discrete time model),
(3.8)
which is the aggregate infected over the interval t ∈ [0, t∗],
V =
1t∗
t∗
0
n
i=1
Ii(t)
dt (continuous time model),
1t∗
t∗t=0
ni=1
Ii(t) (discrete time model),
(3.9)
which is the average infected over the interval t ∈ [0, t∗],
V =
ni=1
Ii(t∗), (3.10)
which is simply the sumof the infected at time t = t∗ > 0.More complicated objective functions could certainly be required,dependingupon the application at hand.However, these simple types of objective functionswill be sufficient for our analysis.Note thatminimization of these ‘‘loss’’ objective functions is dual tomaximizing ‘‘gain’’ objective functions involving S. Morecomplicated functional forms for V , involving both I and S, are possible.
In the event where the relative values of the populations in each patch differ, one may replace the summations presentin (3.7)–(3.10) with weighted summations; i.e.,
ni=1
ωiIi(t) = ω · I(t), (3.11)
where ω = (ω1, . . . , ωn) ∈ Rn+is a weighting vector, and ω · I(t) denotes the dot product of the two vectors.
3.2. Functional forms of the vector fields F ,G,H
The simplest examples are just the multiplicative ones (interactions of the form IS). These can be normalized (in termsof their dependence on L) so that
for all t > 0 (with a similar relation for the SIR case), which permits the population size to remain at N (as we neglectbirths and deaths). Through appropriate normalization, we can ensure that an individual is never double counted. Often,this normalization is achieved through proper selection of dS and dI .
3.3. Time-dependent mobility matrix
While we have considered selecting elements Lij ∈ 0, 1 of the mobility matrix L (with associated parameter vector L),we assume that the optimal choice,L = L∗, holds for all t > 0. It could very well be the case that a better solution exists forsome of the relevant control problems if we allow L to vary in time. That said, finding n2 optimizing Lij’s which depend ontime is a far more complicated and computationally taxing problem, and thus in the present study we restrict our attentionto Lij’s which are constant in time.
4. Experimentation
Optimization of networkmovement and resource allocation forminimization of infection spread are of interest in our ex-perimentation. Both cases have a vast possible pool of solutions.While theoretical work has shown good resource allocationsolutions [4], finding solutions for network movement in patch models still has many possibilities. Genetic algorithms willbe used to search through the large solution spaces of two problems mobility matrix representation of network movementand resource allocation in a network. Genetic algorithms have been useful in the search of large solution spaces [15].
Genetic algorithms are inspired by the natural process of evolution. Certain behaviors are represented by a fitnessfunction f (x) which determines the strength of an individual solution to a problem. Each solution is known as an
J. Sanders et al. / Physica A 391 (2012) 6256–6267 6261
individual and the group of solutions is the population. Individual solutions can be viewed as chromosomes with genesfor representation of problem aspects. The degree to which a solution is able to solve the problem is that individual’s fitness.A population at any given time step is called a generation. At each time step, selection is used to pass on information fromthe stronger solutions to the fitness function. Selection is done randomly from adjusted distributions of the individuals.Some selection methods include select proportionally to fitness values or proportionally to rank (order of fitness valueshighest to lowest). The amount and type of information that is propagated is a function of certain genetic operators, themost common of which are crossover and mutation. In crossover, information of one individual parent solution is swappedwith the corresponding section in another parent solution. This produces two offspring.
In mutation part of the genetic information in an individual has some probability of being randomly changed. This helpsto preserve population diversity, or the capacity of a population to generate unique problem solutions. For a more in-depthdiscussion of genetic algorithms see Refs. [16,15].
4.1. Problem representation
More formally this movement can be described by a mobility matrix
L =
L1,1 L1,2 · · · L1,kL2,1 L2,2 · · · L2,k...
.... . .
...Lj,1 Lj,2 · · · Lj,k
, (4.1)
where Lj,k represents the degree of information flow frompatch k to patch j (this is not to be confusedwith the term softwarepatch used previously). The SIS model that will be used in our experimentation is a discrete time simulation of a modelpresented in Ref. [2] as follows:
Let n ≥ 2 be the number of patches and Ω = 1, 2, . . . , n. Consider the SIS patch model
dSjdt
=
Interpatch Travel dS
k∈Ω
(LjkSk − LkjSj) −
Transmission βjSj IjSj + Ij
+
Recoveryγj Ij , j ∈ Ω, (4.2)
dIjdt
= dI
k∈Ω
(Ljk Ik − Lkj Ij) +βjSj IjSj + Ij
− γj Ij, j ∈ Ω, (4.3)
where Sj and Ij are the number of susceptible and infected individuals (data) in patch j; and dS and dI are diffusion coefficientsfor susceptible and infected subpopulations. Ljk is the k to j interpatchmobility as described above andβj and γj are constantsthat represent the rate of disease transmission and recovery in patch j respectively.
Formalized, our goal is to minimize the global infected population in the shortest possible amount of time. As mentionedbefore we can do this two ways:1. Given fixed recovery resources λ across the network, choose L such that the flow of information across the network
minimizes infection spread;2. Given a fixed network connectivity L, allocate from a finite set of resources λ individual λj to each patch j such that the
infection spread is minimized.
It is of note that for the equations used in this particularmodel only numerical solutions can be found. Finding an optimumusing traditional methods then is out of the question. Instead we use a genetic algorithm to evolve optimal solutions todiscrete time simulations of each of the two cases.
4.1.1. GA parametersA base genetic algorithm will be used for both problems with some problem specific differences. In both cases the
number of nodes was held at 5. Thus, with binary connection representation the search space was 225 for each run. Totalpopulation among all the nodes was 500,500. This population was initially distributed uniformly with approximately100,000 individuals inhabiting each node, such that: Si = 100,000 for all i and I1 ∈ I = 500. This distribution of thepopulationwas chosen to explore relatively healthy node populationswith one node acting as an epidemic generator. Fitnessfor the genetic algorithm was modeled in three ways:1. Lowest Peak—Individuals are more fit who successfully minimize the maximum number of infected at any time;2. Equilibrium—Individuals are more fit who minimize the total infected population when it has reached a steady state;3. Average—Individuals are more fit who successfully minimize the infected population averaged over time until the
population reaches steady state.
Parameters of the genetic algorithm were the same for both problems. Population sizes of 50 are used and proved to besufficient in preliminary work. First generations are randomly initialized. Fifty runs are run for 25 generations. Crossoverrate was set to 0.9; mutation rate, 0.005. Solutions were selected based on rank proportional distributions.
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Fig. 1. Graphical representation of the binary string 01111-10000-10000-10000-10000 for mobility matrix L.
Fig. 2. Minimizing average infected with mobility: average fitness per generation.
4.1.2. Gene encodingIndividuals had the same size of binary string chromosomes in both problems. Gene representationwithin the stringwas
different. For themobility matrix, a gene represents connections to other patches in the network. One signifies a connectionand 0 signifies a lack of connection to the corresponding patch. Fig. 1 is a graphical depiction of the string 01111-10000-10000-10000-10000. Gene length is equivalent to the number of patches in the network, 5, with the self-reference bitignored. Using this encoding, we evolve the network itself with crossover.
Encoding for the resource usage problem has a function to determine the values of each patch’s recovery rate. The totalresource available is held at a constant value. In this way, the problem represents a resource allocation problem. Recoveryrates are improvedwithmore resources at a patch, but limiting the sumof the rates represents a limited resource amount. Tocalculate the values, the function uses the gene to calculate the recovery value, γ , at each patch. An individual’s chromosomeis made up of one gene. Thus, for our 5 patch problem there are 225 possible input values to the function.
4.2. Mobility matrix optimization
In the first set of experiments the resource allocation was held fixed with an infection transmission rate of 0.33 for everypatch and a recovery rate of 0.33 for all but one patch whose recovery rate was 0.3. In this way we hoped to show that evengiven a situation where the total infection rate is greater than the total recovery rate the number of infected individuals canbe successfullyminimized viamodifications to themobilitymatrix L. The results for themobility experiment are surprisinglydifferent from fitness function to fitness function and provide some insight towards infection propagation in SIS models.
Fig. 2 shows the average, best, and standard deviation of solutions the GA produced when using the average numberof infected individuals as a fitness function. The immediate things to notice are the large differences between the averagefitness and the best fitness even from the first generation. This demonstrates that changing network connectivity in an SISmodel can have significant effects on the susceptible/infected population distributions. As the initial population is a randomsampling this also demonstrates a high probability that most of the connectivities which can be chosen have extremely lowfitness relative to the best connectivities, thus the improvements that can be made are substantial. Taking a closer look atthe best individual we can see the population distribution for this experiment in Fig. 3.
Patch 1 (curve I(1)) is the initial infected patch in this simulation. Comparing this to curve I(T ) representing the totalinfected in all patches shows that as infected individuals from I(1) travel to other patches their populations become infected
J. Sanders et al. / Physica A 391 (2012) 6256–6267 6263
Fig. 3. Minimizing average infected with mobility: simulated population distribution.
Fig. 4. Minimizing equilibrium value with mobility: average fitness per generation.
Fig. 5. Minimizing equilibrium value with mobility: simulated population distribution.
as well. In other words the infection not only persists, but spreads and grows even in patches that have a recovery rateequal to the infection rate. This result is not trivial as it tells us that given a total infected population and an arbitraryinfection/recovery rate a patch still has potential for infection growth. The solution therefore is not as simple as sendingevery individual to the patch with the highest recovery to infection ratio as this may in fact overwhelm the patch and causea catastrophic growth in infected individuals which will then trickle out into less capable patches. Judging from the figure,the GA’s solution in this case seems to be to move individuals out of the highly infected patch as quickly as possible andspread them amongst the other patches relatively evenly. Note this may mean that the population can never reach zeroinfected individuals as we see there are already several patches who have reached a steady state by time step 700, howeverthe GA was not searching for zero infected individuals. It was simply searching for the lowest average number of infectedindividuals before equilibrium. In the graph this represents the area under the I(t) curve from time zero to approximatelytime 2000.
Figs. 4 and 5 tell us a much different story.
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Fig. 6. Minimizing maximum infected with mobility: average fitness per generation.
Fig. 7. Minimizing maximum infected with mobility: simulated population distribution.
Wecan see from the average fitness per generation that fromaGA standpoint the problems ofminimizing the equilibriuminfected and average infected populations are in fact verymuch the same. They have the same fitness at the same generationswhich suggests perhaps both fitness functions are finding the same solution; however looking at 5 we can see that this isnot the case. The GAwas able to find a different and in fact slightly better solution. Recall that whenminimizing the averagenumber of infected individuals the best policy seemed to be to disperse individuals from the sickest patches to the healthiesteven if thismeant endingwith patches thatwere in a steady state of sick individuals. In this solutionwe can see both from thegraph and the binary representation 00000-00010-00011-00100-10100 that the best policy is actually filtering individualsfrompatch to patch in a series. The first gene, the leftmost string of five binary numbers represents the incoming connectionsto patch one. There are none. Each subsequent string represents that patch’s incoming connections. Notice that peoplemoveout of patch one only through patch five, and out of patch five only through patch three and out of three only from four,then out of four only from two. Since the GA essentially only cares about the lowest ending number of infected in the case,we do not mind keeping the infected in one patch for a little longer (increasing the average number of sick individuals) aslong as this means they will eventually all be healed. In other words the GA took a hard problem (patch 1 with sick peopleand a high sick rate), moved it to an easier problem (patch 5 with no sick people and a lower sick rate) which then grew intoa hard problem (patch 5 with sick people) and then moved them to an easier problem (patch 3) and so on. The end result isactually a lower total infected population even though the road was rocky getting there.
If the goal however was to not have a rocky road, that is if the goal was to keep the maximum infected at any given timestep to a minimum then one would see the behavior in Figs. 6 and 7.
It is worth mentioning that in 6 the graph behavior is the same as in the other experiments but scaled to higher values.This is because what the GA is minimizing in this case is not just any number of infected people. It is the highest number ofinfected people a solution reached through any time step. We would ideally like this to be low (no sense in having infectedindividuals that don’t have to be) but we can see from 7 that this can have significant drawbacks in the long term. While it’simpressive that this graph contains a lower peak than either of the first two experiments it also contains well over twice asmany infected individuals by the final time step and is not improving. It may not be as apparent in the graph but considerthe best solution 00000-11010-11100-10010-10011. Patch one is outwardly connected to every other patch as is the casewhere average infection was minimized, i.e. 10000-10110-11011-11101-10010. Note the inward connection to itself doesnot affect the result. Also note the difference between the two solutions. In average minimization an attempt is made bythe GA to maximize the number of incoming connections on every patch but the sick patch (in which incoming connectionsseem to be minimized on all experiments) which maximizes travel. In the case where the lowest peak is important theGA minimizes the number of incoming connections on every patch while still allowing incoming from the sick patch, thus
J. Sanders et al. / Physica A 391 (2012) 6256–6267 6265
Fig. 8. Tiered patch connection.
slowing travel. This is an intuitive solution similar to a quarantine but as we see in 7 not the best solution in the long run.As a side note it would be an attractive solution in the case where for example you could not afford to have many infectedpeople at any given time such as in a country’s work force.
4.3. Resource allocation experiments
Our resource allocation experiments aim to investigate good solutions by distributing a set amount of resources.Minimization of infection across the matrix is the goal. Expected solutions are to spend most resources at higher infectednodes. The experiment will give insight on the expected results and possibly unique solutions.
Resource allocation experiments have the same general set-up as the mobility matrix experiments. However, the matrixis set and the γ values are evolved. Two mobility matrices are used: a fully connected matrix and a tiered connectedmatrix based on the solutions found in themobility experiments. The fully connectedmatrix demonstrates scenarios wherethe resource allocation is the only reliance to contain infections. Arbitrarily, patch one has a higher amount of infectedindividuals with 500 at the start for the fully connected model. Tiered connection, seen in Fig. 8 demonstrates containinginfection from a central sick hub.
Node one starts with the 500 infected. Two outgoing connections from this node propagate to nodes two and three. Eachof these nodes is outwardly connected to one of the last two nodes.
Proper resource allocation is important to hinder the spread of infection and hopefully eliminate it. To represent realscenarios a limit on the amount of resources available to the entire connectivity matrix is set. Two amounts are used inthe experiments for the total resource available:
k∈Ω γk is 2 or 1.4. A total of two for the resource represents plenty of
resources, while a total of 1.4 is a limiting amount since the sumof the infection rates is 1.65. One constraint on the allocationof resources is that each patch can only have at most one unit of resource. In other words, the recovery rate is at most 100%for any patch.
These four experiment combinations, two on a fully connected graph and two on a tiered graph, are each runwith two ofthe three objective functions: minimum average and minimum equilibrium. The minimum peak is excluded because theseexperiments produced solutions inwhich the infected population instantly decreased from the initial set-up. This effectivelymakes the experiments equivalent to the minimum peak experiments. Most solutions found by the genetic algorithm areas expected with all three objective functions. These expected solutions offer more support to the resource allocation workin literature [4].
Fig. 9 shows the typical behavior on a fully connected graph. As expected the fully connected graph increases the numberof patches affected by the epidemic. This in turn increases the amount of resources needed to solve the problem. On a tieredgraph we typically see the behavior in 10. The problem requires fewer resources to solve because the effective radius ofthe infection is minimized. In other words if only a fraction of infected individuals can move from the hub to the first tierof outlying patches, then only a fraction of a fraction of people can move to the second and so on. The infection can onlypropagate so far (unless it is not containable with the allotted amount of resources) and the GA takes advantage of this bydistributing resources relatively evenly amongst the patches in the affected radius and not at all amongst the other patches.If however there are not enough resources to contain the infection in the initial affected radius the GA behaves differently.If not contained the affected radius can grow. Patches on the initial outlying boundary can become epidemic generatorsthemselves and generate enough infected individuals that a fraction travel to new patches. In this case the GA does notattempt to distribute resources evenly amongst the initial radius but rather distributes them evenly across every patch. Themost effective resource distribution in a no-win situation from the GA’s perspective is to keep everyone the least sick byallocating resources evenly amongst the global population (see Fig. 10).
5. Conclusion
This work has just scratched the surface of a much larger investigation of the multi-parameter minimization of infectionspread in epidemicmodels.Much ofwhat has been used here can be applied towards other epidemicmodels aswell. Perhapsone would see radically different behavior if the GA was minimizing on a population that had recovered individuals whocould no longer become infected or onmodelswhere births and deathswere taken into account.Many questions still remainto be answered; for instance:
1. Can the infected individuals be further minimized by evolving across both the resource allocation andmobility matrix atone time?
2. How different are solutions if we impose constraints on the mobility optimization such as keeping certain patches or acertain number of patches connected?
6266 J. Sanders et al. / Physica A 391 (2012) 6256–6267
Fig. 9. Minimizing average infected with resource allocation (fully connected): simulated population distribution.
Fig. 10. Minimizing average infected with resource allocation (tiered): simulated population distribution.
3. In what way will the solutions change if we place more weight on certain patches health than others?4. With a larger number of patches is it possible that the GA may evolve scale free networks in order to become robust to
infection spread?5. Is the effective infection radius at time t calculable given that we know the initial population distribution and the γ and
β values?
This work can be applied to both human and computer epidemiological problems. In situations where resources may bescarcewehave shown that infection can be effectivelyminimized utilizing only themobility fromcommunity to community.Additionally we have shown that genetic algorithms can be effective in optimizing real world problems with large searchspaces in a relatively small amount of time. Finally we have verified previous theoretical work [2] that indicates more travelmay be better in the long run than less travelwhen it comes tominimizing infection. This is a surprising result as the commonintuition is to quarantine individual patches. Overall the problemmerits further investigation and shows promise in helpingunderstand infection spread in many types of real world communities.
Acknowledgment
RAV was supported in part by an NSF research fellowship.
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