Parameter Estimation in Ordinary Differential Equations ...downloads.hindawi.com/journals/jam/2018/9160793.pdf · Parameter Estimation in Ordinary Differential Equations Modeling

Research ArticleParameter Estimation in Ordinary Differential EquationsModeling via Particle Swarm Optimization

Devin Akman1 Olcay Akman 2 and Elsa Schaefer3

1University of Illinois Urbana-Champaign USA2Illinois State University USA3Marymount University USA

Correspondence should be addressed to Olcay Akman oakmanilstuedu

Received 25 May 2018 Accepted 29 July 2018 Published 2 September 2018

Academic Editor Theodore E Simos

Copyright copy 2018 Devin Akman et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Researchers using ordinary differential equations to model phenomena face two main challenges among others implementingthe appropriate model and optimizing the parameters of the selected model The latter often proves difficult or computationallyexpensive Here we implement Particle Swarm Optimization which draws inspiration from the optimizing behavior of insectswarms in nature as it is a simple and efficient method for fitting models to data We demonstrate its efficacy by showing that itoutstrips evolutionary computing methods previously used to analyze an epidemic model

1 Introduction

In mathematical biology parameter estimation is one ofthe most important components of model fitting procedureWith poorly estimated parameters even the most appro-priate models perform poorly Although there is alreadyan abundance of traditional algorithms in the literaturesuch as Gauss-Newton methods the Nelder-Mead methodand simulated annealing (see [1] for a thorough review) asmodels get more complex researchers need more versatileand capable optimization methods Evolutionary computingmethods are becoming more frequently used tools reducingthe computational cost ofmodel fitting and starting to attractinterest among mathematical biologists

Compartmental models that are frequently used in infec-tious disease modeling have not escaped the computationalcost versus complex model dilemma either Since parameterestimation for compartmental models can be tackled by suchevolutionary computing methods it is reasonable to examinethe goodness of fit versus the price of fit for these commonlyemployed modelsTherefore we will focus on the lesser-usedevolutionary algorithms in comparison to Particle SwarmOptimization (PSO) In particular Genetic Algorithms (GA)have been frequently used to optimize the parameters of

ordinary differential equations (ODE)models [2ndash7] PSO hasthus far been underutilized in this area It is an optimizationalgorithm inspired by swarms of insects birds and fish innature Although PSO has been applied in a number ofdifferent scenarios [8] its performance on compartmentalmodels has not yet been studied

In optimization the computational cost of an algorithmis just as important as the quality of its output Unfortunatelyfor us cost increases with quality Evolutionary computingalgorithms are especially susceptible to this effect whereinsmall reductions in error must be paid for by disproportion-ate amounts of additional computation To reduce cost orimprove accuracy without compromising the other requiresthe application of an innovative technique to the problemFor example Hallam et al [9] implemented progressivepopulation culling in a genetic algorithm to hit a given errortarget in fewer CPU cycles

Here we intend to show that PSO is not only a viabletechnique for fitting ODE models to data but also that it hasthe potential to outperformGA which was proposed in [2] asa superior optimization method to the traditional ones for fit-ting ODEmodels Hence with this study we aim to establishan even more viable tool for ODE model fitting Specificallywe apply PSO to kinetic parameter optimizationfitting in

HindawiJournal of Applied MathematicsVolume 2018 Article ID 9160793 9 pageshttpsdoiorg10115520189160793

2 Journal of Applied Mathematics

the context of ODEs and we contrast the PSO and GAalgorithms using the cholera SIRB example as commonground The organization of this article is as follows InSection 2 we introduce PSO Section 3 contains a descriptionof how PSO can be applied to ODE models We illustrate animplementation of PSO to optimize anODEmodel of cholerainfections during the recent Haitian epidemic in Section 4Finally we provide concluding remarks in Section 5

2 Particle Swarm Optimization

We are given an objective function 119891 R119863 997888rarr R with thegoal ofminimization Based on reasonable parameter boundsfor the problem at hand we constrain our search space to theCartesian product

Σ =119863

prod119894=1

[119871 119894 119880119894] (1)

for lower bounds 119871 119894 and upper bounds 119880119894 We release aldquoswarmrdquo of point particles within the space and iterativelyupdate their positions The simulation rules allow the parti-cles to explore and exploit the space all the while communi-cating with one another about their discoveries Since PSOdoes not rely on gradients or smoothness in general it iswell-suited to optimizing chaotic and discontinuous systemsThis flexibility is an essential property in the domain ofODEmodels whose prediction errors may change drasticallywith minute parameter variations The swarm nature ofPSO makes it amenable to parallelization giving it a sizableadvantage over sequential methods The algorithm used forour computations is largely based on SPSO 2011 as found in[10] We provide implementation details in Section 2 Thenwe discuss ODE models as they are ubiquitous in the realmofmodeling natural phenomenaWe provide a justification ofour procedure for hyperspherical sampling in the Appendix

21 Particle Swarm Optimization Implementation

211 Initialization First we define a new objective function119892 [0 1]119863 997888rarr R by

119892 (119909) = 119891 (119887 + 119872119909) (2)

where 119887 = (1198711 sdot sdot sdot 119871119863)119879 and 119872 = diag(1198801 minus 1198711 sdot sdot sdot 119880119863 minus

119871119863) Hence the particles search within the unit hypercubewhich provides better hyperspherical sampling performancethan does a search space whose dimensions could be ordersof magnitude apart The particlesrsquo positions are then affinelymapped to 119878 before evaluation by the objective function

A swarm consists of an ordered list of 119878 particles Thisvalue is manually adjusted to fit the problem at handalthough more advanced methods may attempt to select itwith ametaoptimizer or adaptively adjust it during executionIf 119878 is too low then the particles will not be able to explore thespace effectively if it is too high then computational costswill rise without yielding a significant improvement in thefinal fitness Each particle has attributes which are updatedat every iteration The attributes are shown in Table 1 and

their initial values (119905 = 0) are given We define 119880(119886 119887) tomean a value sampled uniformly from [119886 119887] The particlesare initially placed in the search space via Latin HypercubeSampling (LHS)This helps avoid the excessive clustering thatcan occur with truly random placement as we want all areasof the search space to be considered The reader is directed to[11] for more details on LHS

The PSO algorithm derives its power from the communi-cation of personal best positions among particles During theinitialization phase each particle randomly and uniformlyselects 119870 natural numbers in [1 119878] (with replacement) andadds its own index 119895 to the list Once again 119870 is a tunableparameter of the algorithm one might use 119870 = 3 as agood starting point These numbers with repeats discardedform a set 119873119895 They tell the particle which of its peers tocommunicate with when it discovers lower values of theobjective function during the execution of the algorithmThiscommunication is also performed once to initialize the 119897119895values before any motion has taken place In addition 119873119895 iscontinually updated based on the performance of the currentconfiguration this aspect of the algorithm is known as itsadaptive random topology

22 Pragmatic Considerations

221 Exploration and Exploitation Thedegree of explorationin a given PSO configuration refers to the propensity of theparticles to travel large distances within the search spaceand visit different regions This must be properly balancedwith exploitation which is the tendency of particles to staywithin a given region and examine local features to seekout even lower values of the objective function The PSOspecification provides a parameter to this end and itmay haveto be manually or automatically tuned for best results on aparticular function landscape

23 Algorithm We present a description of the algorithmafter initialization below Each step is followed by a commentthat may describe its motivation practical implementationdetails or pitfalls Table 1 contains descriptions of the particleattributes The parameters 119888 and 120596 (not to be confusedwith the 120596 in the SIRB model) used in the algorithmmust be selected beforehand They control the exploration-exploitation balance and the ldquoinertiardquo of the particles respec-tively

(1) Apply a random permutation to the indices of theparticlesPermutations can be selected uniformly at randomwith the simple Fisher-Yates shuffle

(2) For 119895 = 0 to 119878 minus 1

(a) Set 119866119895 = 119909119895 + 119888(119901119895 + 119897119895 minus 2119909119895)3We define 119866119895 to be the ldquocenter of gravityrdquo of thesepoints 119909119895 119909119895 + 119888(119901119895 minus 119909119895) and 119909119895 + 119888(119897119895 minus 119909119895)This provides a mean position around which theparticle can search for new locationsThe larger 119888is the farther the particle will venture on averagePrevious versions of SPSO had detectable biases

Journal of Applied Mathematics 3

Table 1 Particle attributes indexed by position in list

Property Domain Meaning Initial Value119909119895 [0 1]119863 position determined via LHSV119895 R119863 velocity V119895119889(0) = 119880(minus119909119895119889(0) 1 minus 119909119895119889(0))119901119895 [0 1]119863 personal best position 119909119895(0)119902119895 R personal best fitness 119892(119909119895(0))119873119895 2N neighborhood described below119897119895 [0 1]119863 local best position described below119898119895 R local best fitness 119892(119897119895(0))

along the axes of the coordinate system Thiscoordinate-free definition elegantly avoids such aproblem

(i) If 119901119895 = 119897119895 let119866119895 = 119909119895 +119888(119901119895 minus119909119895)2 insteadIf the local best position is equal to theparticlersquos personal best then the originalformula would give a weight of 23 to 119909119895 +119888(119901119895 minus 119909119895) and only 13 to 119909119895 To avoidthis we revise the definition of 119866119895 to be themean of only 119909119895 and 119909119895 + 119888(119901119895 minus 119909119895) inthis case Here we are testing the coordinatesfor bitwise equality as opposed to the usualfloating point comparison This is because aparticle will have 119901119895 exactly equal to 119897119895 whenit has the lowest personal best fitness in itsnetwork of peers

(b) Sample 1199091015840119895 uniformly from the volume of ahypersphere with center 119866119895 and radius 119903 =119909119895 minus 119866119895One might naively attempt this by generating avector representing displacement from 119866119895 whosecoordinates are 119880(0 1) each and scaling it to alength of 119880(0 119903) However this does not producea constant probability densityThe proper methodis to generate a vector whose coordinates arestandard normal variates and scale it to a lengthof 119903119880(0 1)1119863 Normal variates can be generatedfrom a uniform source with methods such as theBox-Muller transform and Ziggurat algorithm

(c) Set V119895(119905 + 1) = 120596V119895(119905) + [1199091015840119895(119905) minus 119909119895(119905)]

The position 1199091015840119895 represents where the particleintends to travel nextThe second term of V119895(119905+1)contains the corresponding displacement from itscurrent position However we would also like toretain some of the particlersquos old velocity to lessenthe chance of getting trapped in local minimumTo this end we add an inertial term with aldquocooldownrdquo factor 120596 that functions analogouslyto friction

(d) Set 119909119895(119905 + 1) = 119909119895(119905) + V119895(119905 + 1)The next position of the particle is its currentposition plus its updated velocity

(e) Clamp position if necessary

If any component of a particlersquos position fallsoutside [0 1] we clamp it to the appropriate end-point and multiply the corresponding componentof its velocity by minus05 This inelastic ldquobouncingrdquocontains particles while still allowing exploitationof the periphery of the search space

(f) Increment 119905(g) If 119892(119909119895(119905)) lt 119902119895(119905 minus 1) set 119901119895(119905) = 119909119895(119905) and

119902119895(119905) = 119892(119909119895(119905)) Else carry forward values fromprevious iterationIf the value of the objective function at the currentposition is better than the personal best valuefrom the previous iteration set the personal bestposition to the current position and the personalbest value to the current value

(h) Communicate with neighborsWhen a particle finds a value of the objectivefunction lower than its 119898119895 (which correspondsto the position 119897119895) it informs all of its neighborsby updating their values of these attributes to itscurrent objective function value and position

(i) Regenerate topology if global fitness min119896119902119896 didnot improve from previous stepIf the global fitness of the swarmhas not improvedat the end of the iteration then the sets 119873119895are updated by regenerating them as during theinitialization phase

(j) Check stopping condition See below for exam-ple conditions

A multitude of conditions can be used to decide when tostop Some common ones are as follows

(i) The iteration count exceeds a threshold 119879 For a fixedthreshold this method gives more computation timeto larger swarms

(ii) The product of the iteration count and the swarmsize (the number of objective function evaluations)exceeds 119879 In this case the threshold is a goodrepresentation of how much total computation timethe swarm utilizes as the computational resourcelimit is nearly independent of swarm size

(iii) The globally optimal objective function value(min119895119902119895) falls below some threshold


Initialize particles

Compute newvelocities as intended

destinations plusinertia

Compute intendeddestinations

Communicate withneighbors

Update positions

No

Yes Global fitnessimproved

Regenerate topology

Yes

No

Stopping rulemetStop

Figure 1 Particle Swarm Optimization process

(iv) The globally optimal objective function value has notimproved by more than 120598 within the last119873 iterations

(v) Any combination of the above

The output of the algorithm is then the 119901119895 value corre-sponding to the minimum 119902119895 value

The flowchart (Figure 1) summarizes the process ofParticle Swarm Optimization

231 Discrete Parameters If some (not necessarily all) of theparameters in the problem are discrete PSO can be adaptedby snapping the appropriate continuous coordinates in thesearch space to the nearest acceptable values during eachiteration This feature gives PSO great versatility in that it cansolve problems with entirely continuous mixed or entirelydiscrete inputs

232 Parallel Computing Most real-world objective func-tions are sufficiently complex so that PSO will spend thevast majority of its time evaluating them as opposed toexecuting the control logic In our case profiling showed thatnumerically solving an ODE at many points consumed morethan 95 of the total CPU time Therefore in order to paral-lelize the algorithm one needs only to evenly distribute theseevaluations among different cores or possibly computersTheremaining logic of the algorithm can be executed in a singlecontrol thread that collates the fitness values and updatesparticle attributes before offloading evaluations for the nextiteration

233 Randomness Far too often in our experience proba-bilistic numerical simulations are run with little thought tothe underlying random number generator (RNG) As Clerc


Table 2 Parameter ranges for cholera model

Name Description Lower bound Upper bound119887 Natural birth rate of humans 0000072 NA119889 Natural death rate of humans 0000044 NA120581 Half-saturation constant of vibrios 106 NA1198610 Starting bacterial concentration 0 1ℎ Proportion of hospitalizations 0001 1120573119861 Ingestion rate of vibrios 0001 1120573119868 Infectivity for susceptible 0001 05120578 Rate of contribution by infected 0001 1120574 Recovery rate of infected 114 12120575 Decay rate of vibrios to non-HI state 140 13120596 Rate of waning immunity 00001 1360

states in [10] the performance of PSO on some problemsis highly dependent on the quality of the RNG The defaultRNG in onersquos favorite programming language is likely notup to the task By default our code uses the fast simpleand statistically robust xorshift128+generator from [12] Itoutputs random 64-bit blocks which are transformed usingstandard algorithms into the various types of random samplesrequired The use of a seed allows for exactly reproduciblesimulations (this is more difficult to ensure if a singlegenerator is being accessed by multiple threads)

3 ODE Models

31 Introduction Suppose we have data and a correspondingsystem of ordinary differential equations which we believemodels the underlying process The system takes parametersand initial values (some known some unknown) as inputand outputs functions which can then be compared to thedata using some goodness of fit (GOF) metric (eg meansquared error) Competing models can be evaluated usingframeworks introduced in [3]

If we take the vector of unknown inputs to be a positionwithin search space then we can apply PSO in order tominimize the modelrsquos error The choice of error functionis an important consideration Those GOF metrics that aresuperlinear in the individual deviations will preferentiallyreduce larger deviations over smaller ones Similarly sub-linear functions will produce good correspondence on mostdata points at the expense of some larger errors

32 Compartmental Models In epidemiology a commontool for studying the spread of an infectious disease isthe compartmental model The prototypical example is theSIR model in which the population is divided into threecompartments Susceptible Infected and Recovering Differ-ential equations describe how individuals flow between thecompartments over time The behavior of a specific diseaseis captured by parameters and it is here that we use modelfitting techniques to derive these parameters from data Thesimplest form of SIR does not include birth or death (thedynamics of the disease are assumed to be sufficiently rapid)so it holds that the population size119873 = 119878 + 119868 + 119877 is constant

More advanced models include more compartments andincorporate factors such as demography In the field it is ofteneasiest to measure the number of infected individuals thus aviable GOF metric may compute the distance between timeseries infection data and the corresponding points on the 119868(119905)curve for a given set of parameters

Chubarova [13] implemented GA for fitting ODE mod-els to cholera epidemic data Although better errors wereachieved than similar studies that used traditional methodsthe computational cost required was substantial This ledto our consideration of a technique which could achievesimilar accuracy with fewer cycles namely PSO One of themodels studied in [13] was SIRB which is a refinement ofSIR it includes birth and death rates for the population and acompartment 119861 representing the concentration of bacteria inthe water supplyWe present equations for SIRB and describethe roles of its parameters in Section 4

4 Application

41 SIRB The SIRBmodel is a system of four nonlinear first-order ODEs that capture how quickly individuals progressthrough the susceptibility-infection-recovery cycle and howbacteria are exchanged between the population and the watersupply

119889119878119889119905

= 119887119873 minus 119889119878 minus 120573119861119861119878

120581 + 119861minus 120573119868

119878119868119873

+ 120596119877

119889119868119889119905

= minus119889119868 + 120573119861119861119878

120581 + 119861+ 120573119868

119878119868119873

minus 120574119868

119889119877119889119905

= minus119889119877 + 120574119868 minus 120596119877

119889119861119889119905

= 120578119868 minus 120575119861

(3)

The parameters and their ranges are listed in Table 2(derived from [2] with some later corrections from theauthors) An upper bound of NA means that the value isconstant and not optimized by PSO The bacteria beingstudied in this case are of the genus Vibrio They decay froma state of hyperinfectivity (HI) to a non-HI state (only the


Cholera Hospitalizations in Department Nord

PredictedObserved

100 200 300 4000Days (t)

0

200

400

Hos

pita

lizat

ions

(hI)

Figure 2 Example run

former is considered infectious in this model) at a rate of 120575The parameter 120581 has units of cells per milliliter while the resthave units of inverse days

42 Model Fitting We apply PSO to fit the SIRB model toepidemic data from [14] using our own C code and the rkf45ODE solver in the GNU Scientific Library Since the datawere obtained from hospitals we must correct for the factthat not every infected person received careWe introduce theparameter ℎ as the proportion of recorded infections Insteadof presuming a certain value we allow ℎ to be optimized byPSO The initial values for the system for Department Nordare 119878(0) = 549486 119868(0) = 1ℎ 119877(0) = 0 and 119861(0) = 1205811198610where 1198610 is another parameter to be optimized by PSO Theparticles explore an 8-dimensional search space with genericposition vector

119909 = (1198610 ℎ 120573119861 120573119868 120578 120574 120575 120596)119879 (4)

We measure GOF using the mean absolute deviation

119891 (119909) = 1119899

119899

sum119894=1

1003816100381610038161003816ℎ119868 (119905119894) minus 1199101198941003816100381610038161003816 (5)

where (119905119894 119910119894) is the time series infection data Figure 2 showsa plot of ℎ119868(119894) and 119910119894 versus 119905 for an example run of the PSOalgorithm The tail behavior of our model is typical in that itfails to account for cyclical recurrence of the disease Akmanet al address this phenomenon in [2]

We performed 12 runs of the algorithm with the param-eters 119888 = 1193 120596 = 0721 119878 = 40 and 119870 = 3(suggested values from [10]) We chose a stopping conditionof two million objective function evaluations We found thiscondition experimentally by observing that fitness valuesrarely if ever improved beyond this point Our results aregiven in Table 3 An individual run only takes around 8minutes on a quad-core Intel Celeron J1900 199 GHz(compiled with icc -O2) The same optimization problemapproached using GA in [13] took between approximately1 and 64 days to run on two hexa-core Intel Xeon X5670s 293 GHz (see Table 4) Considering the similarity ofthe fitness values to the evolutionary computing resultsPSO appears to be an attractive method for model fitting

especially for the case of expensive objective functions Ituses smaller populations and fewer timesteps due to the fastconvergence of the fitness values which distinguishes it fromGA

Code for PSO and the SIRB model with sample data isavailable at [11]

5 Concluding Remarks

The use of PSO is a novel approach in ODE modelingparticularly in the field of epidemiology Where previousalgorithms such as GA have taken excessively long to run inthe face of complex models PSO demonstrates its effective-ness by providing comparable results in an impressively smallfraction of the time It has a structural advantage over GA thatcannot be attributed to the complexity of the control logicPSO requires many fewer objective function calls to reachthe same level of accuracy We implemented PSO to optimizean ODE model given cholera data Although there are otherglobal optimization methods that are also used in fittingparameters of ODE models such as simulated annealingor scatter search we restricted our comparison only to theperformance of GA since it was extensively used for the sameinfectious disease model We achieved similar model errorsin a substantially shorter period of time as explained abovedue to the fundamental differences of operation between PSOand GA Additionally we compared the performance of PSOwith that of DIRECT [15ndash19] per the suggestion of a refereeDIRECT evaluated the objective function approximately sixthousand times in the same amount of time it took PSO toperform twomillion evaluations and it appeared to convergeto a worse fitness value

As with most optimization algorithms the remainingchallenge of PSO is algorithm parameter selection In thecontext of our study this is similar to selecting the right GAparameters such as population size number of generationscross-over rate and mutation rates that all may have animpact on the accuracy and runtime One possible improve-ment is to add a metaoptimizer to choose the algorithmparameters without human intervention A fruitful potentialtopic of investigation is to extend PSO to approximatingunknown functions (such as probability distributions) ratherthan merely selecting parameters

Appendix

Hyperspherical Sampling

In this appendix we justify our procedure for uniformhyperspherical sampling Our goal is to provide a practicalconstruction for a random variable 119882 which takes on valuesfrom the unit119863-ball with a uniform probability densityThen119903119882 will provide the appropriate distribution for the PSOalgorithm Let 119885 be a random variable distributed accordingto the 119863-variate normal distribution of identity covariancecentered at the origin We can sample from 119885 in our codeby generating a vector with 119863 independent standard normalcomponents It is well known that the distribution of 119885 isrotationally symmetric Therefore defining 119891(119911) = 119911119911 we


Table3Parameter

values

from

PSOandcorrespo

ndingfitness

119861 0ℎ

120573 119861120573 119868

120578120574

120575120596

Fitness

Time(s)

000

6170

000

3569

0012

9567

000

1522

087

5338

008

0457

007

8863

000

2336

2719

481

000

6424

000

8251

010

2802

007

7925

079

6273

018

2466

004

8924

000

2660

2826

484

000

7413

000

3196

64015

9124

002

1619

061

7604

007

3014

010

0941

000

2739

2718

477

000

5186

000

3286

015

0995

004

2600

100

0000

007

1429

022

7019

000

2769

2825

477

000

7859

000

3576

009

6382

013

8211

032

1430

007

9145

004

9010

000

2703

2776

473

005

3613

000

3348

16000

9927

002

0090

066

4410

007

1454

002

6494

000

2743

2769

477

000

9432

000

4611

012

2929

003

3397

5811

53010

3619

004

8074

000

2384

2754

482

000

5751

000

3202

018

4915

000

1033

093

3074

007

1438

022

2235

000

2777

2773

485

000

8324

000

3625

010

0021

005

9395

078

1264

008

1375

007

2428

000

2535

2729

478

001

0934

000

3549

55006

5875

014

8408

4260

06007

8190

004

8143

000

2676

2780

475

000

5570

000

3256

024

7551

000

9451

3375

38007

3270

007

5757

000

2577

2733

476

001

9549

000

3365

003

2144

018

0211

4615

70007

3522

004

0767

000

2756

027

88

477


Table4GArunsta

tistic

sTimetomakee

lites

(s)

Timetomakep

aram

eter

sets(s)

Totaltim

e(s)

Samples

ize

Stop

ping

threshold

Num

bero

felites

Max

generatio

nsParameter

setscreated

Fitness

8680

550

9230

5000

1040

010

020

2755

8643

528

9171

5000

1040

010

020

2755

8615

546

9161

5000

1040

010

020

2754

7446

543

7989

4000

1040

010

020

2751

7480

518

7998

4000

1040

010

020

2755

6306

571

6877

3000

1040

010

020

2754

6192

528

6720

3000

1040

010

020

2752

5115

550

5665

2000

1040

010

020

2754

5074

500

5574

2000

1040

010

020

2755

4285

517

4802

1000

1040

010

020

2748

4160

463

4623

1000

1040

010

020

2756

2359

562

2921

500

1040

010

020

2758

2780

556

3336

500

1040

010

020

2759

1963

652

2615

100

1040

010

020

2751

1892

612

2504

100

1040

010

020

2757

748

949

1697

5010

400

100

202788

561

941

1502

5010

400

100

2027

62

386

1052

1438

5010

200

100

2027

97

381

1118

1499

5010

200

100

2027

73

960

960

1920

100

1020

010

020

2756

951

849

1800

100

1020

010

020

2767


find that 119891(119885) must be uniformly distributed on the surfaceof the unit 119863-sphere Since a 119863-ball with radius 119904 le 1 hasvolume proportional to 119904119863 we would like to find a randomvariable 119884 supported on [0 1] such that

119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)

It is easily shown that 119884 = 119880(0 1)1119863 has a cumulativedistribution function of 119866(119904) = 119904119863 Setting 119882 = 119884119891(119885)produces the desired result

The same distribution can also be produced in a simplerfashion by repeatedly generating a vector 119911 with componentsof 119880(minus119903 119903) and rejecting it if 119911 gt 119903 However the expectednumber of tries per sample is equal to the ratio of the volumeof a 119863-cube to that of an inscribed 119863-ball which grows toorapidly with 119863

Data Availability

Thecholera infection data used to support the findings of thisstudy are readily available on the Haiti Department of Healthwebsite

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] J M Krueger Parameter Estimation Methods for OrdinaryDifferential Equation Models with Applications to Microbiology[PhD thesis] Virginia Tech 2017

[2] O Akman and E Schaefer ldquoAn evolutionary computingapproach for parameter estimation investigation of a model forcholerardquo Journal of Biological Dynamics vol 9 pp 147ndash158 2015

[3] O Akman M R Corby and E Schaefer ldquoExamination ofmodels for cholera insights into model comparison methodsrdquoLetters in Biomathematics vol 3 no 1 pp 93ndash118 2016

[4] R Scitovski and D Jukic ldquoA method for solving the parameteridentification problem for ordinary differential equations of thesecond orderrdquo Applied Mathematics and Computation vol 74no 2 pp 273ndash291 1996

[5] N Tutkun ldquoParameter estimation in mathematical modelsusing the real coded genetic algorithmsrdquo Expert Systems withApplications vol 36 no 2 pp 3342ndash3345 2009

[6] M A Khalik M Sherif S Saraya and F Areed ldquoParame-ter identification problem Real-coded GA approachrdquo AppliedMathematics and Computation vol 187 no 2 pp 1495ndash15012007

[7] E K Nyarko and R Scitovski ldquoSolving the parameter identi-fication problem of mathematical models using genetic algo-rithmsrdquo Applied Mathematics and Computation vol 153 no 3pp 651ndash658 2004

[8] A Lazinica Particle Swarm Optimization Kirchengasse 2009[9] J W Hallam O Akman and F Akman ldquoGenetic algorithms

with shrinking population sizerdquo Computational Statistics vol25 no 4 pp 691ndash705 2010

[10] M Clerc Standard Particle Swarm Optimisation (Rep)Retrieved July 1 2016 from httpclercmauricefreefrpsoSPSO descriptionspdf

[11] R L Iman ldquoLatin hypercube samplingrdquo in Encyclopediaof Quantitative Risk Analysis and Assessment Wiley OnlineLibrary 2008

[12] S Vigna ldquoFurther scramblings of Marsagliarsquos generatorsrdquo Jour-nal of Computational andAppliedMathematics vol 315 pp 175ndash181 2017

[13] I N Chubarova Modeling cholera using engineered geneticalgorithm [PhD thesis] Illinois State University 2001

[14] ldquoRepublique drsquohaiti ministere de la sante publique et de la popu-lation Rap-ports journaliers dumspp sur lrsquoevolution du choleraen haitirdquo Retrieved September 6 2014 from httpmsppgouvhtsitedownloads

[15] D E FinkelDIRECTOptimizationAlgorithmUserGuide vol 2Center for Research in Scientific Computation North CarolinaState University 2003

[16] D E Finkel and C T Kelley ldquoAdditive scaling and the DIRECTalgorithmrdquo Journal of Global Optimization vol 36 no 4 pp597ndash608 2006

[17] J M Gablonsky and C T Kelley ldquoA Locally-Biased form of theDIRECTAlgorithmrdquo Journal of Global Optimization vol 21 no1 pp 27ndash37 2001

[18] R Grbic E K Nyarko and R Scitovski ldquoA modification ofthe DIRECT method for Lipschitz global optimization for asymmetric functionrdquo Journal of Global Optimization vol 57 no4 pp 1193ndash1212 2013

[19] D R Jones C D Perttunen and B E Stuckman ldquoLipschitzianoptimization without the Lipschitz constantrdquo Journal of Opti-mizationTheory andApplications vol 79 no 1 pp 157ndash181 1993

Hindawiwwwhindawicom Volume 2018

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Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Submit your manuscripts atwwwhindawicom


the context of ODEs and we contrast the PSO and GAalgorithms using the cholera SIRB example as commonground The organization of this article is as follows InSection 2 we introduce PSO Section 3 contains a descriptionof how PSO can be applied to ODE models We illustrate animplementation of PSO to optimize anODEmodel of cholerainfections during the recent Haitian epidemic in Section 4Finally we provide concluding remarks in Section 5

2 Particle Swarm Optimization

We are given an objective function 119891 R119863 997888rarr R with thegoal ofminimization Based on reasonable parameter boundsfor the problem at hand we constrain our search space to theCartesian product

Σ =119863

prod119894=1

[119871 119894 119880119894] (1)

for lower bounds 119871 119894 and upper bounds 119880119894 We release aldquoswarmrdquo of point particles within the space and iterativelyupdate their positions The simulation rules allow the parti-cles to explore and exploit the space all the while communi-cating with one another about their discoveries Since PSOdoes not rely on gradients or smoothness in general it iswell-suited to optimizing chaotic and discontinuous systemsThis flexibility is an essential property in the domain ofODEmodels whose prediction errors may change drasticallywith minute parameter variations The swarm nature ofPSO makes it amenable to parallelization giving it a sizableadvantage over sequential methods The algorithm used forour computations is largely based on SPSO 2011 as found in[10] We provide implementation details in Section 2 Thenwe discuss ODE models as they are ubiquitous in the realmofmodeling natural phenomenaWe provide a justification ofour procedure for hyperspherical sampling in the Appendix

21 Particle Swarm Optimization Implementation

211 Initialization First we define a new objective function119892 [0 1]119863 997888rarr R by

119892 (119909) = 119891 (119887 + 119872119909) (2)

where 119887 = (1198711 sdot sdot sdot 119871119863)119879 and 119872 = diag(1198801 minus 1198711 sdot sdot sdot 119880119863 minus

119871119863) Hence the particles search within the unit hypercubewhich provides better hyperspherical sampling performancethan does a search space whose dimensions could be ordersof magnitude apart The particlesrsquo positions are then affinelymapped to 119878 before evaluation by the objective function

A swarm consists of an ordered list of 119878 particles Thisvalue is manually adjusted to fit the problem at handalthough more advanced methods may attempt to select itwith ametaoptimizer or adaptively adjust it during executionIf 119878 is too low then the particles will not be able to explore thespace effectively if it is too high then computational costswill rise without yielding a significant improvement in thefinal fitness Each particle has attributes which are updatedat every iteration The attributes are shown in Table 1 and

their initial values (119905 = 0) are given We define 119880(119886 119887) tomean a value sampled uniformly from [119886 119887] The particlesare initially placed in the search space via Latin HypercubeSampling (LHS)This helps avoid the excessive clustering thatcan occur with truly random placement as we want all areasof the search space to be considered The reader is directed to[11] for more details on LHS

The PSO algorithm derives its power from the communi-cation of personal best positions among particles During theinitialization phase each particle randomly and uniformlyselects 119870 natural numbers in [1 119878] (with replacement) andadds its own index 119895 to the list Once again 119870 is a tunableparameter of the algorithm one might use 119870 = 3 as agood starting point These numbers with repeats discardedform a set 119873119895 They tell the particle which of its peers tocommunicate with when it discovers lower values of theobjective function during the execution of the algorithmThiscommunication is also performed once to initialize the 119897119895values before any motion has taken place In addition 119873119895 iscontinually updated based on the performance of the currentconfiguration this aspect of the algorithm is known as itsadaptive random topology

22 Pragmatic Considerations

221 Exploration and Exploitation Thedegree of explorationin a given PSO configuration refers to the propensity of theparticles to travel large distances within the search spaceand visit different regions This must be properly balancedwith exploitation which is the tendency of particles to staywithin a given region and examine local features to seekout even lower values of the objective function The PSOspecification provides a parameter to this end and itmay haveto be manually or automatically tuned for best results on aparticular function landscape

23 Algorithm We present a description of the algorithmafter initialization below Each step is followed by a commentthat may describe its motivation practical implementationdetails or pitfalls Table 1 contains descriptions of the particleattributes The parameters 119888 and 120596 (not to be confusedwith the 120596 in the SIRB model) used in the algorithmmust be selected beforehand They control the exploration-exploitation balance and the ldquoinertiardquo of the particles respec-tively

(1) Apply a random permutation to the indices of theparticlesPermutations can be selected uniformly at randomwith the simple Fisher-Yates shuffle

(2) For 119895 = 0 to 119878 minus 1

(a) Set 119866119895 = 119909119895 + 119888(119901119895 + 119897119895 minus 2119909119895)3We define 119866119895 to be the ldquocenter of gravityrdquo of thesepoints 119909119895 119909119895 + 119888(119901119895 minus 119909119895) and 119909119895 + 119888(119897119895 minus 119909119895)This provides a mean position around which theparticle can search for new locationsThe larger 119888is the farther the particle will venture on averagePrevious versions of SPSO had detectable biases







(c) Set V119895(119905 + 1) = 120596V119895(119905) + [1199091015840119895(119905) minus 119909119895(119905)]




















Update positions

No


Regenerate topology

Yes

No














3 ODE Models






4 Application


119889119878119889119905

= 119887119873 minus 119889119878 minus 120573119861119861119878

120581 + 119861minus 120573119868

119878119868119873

+ 120596119877

119889119868119889119905

= minus119889119868 + 120573119861119861119878

120581 + 119861+ 120573119868

119878119868119873

minus 120574119868

119889119877119889119905

= minus119889119877 + 120574119868 minus 120596119877

119889119861119889119905

= 120578119868 minus 120575119861

(3)




PredictedObserved

100 200 300 4000Days (t)

0

200

400

Hos

pita

lizat

ions

(hI)




119909 = (1198610 ℎ 120573119861 120573119868 120578 120574 120575 120596)119879 (4)


119891 (119909) = 1119899

119899

sum119894=1

1003816100381610038161003816ℎ119868 (119905119894) minus 1199101198941003816100381610038161003816 (5)








Appendix




Table3Parameter

values

from

PSOandcorrespo

ndingfitness

119861 0ℎ

120573 119861120573 119868

120578120574

120575120596

Fitness

Time(s)

000

6170

000

3569

0012

9567

000

1522

087

5338

008

0457

007

8863

000

2336

2719

481

000

6424

000

8251

010

2802

007

7925

079

6273

018

2466

004

8924

000

2660

2826

484

000

7413

000

3196

64015

9124

002

1619

061

7604

007

3014

010

0941

000

2739

2718

477

000

5186

000

3286

015

0995

004

2600

100

0000

007

1429

022

7019

000

2769

2825

477

000

7859

000

3576

009

6382

013

8211

032

1430

007

9145

004

9010

000

2703

2776

473

005

3613

000

3348

16000

9927

002

0090

066

4410

007

1454

002

6494

000

2743

2769

477

000

9432

000

4611

012

2929

003

3397

5811

53010

3619

004

8074

000

2384

2754

482

000

5751

000

3202

018

4915

000

1033

093

3074

007

1438

022

2235

000

2777

2773

485

000

8324

000

3625

010

0021

005

9395

078

1264

008

1375

007

2428

000

2535

2729

478

001

0934

000

3549

55006

5875

014

8408

4260

06007

8190

004

8143

000

2676

2780

475

000

5570

000

3256

024

7551

000

9451

3375

38007

3270

007

5757

000

2577

2733

476

001

9549

000

3365

003

2144

018

0211

4615

70007

3522

004

0767

000

2756

027

88

477


Table4GArunsta

tistic

sTimetomakee

lites

(s)

Timetomakep

aram

eter

sets(s)

Totaltim

e(s)

Samples

ize

Stop

ping

threshold

Num

bero

felites

Max

generatio

nsParameter

setscreated

Fitness

8680

550

9230

5000

1040

010

020

2755

8643

528

9171

5000

1040

010

020

2755

8615

546

9161

5000

1040

010

020

2754

7446

543

7989

4000

1040

010

020

2751

7480

518

7998

4000

1040

010

020

2755

6306

571

6877

3000

1040

010

020

2754

6192

528

6720

3000

1040

010

020

2752

5115

550

5665

2000

1040

010

020

2754

5074

500

5574

2000

1040

010

020

2755

4285

517

4802

1000

1040

010

020

2748

4160

463

4623

1000

1040

010

020

2756

2359

562

2921

500

1040

010

020

2758

2780

556

3336

500

1040

010

020

2759

1963

652

2615

100

1040

010

020

2751

1892

612

2504

100

1040

010

020

2757

748

949

1697

5010

400

100

202788

561

941

1502

5010

400

100

2027

62

386

1052

1438

5010

200

100

2027

97

381

1118

1499

5010

200

100

2027

73

960

960

1920

100

1020

010

020

2756

951

849

1800

100

1020

010

020

2767



119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)



Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018














(c) Set V119895(119905 + 1) = 120596V119895(119905) + [1199091015840119895(119905) minus 119909119895(119905)]




















Update positions

No


Regenerate topology

Yes

No














3 ODE Models






4 Application


119889119878119889119905

= 119887119873 minus 119889119878 minus 120573119861119861119878

120581 + 119861minus 120573119868

119878119868119873

+ 120596119877

119889119868119889119905

= minus119889119868 + 120573119861119861119878

120581 + 119861+ 120573119868

119878119868119873

minus 120574119868

119889119877119889119905

= minus119889119877 + 120574119868 minus 120596119877

119889119861119889119905

= 120578119868 minus 120575119861

(3)




PredictedObserved

100 200 300 4000Days (t)

0

200

400

Hos

pita

lizat

ions

(hI)




119909 = (1198610 ℎ 120573119861 120573119868 120578 120574 120575 120596)119879 (4)


119891 (119909) = 1119899

119899

sum119894=1

1003816100381610038161003816ℎ119868 (119905119894) minus 1199101198941003816100381610038161003816 (5)








Appendix




Table3Parameter

values

from

PSOandcorrespo

ndingfitness

119861 0ℎ

120573 119861120573 119868

120578120574

120575120596

Fitness

Time(s)

000

6170

000

3569

0012

9567

000

1522

087

5338

008

0457

007

8863

000

2336

2719

481

000

6424

000

8251

010

2802

007

7925

079

6273

018

2466

004

8924

000

2660

2826

484

000

7413

000

3196

64015

9124

002

1619

061

7604

007

3014

010

0941

000

2739

2718

477

000

5186

000

3286

015

0995

004

2600

100

0000

007

1429

022

7019

000

2769

2825

477

000

7859

000

3576

009

6382

013

8211

032

1430

007

9145

004

9010

000

2703

2776

473

005

3613

000

3348

16000

9927

002

0090

066

4410

007

1454

002

6494

000

2743

2769

477

000

9432

000

4611

012

2929

003

3397

5811

53010

3619

004

8074

000

2384

2754

482

000

5751

000

3202

018

4915

000

1033

093

3074

007

1438

022

2235

000

2777

2773

485

000

8324

000

3625

010

0021

005

9395

078

1264

008

1375

007

2428

000

2535

2729

478

001

0934

000

3549

55006

5875

014

8408

4260

06007

8190

004

8143

000

2676

2780

475

000

5570

000

3256

024

7551

000

9451

3375

38007

3270

007

5757

000

2577

2733

476

001

9549

000

3365

003

2144

018

0211

4615

70007

3522

004

0767

000

2756

027

88

477


Table4GArunsta

tistic

sTimetomakee

lites

(s)

Timetomakep

aram

eter

sets(s)

Totaltim

e(s)

Samples

ize

Stop

ping

threshold

Num

bero

felites

Max

generatio

nsParameter

setscreated

Fitness

8680

550

9230

5000

1040

010

020

2755

8643

528

9171

5000

1040

010

020

2755

8615

546

9161

5000

1040

010

020

2754

7446

543

7989

4000

1040

010

020

2751

7480

518

7998

4000

1040

010

020

2755

6306

571

6877

3000

1040

010

020

2754

6192

528

6720

3000

1040

010

020

2752

5115

550

5665

2000

1040

010

020

2754

5074

500

5574

2000

1040

010

020

2755

4285

517

4802

1000

1040

010

020

2748

4160

463

4623

1000

1040

010

020

2756

2359

562

2921

500

1040

010

020

2758

2780

556

3336

500

1040

010

020

2759

1963

652

2615

100

1040

010

020

2751

1892

612

2504

100

1040

010

020

2757

748

949

1697

5010

400

100

202788

561

941

1502

5010

400

100

2027

62

386

1052

1438

5010

200

100

2027

97

381

1118

1499

5010

200

100

2027

73

960

960

1920

100

1020

010

020

2756

951

849

1800

100

1020

010

020

2767



119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)



Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018














Update positions

No


Regenerate topology

Yes

No














3 ODE Models






4 Application


119889119878119889119905

= 119887119873 minus 119889119878 minus 120573119861119861119878

120581 + 119861minus 120573119868

119878119868119873

+ 120596119877

119889119868119889119905

= minus119889119868 + 120573119861119861119878

120581 + 119861+ 120573119868

119878119868119873

minus 120574119868

119889119877119889119905

= minus119889119877 + 120574119868 minus 120596119877

119889119861119889119905

= 120578119868 minus 120575119861

(3)




PredictedObserved

100 200 300 4000Days (t)

0

200

400

Hos

pita

lizat

ions

(hI)




119909 = (1198610 ℎ 120573119861 120573119868 120578 120574 120575 120596)119879 (4)


119891 (119909) = 1119899

119899

sum119894=1

1003816100381610038161003816ℎ119868 (119905119894) minus 1199101198941003816100381610038161003816 (5)








Appendix




Table3Parameter

values

from

PSOandcorrespo

ndingfitness

119861 0ℎ

120573 119861120573 119868

120578120574

120575120596

Fitness

Time(s)

000

6170

000

3569

0012

9567

000

1522

087

5338

008

0457

007

8863

000

2336

2719

481

000

6424

000

8251

010

2802

007

7925

079

6273

018

2466

004

8924

000

2660

2826

484

000

7413

000

3196

64015

9124

002

1619

061

7604

007

3014

010

0941

000

2739

2718

477

000

5186

000

3286

015

0995

004

2600

100

0000

007

1429

022

7019

000

2769

2825

477

000

7859

000

3576

009

6382

013

8211

032

1430

007

9145

004

9010

000

2703

2776

473

005

3613

000

3348

16000

9927

002

0090

066

4410

007

1454

002

6494

000

2743

2769

477

000

9432

000

4611

012

2929

003

3397

5811

53010

3619

004

8074

000

2384

2754

482

000

5751

000

3202

018

4915

000

1033

093

3074

007

1438

022

2235

000

2777

2773

485

000

8324

000

3625

010

0021

005

9395

078

1264

008

1375

007

2428

000

2535

2729

478

001

0934

000

3549

55006

5875

014

8408

4260

06007

8190

004

8143

000

2676

2780

475

000

5570

000

3256

024

7551

000

9451

3375

38007

3270

007

5757

000

2577

2733

476

001

9549

000

3365

003

2144

018

0211

4615

70007

3522

004

0767

000

2756

027

88

477


Table4GArunsta

tistic

sTimetomakee

lites

(s)

Timetomakep

aram

eter

sets(s)

Totaltim

e(s)

Samples

ize

Stop

ping

threshold

Num

bero

felites

Max

generatio

nsParameter

setscreated

Fitness

8680

550

9230

5000

1040

010

020

2755

8643

528

9171

5000

1040

010

020

2755

8615

546

9161

5000

1040

010

020

2754

7446

543

7989

4000

1040

010

020

2751

7480

518

7998

4000

1040

010

020

2755

6306

571

6877

3000

1040

010

020

2754

6192

528

6720

3000

1040

010

020

2752

5115

550

5665

2000

1040

010

020

2754

5074

500

5574

2000

1040

010

020

2755

4285

517

4802

1000

1040

010

020

2748

4160

463

4623

1000

1040

010

020

2756

2359

562

2921

500

1040

010

020

2758

2780

556

3336

500

1040

010

020

2759

1963

652

2615

100

1040

010

020

2751

1892

612

2504

100

1040

010

020

2757

748

949

1697

5010

400

100

202788

561

941

1502

5010

400

100

2027

62

386

1052

1438

5010

200

100

2027

97

381

1118

1499

5010

200

100

2027

73

960

960

1920

100

1020

010

020

2756

951

849

1800

100

1020

010

020

2767



119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)



Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018












3 ODE Models






4 Application


119889119878119889119905

= 119887119873 minus 119889119878 minus 120573119861119861119878

120581 + 119861minus 120573119868

119878119868119873

+ 120596119877

119889119868119889119905

= minus119889119868 + 120573119861119861119878

120581 + 119861+ 120573119868

119878119868119873

minus 120574119868

119889119877119889119905

= minus119889119877 + 120574119868 minus 120596119877

119889119861119889119905

= 120578119868 minus 120575119861

(3)




PredictedObserved

100 200 300 4000Days (t)

0

200

400

Hos

pita

lizat

ions

(hI)




119909 = (1198610 ℎ 120573119861 120573119868 120578 120574 120575 120596)119879 (4)


119891 (119909) = 1119899

119899

sum119894=1

1003816100381610038161003816ℎ119868 (119905119894) minus 1199101198941003816100381610038161003816 (5)








Appendix




Table3Parameter

values

from

PSOandcorrespo

ndingfitness

119861 0ℎ

120573 119861120573 119868

120578120574

120575120596

Fitness

Time(s)

000

6170

000

3569

0012

9567

000

1522

087

5338

008

0457

007

8863

000

2336

2719

481

000

6424

000

8251

010

2802

007

7925

079

6273

018

2466

004

8924

000

2660

2826

484

000

7413

000

3196

64015

9124

002

1619

061

7604

007

3014

010

0941

000

2739

2718

477

000

5186

000

3286

015

0995

004

2600

100

0000

007

1429

022

7019

000

2769

2825

477

000

7859

000

3576

009

6382

013

8211

032

1430

007

9145

004

9010

000

2703

2776

473

005

3613

000

3348

16000

9927

002

0090

066

4410

007

1454

002

6494

000

2743

2769

477

000

9432

000

4611

012

2929

003

3397

5811

53010

3619

004

8074

000

2384

2754

482

000

5751

000

3202

018

4915

000

1033

093

3074

007

1438

022

2235

000

2777

2773

485

000

8324

000

3625

010

0021

005

9395

078

1264

008

1375

007

2428

000

2535

2729

478

001

0934

000

3549

55006

5875

014

8408

4260

06007

8190

004

8143

000

2676

2780

475

000

5570

000

3256

024

7551

000

9451

3375

38007

3270

007

5757

000

2577

2733

476

001

9549

000

3365

003

2144

018

0211

4615

70007

3522

004

0767

000

2756

027

88

477


Table4GArunsta

tistic

sTimetomakee

lites

(s)

Timetomakep

aram

eter

sets(s)

Totaltim

e(s)

Samples

ize

Stop

ping

threshold

Num

bero

felites

Max

generatio

nsParameter

setscreated

Fitness

8680

550

9230

5000

1040

010

020

2755

8643

528

9171

5000

1040

010

020

2755

8615

546

9161

5000

1040

010

020

2754

7446

543

7989

4000

1040

010

020

2751

7480

518

7998

4000

1040

010

020

2755

6306

571

6877

3000

1040

010

020

2754

6192

528

6720

3000

1040

010

020

2752

5115

550

5665

2000

1040

010

020

2754

5074

500

5574

2000

1040

010

020

2755

4285

517

4802

1000

1040

010

020

2748

4160

463

4623

1000

1040

010

020

2756

2359

562

2921

500

1040

010

020

2758

2780

556

3336

500

1040

010

020

2759

1963

652

2615

100

1040

010

020

2751

1892

612

2504

100

1040

010

020

2757

748

949

1697

5010

400

100

202788

561

941

1502

5010

400

100

2027

62

386

1052

1438

5010

200

100

2027

97

381

1118

1499

5010

200

100

2027

73

960

960

1920

100

1020

010

020

2756

951

849

1800

100

1020

010

020

2767



119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)



Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018










PredictedObserved

100 200 300 4000Days (t)

0

200

400

Hos

pita

lizat

ions

(hI)




119909 = (1198610 ℎ 120573119861 120573119868 120578 120574 120575 120596)119879 (4)


119891 (119909) = 1119899

119899

sum119894=1

1003816100381610038161003816ℎ119868 (119905119894) minus 1199101198941003816100381610038161003816 (5)








Appendix




Table3Parameter

values

from

PSOandcorrespo

ndingfitness

119861 0ℎ

120573 119861120573 119868

120578120574

120575120596

Fitness

Time(s)

000

6170

000

3569

0012

9567

000

1522

087

5338

008

0457

007

8863

000

2336

2719

481

000

6424

000

8251

010

2802

007

7925

079

6273

018

2466

004

8924

000

2660

2826

484

000

7413

000

3196

64015

9124

002

1619

061

7604

007

3014

010

0941

000

2739

2718

477

000

5186

000

3286

015

0995

004

2600

100

0000

007

1429

022

7019

000

2769

2825

477

000

7859

000

3576

009

6382

013

8211

032

1430

007

9145

004

9010

000

2703

2776

473

005

3613

000

3348

16000

9927

002

0090

066

4410

007

1454

002

6494

000

2743

2769

477

000

9432

000

4611

012

2929

003

3397

5811

53010

3619

004

8074

000

2384

2754

482

000

5751

000

3202

018

4915

000

1033

093

3074

007

1438

022

2235

000

2777

2773

485

000

8324

000

3625

010

0021

005

9395

078

1264

008

1375

007

2428

000

2535

2729

478

001

0934

000

3549

55006

5875

014

8408

4260

06007

8190

004

8143

000

2676

2780

475

000

5570

000

3256

024

7551

000

9451

3375

38007

3270

007

5757

000

2577

2733

476

001

9549

000

3365

003

2144

018

0211

4615

70007

3522

004

0767

000

2756

027

88

477


Table4GArunsta

tistic

sTimetomakee

lites

(s)

Timetomakep

aram

eter

sets(s)

Totaltim

e(s)

Samples

ize

Stop

ping

threshold

Num

bero

felites

Max

generatio

nsParameter

setscreated

Fitness

8680

550

9230

5000

1040

010

020

2755

8643

528

9171

5000

1040

010

020

2755

8615

546

9161

5000

1040

010

020

2754

7446

543

7989

4000

1040

010

020

2751

7480

518

7998

4000

1040

010

020

2755

6306

571

6877

3000

1040

010

020

2754

6192

528

6720

3000

1040

010

020

2752

5115

550

5665

2000

1040

010

020

2754

5074

500

5574

2000

1040

010

020

2755

4285

517

4802

1000

1040

010

020

2748

4160

463

4623

1000

1040

010

020

2756

2359

562

2921

500

1040

010

020

2758

2780

556

3336

500

1040

010

020

2759

1963

652

2615

100

1040

010

020

2751

1892

612

2504

100

1040

010

020

2757

748

949

1697

5010

400

100

202788

561

941

1502

5010

400

100

2027

62

386

1052

1438

5010

200

100

2027

97

381

1118

1499

5010

200

100

2027

73

960

960

1920

100

1020

010

020

2756

951

849

1800

100

1020

010

020

2767



119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)



Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018









Table3Parameter

values

from

PSOandcorrespo

ndingfitness

119861 0ℎ

120573 119861120573 119868

120578120574

120575120596

Fitness

Time(s)

000

6170

000

3569

0012

9567

000

1522

087

5338

008

0457

007

8863

000

2336

2719

481

000

6424

000

8251

010

2802

007

7925

079

6273

018

2466

004

8924

000

2660

2826

484

000

7413

000

3196

64015

9124

002

1619

061

7604

007

3014

010

0941

000

2739

2718

477

000

5186

000

3286

015

0995

004

2600

100

0000

007

1429

022

7019

000

2769

2825

477

000

7859

000

3576

009

6382

013

8211

032

1430

007

9145

004

9010

000

2703

2776

473

005

3613

000

3348

16000

9927

002

0090

066

4410

007

1454

002

6494

000

2743

2769

477

000

9432

000

4611

012

2929

003

3397

5811

53010

3619

004

8074

000

2384

2754

482

000

5751

000

3202

018

4915

000

1033

093

3074

007

1438

022

2235

000

2777

2773

485

000

8324

000

3625

010

0021

005

9395

078

1264

008

1375

007

2428

000

2535

2729

478

001

0934

000

3549

55006

5875

014

8408

4260

06007

8190

004

8143

000

2676

2780

475

000

5570

000

3256

024

7551

000

9451

3375

38007

3270

007

5757

000

2577

2733

476

001

9549

000

3365

003

2144

018

0211

4615

70007

3522

004

0767

000

2756

027

88

477


Table4GArunsta

tistic

sTimetomakee

lites

(s)

Timetomakep

aram

eter

sets(s)

Totaltim

e(s)

Samples

ize

Stop

ping

threshold

Num

bero

felites

Max

generatio

nsParameter

setscreated

Fitness

8680

550

9230

5000

1040

010

020

2755

8643

528

9171

5000

1040

010

020

2755

8615

546

9161

5000

1040

010

020

2754

7446

543

7989

4000

1040

010

020

2751

7480

518

7998

4000

1040

010

020

2755

6306

571

6877

3000

1040

010

020

2754

6192

528

6720

3000

1040

010

020

2752

5115

550

5665

2000

1040

010

020

2754

5074

500

5574

2000

1040

010

020

2755

4285

517

4802

1000

1040

010

020

2748

4160

463

4623

1000

1040

010

020

2756

2359

562

2921

500

1040

010

020

2758

2780

556

3336

500

1040

010

020

2759

1963

652

2615

100

1040

010

020

2751

1892

612

2504

100

1040

010

020

2757

748

949

1697

5010

400

100

202788

561

941

1502

5010

400

100

2027

62

386

1052

1438

5010

200

100

2027

97

381

1118

1499

5010

200

100

2027

73

960

960

1920

100

1020

010

020

2756

951

849

1800

100

1020

010

020

2767



119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)



Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018









Table4GArunsta

tistic

sTimetomakee

lites

(s)

Timetomakep

aram

eter

sets(s)

Totaltim

e(s)

Samples

ize

Stop

ping

threshold

Num

bero

felites

Max

generatio

nsParameter

setscreated

Fitness

8680

550

9230

5000

1040

010

020

2755

8643

528

9171

5000

1040

010

020

2755

8615

546

9161

5000

1040

010

020

2754

7446

543

7989

4000

1040

010

020

2751

7480

518

7998

4000

1040

010

020

2755

6306

571

6877

3000

1040

010

020

2754

6192

528

6720

3000

1040

010

020

2752

5115

550

5665

2000

1040

010

020

2754

5074

500

5574

2000

1040

010

020

2755

4285

517

4802

1000

1040

010

020

2748

4160

463

4623

1000

1040

010

020

2756

2359

562

2921

500

1040

010

020

2758

2780

556

3336

500

1040

010

020

2759

1963

652

2615

100

1040

010

020

2751

1892

612

2504

100

1040

010

020

2757

748

949

1697

5010

400

100

202788

561

941

1502

5010

400

100

2027

62

386

1052

1438

5010

200

100

2027

97

381

1118

1499

5010

200

100

2027

73

960

960

1920

100

1020

010

020

2756

951

849

1800

100

1020

010

020

2767



119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)



Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018










119875 (1003817100381710038171003817119884119891 (119885)1003817100381710038171003817 le 119904) prop 119904119863 997904rArr

119875 (119884 le 119904) prop 119904119863(A1)



Data Availability




References



























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Parameter Estimation in Ordinary Differential Equations ...downloads.hindawi.com/journals/jam/2018/9160793.pdf · Parameter Estimation in Ordinary Differential Equations Modeling