ampliﬁcation Mathematical analysis of a two-strain disease model with Chaos, Solitons and Fractals

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ContentslistsavailableatScienceDirect

Chaos, Solitons and Fractals

Nonlinear Science,andNonequilibrium andComplexPhenomena journalhomepage:www.elsevier.com/locate/chaos

Mathematical analysis of a two-strain disease model with ampliﬁcation

Md Abdul Kuddus

^a^,^b^,^e^,^∗

, Emma S. McBryde

^a^,^b

, Adeshina I. Adekunle

^a

, Lisa J. White

^c^,^d

, Michael T. Meehan

^a

aAustralian Institute of Tropical Health and Medicine, James Cook University, Townsville, QLD

bCollege of Medicine and Dentistry, James Cook University, Townsville, QLD

cMahidol Oxford Tropical Medicine Research Unit (MORU), Bangkok, Thailand

dCentre for Tropical Medicine and Global Health, Nuﬃeld Department of Clinical Medicine, University of Oxford, Oxford, United Kingdom

eDepartment of Mathematics, University of Rajshahi, Rajshahi, 6205, Bangladesh

a rt i c l e i nf o

Article history:

Received 19 February 2020 Revised 19 November 2020 Accepted 14 December 2020

Keywords:

drug resistance multi-strain stability analysis

a b s t r a c t

We investigate a two-strain disease model with amplification to simulate the prevalence of drug- susceptible (s) and drug-resistant (m) disease strains. Drug resistance first emerges when drug- susceptible strains mutateand becomedrug-resistant, possiblyas aconsequenceof inadequatetreat- ment,i.e.amplification.Inthiscase,thedrug-susceptibleanddrug-resistantstrainsarecoupled.Weper- formadynamicalanalysisoftheresultingsystemand findthatthemodel containsthreeequilibrium points:adisease-freeequilibrium;amono-existentdisease-endemicequilibriumatwhichonlythedrug- resistantstrainpersists;andaco-existentdisease-endemicequilibriumwhereboththedrug-susceptible and drug-resistantstrainspersist.Wefoundtwo basicreproductionnumbers:one associatedwiththe drug-susceptiblestrain (^R0s)^;^theôther^with^thedrug-resistantstrain (^R0m),andshowedthatatleast oneofthestrainscanspreadinapopulationifmax[R0s,R0m]>1.Furthermore,wealsoshowedthatif R0m>max[R0s,1],thedrug-susceptiblestraindiesoutbutthedrug-resistantstrainpersistsinthepop- ulation(mono-existentequilibrium);howeverif R_0s>max[R_0m,1],thenboththedrug-susceptibleand drug-resistantstrainspersistinthepopulation(co-existentequilibrium).Weconductedalocalstability analysisofthesystemequilibriumpointsusingtheRouth-Hurwitzconditionsandaglobalstabilityanal- ysisusingappropriateLyapunovfunctions.Sensitivityanalysiswasusedtoidentifythekeymodelparam- etersthatdrivetransmissionthroughcalculationofthepartialrankcorrelationcoefficients(PRCCs).We foundthatthecontactrateofbothstrainshadthelargestinfluenceonprevalence.Wealsoinvestigated theimpactofamplificationand treatment/recoveryratesofbothstrainsontheequilibrium prevalence ofinfection;resultssuggestthatpoorqualitytreatment/recoverymakescoexistencemorelikelyandin- creasestherelativeabundanceofresistantinfections.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Many pathogenshave severalcirculatingstrains.The presence of drug-resistant strains of a pathogen often follows soon af- ter a new treatment becomesavailable. This can be dueto sub- therapeutic druglevelswhichmayeﬃcientlykilldrug-susceptible pathogens whilstallowing drug-resistantsub-populationstogrow [1].Thisacquiredresistance,whichcanresultfromincorrecttreat-

∗Corresponding author. Md. Abdul Kuddus, Australian Institute of Tropical Health and Medicine, James Cook University, Townsville, QLD.

E-mail address: [email protected] (M.A. Kuddus).

ment, poor adherence or malabsorption, is called ampliﬁcation [2–4].Oneofthemajorchallengesinpreventingthespreadofin- fectious diseasesis tocontrol thegenetic variationsofpathogens throughpropertreatmentregimens[5,6].

Mathematical models can improve our understanding of ge- netic variations ofinfectious pathogens aswell asthose compo- nentsthataresigniﬁcanttoinfectiousdiseasediagnosis,andtreat- ment [7–13]. Mathematical models can also be used to improve healthpolicy andinfectious diseasemonitoringplansbyidentify- ing thresholds whichmust be reachedin orderto achieve elimi- nation[13–15].For example,analytical solutions, numericalsolu- tions and stability analyses of mathematical models can identify

https://doi.org/10.1016/j.chaos.2020.110594

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regionsintheparameterspacewherethevariousasymptoticstates are stable orunstable, thus allowing usto predict the long-term behaviour of thesystem[13,14,16]. Further,sensitivity analysisof a mathematical model allows usto discover the parameters that havethegreatestinﬂuenceonthemodeloutputs[14,17].

The growingthreat ofdrug-resistantpathogenstrainspresents a signiﬁcant challenge throughout the world, particularly in de- veloping countries and those with lower socio-economic status [18]. Once drug-resistant strains have emerged in a population, primary transmissionof thesestrains may alsocontribute to the disease burden (in addition to ampliﬁcation) [19].Recent studies [20–24] haveshownthat drug-resistantstrainscaninsomecases possesshighervirulencetotransmitdiseasethandrug-susceptible strains, and those individuals infected with a drug-resistant strain have the highest mortality rate, e.g. tuberculosis and HIV [25,26].

Toexaminethethreatposedbygeneticvariationsofpathogens, we present a two-strain (drug-susceptible, and drug-resistant) Susceptible-Infected-Recovered(SIR)epidemicmodelwithcoupled infectious compartments anduseit to investigatethe emergence andspreadofmutatedstrainsofinfectiousdiseases. Weconsider the possibility that an individual’s position changes from drug- susceptibleatinitialpresentationtoresistantatfollow-up.Thisis the modeby which drugresistanceﬁrst emergesin apopulation andisdesignedtoreproducethephenotypicphenomenonofam- pliﬁcation.Themodelcanbeappliedtoinvestigatetheco-existent orcompetitiveexclusivephenomenaamongthestrains.Wechoose the SIRmodel inthisstudyinordertomodel themanydiseases that haveaprotracted infectiousperiodwithtreatment–including hepatitis C andHIV. Here theremovedcompartment“R” is tobe appliedbroadlytothosepeoplewhoareneitherinfectiousnorsus- ceptible, including people in treatment, isolation, no longer con- tactingothersordead.Inthisway,webelievethatourmodelcap- turesmanyoftheinfectiousagentsthataretraditionallymodelled bySusceptible(S)toInfected(I)models.

Explicitly, in this paper we perform a rigorous analytical and numerical analysis of the proposed two- strain model properties and solutions from both the mathematical and biological view- points.Foreach,weusethenext-generationmatrixmethodtode- termineanalyticexpressionsforthebasicreproductionnumbersof thedrug-susceptibleanddrug-resistantstrainsandﬁndthatthese are importantdeterminantsforregulatingsystemdynamics.With afocusontheearlyandlate-timebehaviourofthesystem,weout- linethe requiredconditionsfordiseasefade-out,infectionmono- existence,andco-existence.

To supplement andvalidate the analytic analysis, we use numerical techniques tosolve the modelequationsand explorethe epidemic trajectory fora range of possibleparameter values and initial conditions. The local stability of the three system equilibria is examined using the Routh-Hurwitz conditions and the globalstabilityofthe disease-free equilibriumandmono-existent disease-endemic equilibrium is examined using appropriate Lya- punovfunctions. Followingthis, we performa sensitivityanalysis to investigatethe modelparameters that have thegreatest inﬂu- enceondiseaseprevalence.

The remainder of this paper is constructed as follows: in section2wepresentthetwo-strainSIRmodelwithdifferentialin- fectivity and ampliﬁcation,and verify the boundedness andpos- itivity of solutions as well as the existence of several equilibria.

Local andglobal stabilityanalyses ofthe equilibriaarepresented in section 3. Insection 4 we discussa sensitivity analysisof the model outputs. We then provide numerical simulations to sup- port analyticresultsinsection 5.Finally,insection6,we provide a summary ofour outcomes, discusstheir importance forpublic health policy and propose guidelines for future disease manage- mentefforts.

Fig. 1. Flow chart of the two-strain SIR model showing the four infection states, and the per-capita transition rates in and out of each state (not shown: the constant recruitment rate into the susceptible compartment S). Subscripts s and m denote drug-susceptible and drug-resistant quantities respectively.

2. Modeldescriptionandanalysis

Modelequations:

We developed a dynamic two-strainSIR modelfor the transmission of drug-susceptible and drug-resistant infections, where the total populationis divided into foursubclasses: S− suscepti- bleindividuals;I_s – individualsinfectedwiththedrug-susceptible strain; Im – individuals infected with the drug-resistant strain;

andR–recovered individuals,who are assumedtohaveimmunity againstbothstrains.ThusthetotalpopulationnumberN(^t)^at^time tis

N

(

^t

)

⁼^S

(

^t

)

⁺^I^s

(

^t

)

⁺^I^m

(

^t

)

⁺^R

(

^t

)

^. ⁽¹⁾

We also introduced the following parameters: ^{– constant} recruitment rate into the susceptible class through birth or im- migration¹;

μ

^–natural ^per-capita ^death ^rate ^across ^the ^entire

population;

β

^s (

β

^m) – effective contact ratebetween individuals with drug-susceptible (drug-resistant) infection andsusceptibles;

ω

s (

ω

m)–per-capita treatment/recovery rate for drug-susceptible (drug-resistant)infectedindividuals;

φ

^s(

φ

^m)−disease-relatedper- capita death rate for drug-susceptible (drug-resistant) infected individuals;

ρ

–proportion of individuals who amplify from the drug-susceptible strain to the drug-resistant strain during treatment/recovery. We assumed the proportion of individuals who amplify–duetoincompletetreatmentorlackofcomplianceinthe use of ﬁrst-line drugs–move directly from the drug-susceptible compartment Is into the drug-resistant compartment Im. The modelstructureisillustratedinFig.1.

Fromtheaforementioned,thepopulationsineachdiseasestate aredeterminedbythefollowingsystemofnonlinearordinarydif- ferentialequations:

S˙ =

−

μ

^S−

β

sIsS−

β

mImS, (2) I˙s=

β

sIsS−

( ω

s+

φ

s+

μ )

^I^s^, ⁽³⁾ I˙m=

ρω

^s^I^s⁺

β

^m^I^m^S⁻

( ω

^m⁺

φ

^m⁺

μ )

^I^m^, ⁽⁴⁾

1Data strongly suggest that the absolute number of births globally has been ap- proximately constant for the last 30 years and is predicted to remain constant for the next 30 years. Therefore, within the timescale of an SIR-type infection it is rea- sonable to assume a constant growth rate ( https://population.un.org/wpp/Graphs/

900 ).

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R˙ =

(

¹⁻

ρ ) ω

^s^I^s⁺

ω

^m^I^m⁻

μ

^R^. ⁽⁵⁾

Given non-negative initial conditions for the system above, it isstraightforwardto showthateach ofthestatevariablesremain non-negative for all t>0. Moreover, summing equations (2)–(5) weﬁndthatthesizeofthetotalpopulation,N(^t)^,^satisﬁes N˙

(

^t

)

^≤

−

μ

^N

(

^t

)

^.

Integratingthisinequalityweﬁnd N

(

^t

)

≤

μ

⁺^N

(

⁰

)

^e⁻^μ^t^.

This shows that the total population size N(^t) ^is ^bounded ⁱⁿ this caseand that it naturallyfollows that each ofthe compart- mentstates(S, Is,Im,R)arealsobounded.

Notethat equations(2)–(4)areindependent ofthesizeofthe recoveredpopulation R(^t)^;^therefore,^if^we^only^wish^to^track^disease incidenceandprevalence, wecanfocus ourattentiononthe followingreducedsystem(6)–(8):

S˙ =

−

μ

^S⁻

β

^s^I^s^S⁻

β

^m^I^m^S^, ⁽⁶⁾

I˙s=

( β

^s^S⁻

χ

^s

)

^I^s^, ⁽⁷⁾

I˙m=

ρω

^s^I^s+

β

^m^I^m^S−

χ

^m^I^m ⁽⁸⁾

where

χ

^s=

ω

^s+

φ

^s+

μ

^and

χ

^m=

ω

^m+

φ

^m+

μ

^represent^the^to-

talremovalratesfromtherespectiveinfectiouscompartments.

Giventhe positivityandboundednessof thesystemsolutions, weﬁndthatthefeasibleregionforequations(6)–(8)isgivenby D=

(

^S, Is, Im

)

∈R³₊:S+Is+Im≤

μ

(c)

whereDispositivelyinvariant.Therefore,inthisstudyweconsider thesystemofequations(6)–(8)intheset D.

2.1. Basicreproductionnumber

Herewe estimatethebasicreproductionnumberofthemodel (6)–(8). Inan epidemicmodel, the basicreproduction numberis theexpectednumberofsecondarycasescreatedbyasingleinfec- tious caseintroduced into a totally susceptiblepopulation. Ifthe basicreproductionnumberisgreaterthanone,thenumberofin- fectedindividuals growsandtheinfectiontypically showspersis- tentbehaviour.Conversely,ifthebasicreproductionnumberisless than one, the number of infective individuals typically tends to zero[12,27,28].Hereweusethenext-generationmatrixtechnique toestimatethebasicreproductionnumber(s)ofoursystem[28].

The reducedmodel(6)–(8) hastwo infectedstates:Is;andIm, andoneuninfectedstate: S.Attheinfection-freesteadystateI^∗_s= I^∗_m =0. Hence,from(6),intheabsenceofinfectionS^∗=_μ.

Linearizingthesystemabouttheinfection-freeequilibrium,we ﬁndthatequations(3)–(4)areclosed,such thatthelinearizedin- fectionsub-modelbecomes

I˙s=

( β

^s^S^∗⁻

χ

^s

)

^I^s^, ⁽⁹⁾

I˙m=

ρω

^s^I^s+

β

^m^I^m^S^∗−

χ

^m^I^m. (10) Here, theODEs (9)and (10)describe theproduction ofnewly infected individualsandchanges inthe statesof alreadyinfected individuals.

Bysetting x^T=(^Is, Im)^T^,^where^T^denotes^transpose,^the^infec- tionsubsystemcanbewritteninthefollowingform:

˙

x=

(

^T⁺

)

^x^. ⁽¹¹⁾

The matrix T contains the transmission component of equations (9) and (10) (i.e. the arrival of susceptible individuals intotheinfectedcompartmentsIsandIm) andthematrix ^con- tainstransitionsbetween,andoutoftheinfectedstates(i.e.recovery,ampliﬁcationanddeath).

Forthesubsystem(9)–(10),thesecomponentsaregivenrespec- tivelyby

T=

β

^s^S^∗ ⁰

0

β

^m^S^∗

and

=

−

χ

^s ⁰

ρω

^s ⁻

χ

^m

. Thenext-generationmatrix,K,isthengivenby[27]

K=−T

⁻¹=

_S∗β^s

χs 0

S^∗βmωsρ χ^sχ^m ^S

∗βm

χ^m

.

ThedominanteigenvaluesofKarethebasicreproductionnum- bersforthedrug-susceptible anddrug-resistantstrains;they rep- resenttheaveragenumberofnewinfectionsfromeachstrainpro- ducedbyoneinfectedindividual.Thelowertriangularstructureof Kallows ustoimmediatelyread off thebasicreproductionnum- bers for the drug-susceptible and drug-resistant strains respec- tivelyas:

R0s=S^∗

β

s

χ

s =

β

s

μχ

s

(a) and

R0m= S^∗

β

^m

χ

m =

β

^m

μχ

m. (b)

Interestinglywe ﬁndthatthebasicreproductionnumbersR₀_s andR0marebothindependentoftheampliﬁcationrate

ρ

^[29]^.

2.1.1. Strainreplacement

ToinvestigatetherelativemagnitudeofR_0s andR_0m,whichis anticipatedtostronglyinfluencethesystemdynamics(seebelow), we introduce the parameters c and∈ which we respectively de- fineasthefitnesscost exactedonthetransmissibilityofstrain m relativetothat ofstrain s,andthereductionintreatment rateof strainmrelativetothatofstrains.Morespecifically,welet

β

^m⁼

(

¹⁻^c

) β

^s

and

ω

^m⁼

(

¹^{− ∈}

) ω

^s^.

Ifwe assume that both R_0s andR_0m are greater than 1, then theconditionforresistantinfectionstoreplacesensitiveinfections isgivenby,

R_0m>R_0s.

Substitutingtheformulae(a)and(b)forthebasicreproduction numbersgives:

β

^m

ω

^m⁺

φ

^m⁺

μ

^>

β

^s

ω

^s⁺

φ

^s⁺

μ

^.

Assuming that

μ

≈0 (since it is very slow compared to the otherrates)andthat

φ

^m≈

φ

^s^yields^the^condition

(

¹−c

) β

s

(

¹− ∈

) ω

^s+

φ

^s ^>

β

s

ω

^s+

φ

^s^,

whichwecanrearrangetoobtain

∈>c

( ω

^s⁺

φ

^s

) ω

^s ^.

Theaboverelationshowsthattheresistant straincanoutcom- petethesusceptiblestrainiftheresistancelevel∈ishigh(which maybe the casefor drug-resistantindividuals). Alternatively, the resistantstrainwillbefitterthanthesusceptibleoneifthefitness costcissufficientlylow.

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2.2. Systemproperties

2.2.1. Existenceofequilibria

Itisclearfromequations(6)–(8)thatadisease-freeequilibrium (denotedby E^∗)alwaysexists:

E^∗=

(

^S^∗^, ^I^∗s, I^∗_m

)

⁼

μ

^, ⁰^, ⁰

. (12)

From equation (6)–(8) we can also derive the mono-existent endemic equilibrium point (denoted by E^#) atwhich the drug- resistantstrainpersistsandthedrug-susceptiblestraindiesout:

E^#=

S^#,0,I^#_m , where

S^#= S^∗ R_0m =

μ

^R0m, I^#_s=0,I^#_m=

μ (

^R0m−1

)

β

^m ^.

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Inspecting(13)weseethatthemono-existentendemicequilib- riumE^#=(^S^#,0,I^#_m)∈D(i.e.exists)if,andonlyifR_0m≥1.

Finally,theco-existentendemicequilibriumofthesystem(6)–

(8)(denotedbyE^†)^is^given^by E^†=

(

^S^†,I^†_s,I_m^†

)

,

where S^†=

μ

^R⁰^s ⁼^S^∗^/R⁰^s^,

I^†_s=

μ (

^R0s−1

) β

^s ^,

I^†_m=

ρ

^R0s

ω

^s

μ (

^R0s−1

) β

^s

χ

^m

(

^R0s−R_0m

)

+

ρ

^R0s

ω

^s

β

^m^,

=

ρω

^s^R^0s

χ

^m

(

^R^0s⁻^R^0m

) μ (

^R^0s⁻¹

) β

^s ^.

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Thevariable inequation(14)isdeﬁnedas

=

1+

ρω

sR0s

β

m

β

s

χ

m

(

^R^0s⁻^R^0m

)

−1

=

1+

ρω

sR0m

χ

s

(

^R^0s⁻^R^0m

)

−1

,

(d) and0< <1forR_0s>R_0m.Therefore,equation (14)showsthat the co-existent endemic equilibriumE^†=(^S^†, I^†_s, I^†_m)∈D(i.e. exists)if,andonlyifR_0s>max[R_0m, 1].

3. Stabilityanalysis

Sinceequations(2)–(4)areindependentofequation(5)(i.e.the evolution of S,Is andIm are independentof R(^t)^),^we^can ^focus our attentionon the reducedsystem(6)–(8) to studythe persis- tence of theinfection. Toinvestigate stabilityofthe equilibria of equations(6)–(8),thefollowingresultsareestablished:

3.1. Infection-freeequilibrium

Lemma1. If R₀=max[R_0s, R_0m]<1,thedisease-freeequilibrium E^∗ of(6)–(8) islocallyandgloballyasymptoticallystable;if, how- ever,R₀=max[R_0s, R_0m]>1,E^∗isunstable.

Proof. :We considerthe Jacobian ofthesystem(6)–(8) whichis givenby

J=

₋

β

sIs−

β

mIm−

μ

−

β

sS −

β

mS

β

^s^I^s

β

^s^S−

χ

^s ⁰

β

^m^I^m

ρω

^s

β

^m^S⁻

χ

^m

.

Attheinfection-freeequilibriumpoint,E^∗,thisreducesto J^∗=

₋

μ

−

β

sS^∗ −

β

mS^∗ 0

χ

^s

(

^R0s−1

)

⁰ 0

ρω

^s

χ

^m

(

^R^0m⁻¹

)

.

ThestructureofJ^∗ allowsustoimmediatelyreadoff thethree eigenvalues,

λ

i,as

λ

1=−

μ

^,

λ

2=

χ

^s

(

^R0s−1

)

^and

λ

3=

χ

^m

(

^R0m−1

)

^. ⁽¹⁵⁾

It iseasy to verify that all theeigenvalues (15) havenegative realparts forR_0s<1 and R_0m<1.Hence, thedisease-free equi- libriumE^∗of(6)–(8)islocallyasymptoticallystableforR_0s<1and R_0m<1. If R_0s>1 or R_0m>1, at least one of the eigenvalues (15)hasapositiverealpartandE^∗isunstable.

Now theglobal stabilityofthedisease-free equilibrium E^∗ for R_0s<1and R_0m<1canbeinvestigated.First,fromequation(7), wehave

I˙s=

( β

sS−

χ

s

)

^I^s^,

whichcanbeintegratedtogive Is

(

^t

)

⁼^I^s

(

⁰

)

^e

∫t

0β^s^S(τ )^dτ−χ^s ^t

(16) forall t≥0.

Substitutingintheupperbound S(^t)≤_μ=S^∗,whichfollows immediatelyfromthedeﬁnitionofD(equation(c)),weobtain Is

(

^t

)

≤Is

(

⁰

)

^e⁽^β^s^S^∗⁻^χ^s⁾^t,

≤Is

(

⁰

)

^e^χ^s⁽^R^0s⁻¹⁾^t^.

It follows then that if R_0s<1 we have Is(^t)→0 as t→∞. Hence thehyperplane I_s=0attracts all solutions of(6)–(8) orig- inatinginDwheneverR_0s<1.

SinceIs(^t)→0ast→∞forR_0s<1,itfollowsthat

ρ ω

^s^I^s(^t)→ 0suchthatequation(8)reducesto

I˙m=

β

mImS−

χ

mIm.

Following the same strategy for Im as we used above for Is

yields

Im

(

^t

)

^≤^I^m

(

⁰

)

^e^χ^m⁽^R^0m⁻¹⁾^t^.

Similarly, if R_0m<1, Im(^t)→0 as t→∞ and the hyperplane Im=0 attracts all solutions of(6)–(8) originatingin D.Fi- nally, it is straightforward to show that if Is→0, and Im→0, then S→S^∗.Therefore E^∗ is globally asymptotically stablewhen R₀=max[R_0s,R_0m]<1.

3.2. Mono-existentendemicequilibrium

Lemma 2. If the boundary equilibrium E^#=(^S^#,0,I^#_m) ^of ^the equations(6)—(8)exists andR_0m>max[1,R_0s],E^# islocallyand globallyasymptoticallystable.

Proof. : We consider the Jacobian of the system (6)—(8) at the mono-existentendemicequilibriumpointE^# whichisgivenby

J^#=

⎛

⎝

⁻

β

^m^I^#m−

μ

⁻

β

^s^S^# ⁻

β

^m^S^#

0 −^χ^s⁽^R^0m_R_0m^−R^0s⁾ 0

β

^m^I^#m

ρω

^s ⁰

⎞

⎠

.

Thestructure ofJ^# allows ustoimmediatelyread off the ﬁrst eigenvalue,

λ

¹=−

χ

^s⁽^R^0mR_0m^−R^0s⁾ whichisnegativewhenever R0m>

R_0s. The remaining eigenvalues can be calculated asthe roots of thefollowingequation

λ

²⁺^a1

λ

⁺^a2 =0 (17)

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where

a1=

β

^m^I^#m+

μ

⁼

μ

^R^0m^,

a2=

β

m²I^#_mS^#=

μχ

^m

(

^R^0m⁻¹

)

^.

Forlocalstabilitywe mustensurethat theRouth-Hurwitzcri- teria[30]aresatisﬁed:

a₁>0,and

a₂>0, whichholdswheneverR_0m>1.

Thus, by theRouth-Hurwitzcriteria, theboundary equilibrium E^# islocally asymptoticallystablewhenever R_0m>max[1, R_0s]. Conversely,forE^#∈D,itisunstablewhenR_0m<R_0s.

Now we prove E^# is globally asymptotically stable if R_0m>

max[1, R_0s].Consideringequation(7)and(8),wehave

I˙s=

( β

sS−

χ

s

)

^I^s^, ⁽¹⁸⁾

I˙m=

ρω

^s^I^s⁺

β

^m^I^m^S⁻

χ

^m^I^m. (19)

Following [31],ﬁrst we divide equation (18)and (19)through byIsandImrespectivelytoobtain

dlogIs

dt =

β

^s^S⁻

χ

^s^, ⁽²⁰⁾

dlogIm

dt =

β

^m^S⁻

χ

^m⁺

ρω

^s_I^I^s

m. (21)

Rearrangingequations(20)and(21)tosolveforS weget S= 1

β

^s^d^log^dt^I^s ⁺

χ

s

β

^s ⁼

β

¹^m^d^log^dt^I^m ⁺

χ

m

β

^m ⁻

ρω

s

β

^m ^I^I^m^s ⁽²²⁾

whichimmediatelyleadstothefollowinginequality:

1

β

s

dlogIs

dt +

χ

s

β

s ≤ 1

β

m

dlogIm

dt +

χ

m

β

m.

Integratingbothsidesoftheequationabovegives

Is

(

^t

)

Is

(

⁰

)

_β¹_s

e^χ^β^s^s^t≤

Im

(

^t

)

Im

(

⁰

)

_β¹_m

e^χ^β^m^m^t

whichwecanrearrangetoobtain

Is

(

^t

)

Is

(

⁰

)

_β¹_s

≤

Im

(

^t

)

Im

(

⁰

)

_β¹_m

e χm

βm−^χ_β^s_s t.

Next, we useequations (a)and (b)forthe basic reproduction numbers,torewritethisinequalityas

Is

(

^t

)

Is

(

⁰

)

_β¹_s

≤

Im

(

^t

)

Im

(

⁰

)

_β¹_m

e^S^∗ 1

R0m−R¹0s

t.

Finally,sinceboth Is(^t)ândÎ^m(^t)âre^bounded,âs^we^take^the limitast→∞wefind:

tlim→∞

Is

(

^t

)

Is

(

⁰

)

_β¹_s

≤lim

t→∞

Im

(

^t

)

Im

(

⁰

)

_β¹_m

e^S^∗ 1

R0m−R¹0s

t→0forR0m>R0s.

Hence the hyperplane Is=0 attracts all solutions of (6)–(8) when R_0m>R_0s.

To complete the globalstability proof, we show the endemic equilibriumE^#isgloballyasymptoticallystableonthehyperplane Is=0byconstructingthefollowingLyapunovfunction[32]: V^#=S−S^#lnS+Im−I^#_mlnIm+C

where C=−

S^#−S^#lnS^#+I^#_m−I^#_mlnI^#_m .

TakingthederivativeofV^#(^t)^along^systemtrajectoriesyields V˙^#=

1−^S_S^# S˙+

1−^I_I^#^m_m

I˙m,

=

1−^S_S^#

(

−

μ

^S⁻

β

^m^I^m^S

)

⁺

1−^I_I^#^m_m

( β

^m^I^m^S⁻

χ

^m^I^m

)

^,

=

−

μ

^S−

β

^m^I^m^S−

^SS^#+

μ

^S^#+

β

^m^I^m^S^#+

β

^m^I^m^S−

χ

^m^I^m

−

β

^m^I^#mS+

χ

^m^I^#m.

First,wesubstituteintheidentity

=

μ

^S^#⁺

β

^m^I^#mS^#, toobtain

V˙^#=

μ

^S^#+

β

^m^I^#mS^#−

μ

^S−

μ

^S^#^S_S^# −

β

^m^I^#mS^#S^# S +

μ

^S^#

+

β

mImS^#−

χ

mIm−

β

mI^#_mS+

χ

mI^#_m,

=

μ

^S^#

2− S S^#−S^#

S

+

β

mI^#_mS^#−

β

mI^#_mS^#S^#

S +

β

mImS^#

−

χ

^m^I^m⁻

β

^m^I^#mS+

χ

^m^I^#m

.

We cansimplify thisexpressionfurther by substitutingin the identity

β

^m^S^#=

χ

^m ^to^get

V˙^#=

μ

^S^#

2− S S^#−S^#

S

+

χ

^m^I^#m−

χ

^m^I^#m

S^#

S +

χ

^m^I^m⁻

χ

^m^I^m

−

χ

mI^#_m_S^S# +

χ

mI^#_m,

=

μ

^S^#

2− S S^#−S^#

S

+

χ

^m^I^#m

2− S S^# −S^#

S

,

=

μ

^S^#⁺

χ

^m^I^#m 2− S S^# −S^#

S

.

Sincethearithmeticmeanisgreater thanorequalto thegeo- metricmean,weobtainV˙^#≤0.

Therefore,the mono-existent endemic equilibrium E^# is glob- allyasymptoticallystableifR_0m>1.

3.3. Co-existentendemicequilibrium

Lemma3. Ifthe endemic equilibriumE^†=(^S^†, I^†_s, I^†_m) ^of ^equations(6)—(8)exists,E^†islocallyasymptoticallystable.

Proof. :WeconsidertheJacobianofthesystem(6)—(8)attheco- existentendemicequilibriumpointE^†whichisgivenby

J^†=

⎛

⎜ ⎝

−

β

sI^†_s−

β

mI^†_m−

μ

−

β

sS^† −

β

mS^†

β

^s^I^†s

β

^s^S^†⁻

χ

^s ⁰

β

mI^†_m

ρω

s

β

mS^†−

χ

m

⎞

⎟ ⎠

^.

Tosimplifythisexpression,weusethefollowingidentities

−

β

^s^I^†s−

β

^m^I^†m−

μ

⁼⁻

μ

^R0s,

−

β

^s^S^†⁼ ⁻^R^0s

χ

^s

S^∗ S^∗

R0s = −

χ

^s^,

−

β

^m^S^†⁼ ⁻^R^0m

χ

^m

S^∗ S^∗

R0s = −

χ

^m^R_R^0m

0s,

β

^s^I^†s=

β

^s

μ (

^R0s−1

)

β

^s ⁼

μ (

^R0s−1

)

,

β

^m^I^†m=

ρω

s

χ

^s

(

^R0s^R−^0mR_0m

) μ (

^R0s−1

)

,

β

sS^†−

χ

s=

χ

s

β

^s^S^∗

R0s

χ

s−1

=

χ

s

_R

0s

R0s −1

=

χ

s

(

¹⁻¹

)

⁼⁰^,

(6)

β

^m^S^†⁻

χ

^m⁼

χ

^m

β

^m^S^∗

χ

^m^R^0s⁻¹

=

χ

^m

R0s

(

^R^0m⁻^R^0s

)

where inthe fourthlinewe havesubstituted inthedeﬁnition of giveninequation(d).ThisallowsustorewritethematrixJ^† in thefollowingform:

J^† =

⎛

⎝

⁻

μ

^R0s −

χ

s −

χ

m R0m R0s

μ (

^R^0s⁻¹

)

⁰ ⁰

μ (

^R0s−1

) (

¹⁻

) ρ ω

^s

χ

^m⁽^R^0mR⁻0s^R^0s⁾

⎞

⎠

.

To determine the stability of this matrix we use the Routh- Hurwitzcriteria,whichstatethattherealpartsoftherootsofthe characteristicpolynomialassociatedwithathreebythreematrixJ^† arenegativeifA₁>0, A₂>0, A₃>0,andA₁A₂>A₃,whereA₁=

−trace(^J^†)^,^A2 representsthesumofthetwobytwoprincipalmi- norsofJ^† andA3=−det(^J^†)^.

Condition1:ForthematrixJ^†,wehave A₁=−trace

J^† =

μ

^R0s+

χ

^m

(

^R0s−R_0m

)

R_0s . Hence,A₁ >0ifR_0s>R_0m.

Condition2:

A2=

ρ ω⁰s χm ( ^R⁰0m−R0s) R0s

⁺

−μ^R^0s −χ^m ^RR^0m_0s

μ (^R^0s−1)(¹− ) χ^m ⁽^R^0mR^−R0s^0s⁾

+

⁻μ^R0s −χ^s μ(^R^0s−1) ⁰

^,

=μχ^m (^R^0s−R0m)+μχ^m^RR^0m_0s(^R^0s−1)(¹− )+μχ^s(^R^0s−1) . Recalling that 0< <1 for R_0s>R_0m, we see that A₂>0 is satisﬁedwhenever R_0s>1andR_0s>R_0m.

Condition3:

A3=det

J^† ,

=−

μ (

^R0s−1

)

⁻

χ

^s

χ

^m^RR^0m0s

ρ ω

^s

χ

^m⁽^R^0mR⁻0s^R^0s⁾

^,

=−

μ (

^R^0s⁻¹

)

_χ_s_χ_m ₍_R_0s_−R_0m₎

R0s +^{ρ ω}^s_R^χ_0s^m^R^0m

,

=

μ (

^R^0s⁻¹

) χ

^s

χ

^m

(

^R^0s⁻^R^0m

)

R0s

1+_χ_s^{ρ ω}₍_R₀_s^s₋^R^0m_R_0m₎

,

=

μ (

^R^0s⁻¹

) χ

^s

χ

^m

(

^R^0s⁻^R^0m

)

R0s ,

=

μ χ

^s

χ

^m

R0s

(

^R^0s⁻¹

) (

^R^0s⁻^R^0m

)

whereintheﬁfthlinewe havesubstituted inthedeﬁnitionof given inequation (d).The conditionA₃>0 istrueifR_0s>1and R_0s>R_0m.

Finally, if we multiply the expressions for A₁ and A₂ it is straightforward to show that the conditionA1A2>A3 issatisﬁed when R_0s>1 and R_0s>R_0m. Thus, by the Routh-Hurwitz crite- ria,theco-existentendemicequilibriumE^†islocallyasymptotically stablewhenR_0s>1andR_0s>R_0m.

4. Sensitivityanalysis

Recognizingthe relativeimportance ofthe variousrisk factors responsibleforthetransmissionofinfectiousdiseasesisessential.

The progressionofthe drug-resistantstrain andits incidenceand prevalencemustbeunderstoodinordertodeterminehowbestto decrease diseaseburden. Forthispurpose,wecalculated thepar- tial rank correlation coeﬃcients(PRCCs)–which is a global sensi- tivityanalysistechniqueusingLatin HypercubeSampling(LHS)–of several keyoutput variables. Ineach casewe assigneda uniform distribution from0 to 3 times the baseline value for each input parametertogenerateatotalof100,000,000computationsofeach

Fig. 2. PRCC values depicting the sensitivities of the model output I swith respect to the input parameters βs, ωs, φs, βm, ωm, φmand ρ, when R 0s> max [ R 0m, 1 ] (i.e. co-existent endemic equilibrium E ^†).

Fig. 3. PRCC values depicting the sensitivities of the model output I mwith respect to the input parameters βs, ωs, φs, βm, ωm, φmand ρ, when R 0s> max [ R 0m, 1 ]]

(i.e. co-existent endemic equilibrium E ^†).

output variable of interest. Here the model outputs we consider arethe numberofinfectiousindividualsIs andIm andtheir total sum (^Is+Im)^atequilibrium.NotethatPRCCvaluesliebetween-1 and+1.Positive(negative)valuesimplyapositive(negative)corre- lationtothemodelparameterandoutcomes.Thebigger(smaller) theabsolutevalueofthePRCC,thegreater(lesser)thecorrelation oftheparametertothemodeloutcome.

Figs 2–4 display the correlation between I_s, I_m and (^Is+I_m) and the corresponding parameters

β

^s,

ω

^s,

φ

^s,

β

^m,

ω

^m,

φ

^m ^and

ρ

^when ^R0s>max[R_0m, 1], that is, at the co-existent endemic equilibrium. From Figs 2–4, it is easy to perceive that I_s and (^I^s+Im) ^have ^a ^strong ^positive correlation with

β

^s ^and ^I^m ^has

a weaker positive correlation with

β

^s^, ^implying ^that ^a ^positive

changeof

β

s willincrease I_s, (Îs+I_m)ândÎm.Parameters

ω

s and

φ

^s^haveâ^negativecorrelationwithIs,Imand(Î^s+Im)^.Înâddition

β

^m ^has â ^negative correlation with Is and (Îs+Im) ^but â ^strong positive correlationwithIm.Parameters

ω

m and

φ

m havea positive correlation withIs and (^I^s+Im) ^but^strong^negative ^correla- tionwithIm.

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Table 1

Description of model parameters

Parameters Description Estimated value References

A demographic parameter which represents the recruitment rate into the population 1

μ Per-capita death rate ₇₀¹ per year [33]

βs The effective contact rate per unit time between susceptible and drug-susceptible infective individuals variable – βm The effective contact rate per unit time between susceptible and drug-resistant infective individuals variable – ωs The per-capita rate at which the drug-susceptible infected population progress to the recovery stage per

unit time as a result of treatment

0.290 per year [34]

ωm The per-capita rate at which the drug-resistant infected population progress to the recovery stage per unit

time as a result of treatment 0.145 per year Assume

ρ The proportion of ampliﬁcation due to treatment default on ﬁrst-line drug therapy 0.035 [35]

ϕs The per-capita rate at which the drug-susceptible infected population die from infection per unit time 0.37 over 3 years [35]

ϕm The per-capita rate at which the drug-resistant infected population die from infection per unit time 0.37 over 3 years [35]

Fig. 4. PRCC values depicting the sensitivities of the model output I s+ I mwith respect to the input parameters βs, ωs, φs, βm, ωm, φmand ρ, R 0s> max [ R 0m, 1 ]]

(i.e. co-existent endemic equilibrium E ^†).

Further, parameter

ρ

^has â ^negative correlation with I_s and (Î^s+Im) ^but â ^strong ^positive correlation with Im. Fig.5 repre- sentsthecorrelationbetweentheequilibriumvalueofIm andthe

Fig. 5. PRCC values depicting the sensitivities of the model output I m with respect to the input parameters βs, ωs, φs, βm, ωm, φmand ρ, when R 0m>

R 0sand R 0m> 1] (i.e. mono-existent endemic equilibrium E ^#).

correspondingmodelparameters

β

s,

ω

s,

φ

s,

β

m,

ω

m,

φ

m and

ρ

when R_0m>R_0s and R_0m>1 (i.e. at the mono-existent endemic equilibrium).Parameters

β

^s^,

β

^m^and

ρ

^(small^value ^not^showing)

Fig. 6. Co-existent endemic equilibrium: R 0s> max [ R 0m, 1 ] . In this case both the drug-susceptible infection and drug-resistant infection persist in the population (black dot). All parameter values assume their baseline values given in Table 1 .

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Fig. 7. Effect of ampliﬁcation ( ρ) on the drug-susceptible (I s)prevalence and drug- resistant prevalence (I m). All parameter values assume their baseline values given in Table 1 .

havepositivePRCCvalues,implyingthatapositivechangeinthese parameterswillincrease Im.Incontrast,

ω

^s,

φ

^s,

ω

^m^and

φ

^m ^have

negativePRCC valuesand,thus, increasingtheses parameterswill consequentlydecrease Im.

5. Numericalsimulations

In this section, we carry out detailed numerical simulations (using theMatlabprogramminglanguage)to supporttheanalytic results and to assess the impact of ampliﬁcation and the drug- susceptible treatment/recovery rateon equilibriumlevels oftotal prevalenceanddrug-resistantprevalence. Forillustrationwe have chosen baseline parametervaluesconsistentwithtuberculosis infection andtransmission [33–35].Inaccordancewiththeanalytic results we found three equilibriumpoints: thedisease-free equilibrium E^∗; a mono-existentendemic equilibrium E^#; anda co- existent endemic equilibrium E^†. We used different initial condi- tionsforbothstrainsofallpopulationsandfoundthatifbothbasic reproduction numbers are less than one (i.e. max[R_0s, R_0m]<1) then the disease-free equilibrium is locally and globally asymptotically stable. If R_0m>max[R_0s, 1], the drug-susceptible strain dies out butthe drug-resistant strain persists in the population.

Furthermore,ifR_0s>max[R_0m, 1],thenboththedrug-susceptible anddrug-resistantstrainspersistinthepopulation.

Fig. 6illustrates the stabilityofthe co-existent endemic equilibrium(i.e.when R_0s>max[R_0m,1]) bydepictingsystemtrajec- tories throughthe Is vsIm plane originatingfromdifferentinitial conditions. Inthissystemboth strains(IsandIm) persistbecause of the ampliﬁcation pathway fromthe drug-susceptible strain to the drug-resistant strain. Fig. 7 depicts the effect of ampliﬁca- tion (

ρ

) ^onequilibriumlevelsofdrug-susceptible prevalenceand drug-resistant prevalenceandshowsthat in theﬁrst region (

ρ

0.6)thedrug-susceptibleprevalenceisinitiallydominantbutthat the drug-resistantprevalencerises withincreasing

ρ

^.Eventually, for

ρ

0.6,the drug-resistant strain becomesdominantcourtesy oftheampliﬁcationpathway.

Fig.8,andFig.9showtheeffectofthedrug-susceptiblestrain treatment/recovery