Mathematical analysis of swine influenza epidemic model …...Mathematical analysis of swine...

Japan J. Indust. Appl. Math. (2016) 33:269–296DOI 10.1007/s13160-016-0210-3

ORIGINAL PAPER Area 1

Mathematical analysis of swine influenza epidemicmodel with optimal control

Mudassar Imran1 · Tufail Malik2 ·Ali R Ansari1 · Adnan Khan3

Received: 23 July 2015 / Revised: 31 December 2015 / Published online: 21 January 2016© The JJIAM Publishing Committee and Springer Japan 2016

Abstract A deterministic model is designed and used to analyze the transmissiondynamics and the impact of antiviral drugs in controlling the spread of the 2009 swineinfluenza pandemic. In particular, the model considers the administration of the antivi-ral both as a preventive as well as a therapeutic agent. Rigorous analysis of the modelreveals that its disease-free equilibrium is globally asymptotically stable under a con-dition involving the threshold quantity-reproduction numberRc. The disease persistsuniformly if Rc > 1 and the model has a unique endemic equilibrium under certaincondition. The model undergoes backward bifurcation if the antiviral drugs are com-pletely efficient. Uncertainty and sensitivity analysis is presented to identify and studythe impact of critical model parameters on the reproduction number. A time depen-dent optimal treatment strategy is designed using Pontryagin’s maximum principleto minimize the treatment cost and the infected population. Finally the reproductionnumber is estimated for the influenza outbreak and model provides a reasonable fit tothe observed swine (H1N1) pandemic data in Manitoba, Canada, in 2009.

Keywords Influenza · Reproduction number · Backward bifurcation · Uncertaintyand sensitivity analysis · Optimal control · Statistical inference

Mathematics Subject Classification 92B08 · 49J15 · 34C23

B Mudassar [email protected]

1 Department of Mathematics and Natural Sciences, Gulf University for Science and Technology,Kuwait City, Kuwait

2 Department of Applied Mathematics and Sciences, Khalifa University, Abu Dhabi, UAE

3 Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan

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270 M. Imran et al.

1 Introduction

Influenza is a viral infectionwhich is primarily transmitted through respiratory dropletsproduced by coughing and sneezing by an infected person [13,41,46]. Fever, cough,headache, muscle and joint pain, and reddened eyes are common symptoms ofinfluenza. The infection, mostly prevalent in young children, elderly, pregnant womenand patients with particular medical conditions such as chronic heart disease, hasan initial incubation period of 1–4days followed by a latency period of 2–10days[33,41,46].

For the past few centuries, influenza remains a serious threat to public health aroundthe globe [2,13,41,46]. It is reported that an influenza epidemicmight have occurred inthe sixteenth century [38]. During the past century, thousands of people lost their livesduring three disastrous pandemics which include the Spanish flu, Asian flu, and HongKong flu during the year 1918, 1958, and 1968 respectively [2,38,46]. Most recentlyin 2014, India had reported 937 cases and 218 deaths from swine flu. The reportedcases and deaths in 2015 had surpassed the previous numbers. The total number ofconfirmed cases crossed 33,000 with death of more than 2000 [43].

In 2009, the world experienced the H1N1 influenza, also known as the SwineInfluenza, an epidemicwhich led to over 16455 deaths globally [51], 25,828 confirmedcases [36] and 428 deaths [37] in Canada alone. In Canada, this epidemic occurredin two waves, first wave occurred during the spring of 2009 whereas the second wavestarted around early October 2009 [5] and spanned about 3 months as it diminishedby the end of January 2010 [31]. It is believed that genetic re-assortment involvingthe swine influenza virus lineages might be the main factor behind the emergence ofH1N1 pandemic [22]. Various behavioral factors and chronic health conditions areconsidered to impose an increased risk of infection severity among H1N1 individuals.In particular, pregnant women, especially in their third trimester and infants are atincreased risk of hospitalization and intensive care unit (ICU) admission [8,10,23,50].Moreover, individuals with pre-existing chronic conditions such as asthma, chronickidney and heart diseases, and obesity are at far greater risk of death and ICUadmissionin comparison to healthy individuals [11,28]. Additionally, geographical location mayalso impart a great risk to the severity of the infection. It is observed that peopleresiding in remote, isolated communities and aboriginals, in the Canadian provinceof Manitoba, experienced severe illness due to the pandemic H1N1 infections [45].Most recent outbreak of 2014, in India, had reported 937 cases and 218 deaths fromswine flu. The reported cases and deaths in 2015 had surpassed the previous numbers.The total number of confirmed cases crossed 33,000 with death of more than 2000[43].

Like seasonal influenza, H1N1 influenza is also transmitted through coughing andsneezing by infected individuals. It may also be transmitted through contact withcontaminated objects. Main symptoms of H1N1 infection include fever, cough, sorethroat, body aches, headache, chills and fatigue [7]. Various preventive measures weretaken tominimize H1N1 infection cases. Thesemeasures included (a) social exclusionsuch as closing schools during the epidemic and banning public gatherings etc., (b)mass vaccination mainly to high risk groups due to limited supply and (c) the use ofantiviral drugs such as oseltamivir (Tamiflu) and zanamivir (Relenza) [6].Although the

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Mathematical analysis of swine influenza epidemic model… 271

epidemic breakout occurred in 2009, H1N1 influenza still poses significant challengesto public health worldwide [9,14,18,47–49].

In view of the serious consequences posed by the H1N1 epidemic on the publichealth, various mathematical models have been proposed and analyzed in order tounderstand the transmission dynamics of the H1N1 influenza [3,4,17,19,21,22,34,39]. In particular, Boëlle et al. [3] estimated the basic reproduction number for theH1N1 influenza outbreak inMexico. Gojovic et al. [19] proposed mitigation strategiesto overcome H1N1 influenza. Sharomi et al. [39] presented an H1N1 influenza modelthat accounts for the role of an imperfect vaccine and antiviral drugs administeredto infected individuals. In the past, the basic reproduction number Rc has also beencalculated by various researchers to control the disease by considering most-sensitiveparameters but these strategies were not time-dependent. Lenhart proposed HIVmod-els [15,27], usingwhich time-dependent optimal controls were proposed. In this paper,wewill propose an optimal treatment strategy to prevent theH1N1 influenza epidemic.

The purpose of the current study is to complement the aforementioned studies byformulating a new model in which the susceptible population is stratified accordingto the risk of infection. Our model also accounts for the impact of singular use ofthe antiviral drugs, administered as a preventive agents only i.e., given to susceptibleindividuals or, in addition, as a therapeutic agent i.e., given to individuals with symp-toms of the pandemic H1N1 infection in the early stage of illness. In this study wewill present a rigorous mathematical analysis of our model and show that our model,under suitable conditions involving the basic reproduction number Rc, exhibits dis-ease free equilibrium as well as the phenomenon of backward bifurcation. We willperform uncertainty and sensitivity analysis of Rc in order to identify crucial modelparameters. Additionally we will also propose optimal strategies that will eliminatethe infection with minimal application of resources.

The paper is organized as follows. Themodel is formulated in Sect. 2 and rigorouslyanalyzed in Sect. 3.Weperformuncertainty and sensitivity analysis in Sect. 4 and usingideas from optimal control theory, we present an optimal treatment strategy in Sects. 5and 6 marks the end of our paper with a discussion on our results.

2 Model formulation

The total human population at time t , denoted by N (t), is sub-divided into ninemutually-exclusive sub-populations comprising of low risk susceptible individuals(SL(t)), high risk susceptible individuals (SH (t)), susceptible individuals who weregiven antiviral drugs/vaccines (P(t)), latent individuals (L(t)), symptomatic individ-uals at early stage (I1(t)), symptomatic individuals at later stage (I2(t)), hospitalizedindividuals (H(t)), treated individuals (T (t)) and recovered individuals (R(t)), sothat

N (t) = SL(t) + SH (t) + P(t) + L(t) + I1(t) + I2(t) + H(t) + T (t) + R(t).

The high risk susceptible population includes pregnant women, children, health-careworkers and providers (including all front-lineworkers), the elderly and other immune-

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272 M. Imran et al.

Fig. 1 Schematic diagram of the model (1)

compromised individuals. The rest of the susceptible population is considered to beat low risk of acquiring H1N1 infection. The model to be considered in this studyis given by the following deterministic system of non-linear differential equations (aschematic diagram of the model is given in Fig. 1, and the associated variables andparameters are described in Table 1).

dSLdt

= π(1 − p) − λSL − σL SL − μSL ,dSHdt

= πp − θHλSH − σH SH − μSH ,dP

dt= σL SL + σH SH − θPλP − μP,

dL

dt= λ(SL + θH SH + θP P) − (α + μ)L ,

123


Table 1 Description and nominal values of the model parameters

Description Value Ref

π Birth rate 1,119,583/80*365 Assumed

1/μ Average human lifespan 80*365 Assumed

p Fraction of susceptible individuals athigh risk for contracting infection

0.4 Assumed

β Effective contact rate for transmittingH1N1 influenza

0.9 (initial) To be estimated

σL Antiviral coverage rate for low risksusceptible individuals

0.3 [32]

σH Antiviral coverage rate for high risksusceptible individuals

0.5 [32]

α Rate at which latent individualsbecome infectious

1/1.9 [32]

τ1 Treatment rate for individuals in theearly stage of infection

1/5 [32]

τ2 Treatment rate for individuals in thelater stage of infection

1/3 [32]

φI2 Recovery rate for symptomatic infec-tious individuals in the

1/5 [32]

Later stage

φH Recovery rate for hospitalized indi-viduals

1/5 [32]

φT Recovery rate treated individuals 1/3 [32]

η1 Modification parameter (see text) 0.1 [32]

η2 Modification parameter (see text) 1/2 [32]

η3 Modification parameter (see text) 1.2 [32]

η4 Modification parameter (see text) 1 [32]

θH Modification parameter for infectionrate of high risk

1.2 [32]

Susceptible individuals

1 − θP Drug efficacy against infection 0.5 [32]ψ Hospitalization rate of individuals in

I2 class0.5 [32]

γ Progression rate from I1 to I2 classes 0.06 [32]

δ Disease-induced death rate of individ-uals in I2 class

1/100 [32]

θ1δ Disease-induced death rate for hospi-talized individuals

1/100 [32]

d I1dt

= αL − (τ1 + γ + μ)I1, (1)d I2dt

= γ I1 − (τ2 + ψ + φI2 + μ + δ)I2,dH

dt= ψ I2 − (φH + μ + θ1δ)H,

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274 M. Imran et al.

dT

dt= τ1 I1 + τ2 I2 − (φT + μ)T,

dR

dt= φI2 I2 + φH H + φT T − μR.

In (1), π represents the recruitment rate into the population (all recruited individualsare assumed to be susceptible) and p is the fraction of recruited individuals who areat high risk of acquiring infection. Low risk susceptible individuals acquire infectionat a rate λ, given by

λ = β(η1L + I1 + η2 I2 + η3H + η4T )N

,

where β is the effective contact rate, ηi (i = 1, . . . , 4) are the modification parametersaccounting for relative infectiousness of individuals in the L , I2, H1 and T classesrespectively in comparison to those in the I1 class. High risk susceptible individualsacquire infection at a rate θHλ, where θH > 1 accounts for the assumption that highrisk susceptible individuals are more likely to get infected in comparison to low risksusceptible individuals. Low (high) risk susceptible individuals receive antiviral at arate σL (σH ), and individuals in all epidemiological classes suffer natural death at arate μ.

Susceptible individuals who received prophylaxis (P) can become infected at areduced rate θPλ, where 1 − θP (0 < θP < 1) is the efficacy of the antiviral in pre-venting infection. Individuals in the latent class become infectious at a rate α andmove to the symptomatic early stage of infection I1. Individuals in the I1 class receiveantiviral treatment at a rate τ1. These individuals progress to the later infectious class(I2) at a rate γ . Similarly, individuals in the I2 class are treated (at a rate τ2), hospi-talized (at a rate ψ), recover (at a rate φI2 ) and suffer disease induced death (at a rateδ). Hospitalized individuals, recover (at a rate φH ) and suffer disease-induced death(at a reduced rate θ1δ, where 0 < θ1 < 1 accounts for the assumption that hospi-talized individuals, in the H class, are less likely to die than uncapitalized infectiousindividuals in the I2 class).

Finally, treated individuals recover (at a rateφT ). It is assumed that recovery conferspermanent immunity against re-infection with H1N1. The model (1) is an extensionof the model presented by Sharomi et al. [39] by:

(i) administering antivirals to susceptible individuals.(ii) stratifying the susceptible population based on the risk of infection.

The basic qualitative properties of the model (1) will now be analyzed.

2.1 Basic properties of the model

Lemma 1 The closed set

D ={(SL , SH , P, L , I1, I2, H, T, R) ∈ R9+ : N ≤

π

μ

}

123


is positively-invariant and attracting with respect to the model (1).

Thus, in the regionD, the model is well-posed epidemiologically and mathematically[20]. Hence, it is sufficient to study the qualitative dynamics of the model (1) in D.

3 Existence and stability of equilibria

3.1 Local stability of disease-free equilibrium (DFE)

The model (1) has a DFE, given by

E0 = (S∗L , S∗H , P∗, L∗, I ∗1 , I ∗2 , H∗, T ∗, R∗)=

(S∗L , S∗H ,

σL S∗L + σH S∗Hμ

, 0, 0, 0, 0, 0, 0,

),

with,

S∗L =π(1 − p)σL + μ and S

∗H =

πp

σH + μ.

Following [44], the linear stability of E0 can be established using the next generationoperator method on system (1). The calculation is presented in Appendix. It followsthat the control reproduction number, denoted by Rc, is given by

Rc = βΩK1{η1 + αK2 +

η4(αγ τ2+αK3τ1)K2K3K5

+ αγ η2K2K3 +αγψη3K2K3K4

}(2)

where, ρ is the spectral radius (dominant eigenvalue in magnitude) of the next gener-ation matrix FV−1 and K1 = α + μ, K2 = τ1 + γ + μ, K3 = τ2 + ψ + φ2 + μ + δ,K4 = ψ2 + φ3 + μ + θ1δ, K5 = φT + μ. Hence, using Theorem 2 of [44], thefollowing result is established.

The epidemiological significance of the control reproduction number, Rc, whichrepresents the average number of new cases generated by a primary infectious individ-ual in a population where some susceptible individuals receive antiviral prophylaxis,is that the H1N1 pandemic can be effectively controlled if the use of antiviral canbring the threshold quantity (Rc) to a value less than unity. Biologically-speaking,Lemma 2 implies that the H1N1 pandemic can be eliminated from the population(when Rc < 1) if the initial sizes of the sub-populations in various compartmentsof the model are in the basin of attraction of the DFE (E0). To ensure that diseaseelimination is independent of the initial sizes of the sub-populations of the model, itis necessary to show that the DFE is globally asymptotically stable (GAS). This isconsidered below, for a special case.

Lemma 2 The DFE, E0, of the model (1), is locally asymptotically stable (LAS) ifRc < 1, and unstable if Rc > 1.

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276 M. Imran et al.

3.2 Global stability of DFE

The GAS property of E0 is presented here. We proved the following result: following:

Theorem 1 The DFE, E0, of the model (1) is GAS in D if Rc ≤ R∗ = ΩθH

and

θH = 1, θP = 0.The proof is presented in the appendix.

Theorem 1 shows that the H1N1 pandemic can be eliminated from the communityif the use of antivirals can lead to Rc ≤ R∗.Theorem 2 IfRc > 1, then the disease is uniformly persistent: there exists an � > 0such that

lim inft→∞ {L(t) + I1(t) + I2(t) + H(t) + T (t)} > �

for all solutions (SL , SH , P, L , I1, I2, H, T, R) of (1) with L(0) + I1(0) + I2(0) +H(0) + T (0) > 0.Proof is presented in the appendix. The epidemiological implication of the result isthat disease will prevail in the community ifRc > 1.

It is convenient to define the following quantities

RP = Rc∣∣τ1=τ2=0 =

βΩ

K1

{η1 + α

K2+ αγ η2

K2K3+ αγψη3

K2K3K4

},

RT = Rc∣∣σL=σH=0 =

βω

K1

{η1+ α

K2+ η4(αγ τ2+ αK3τ1)

K2K3K5+ αγ η2

K2K3+ αγψη3

K2K3K4

}

(3)

where,ω = Ω∣∣σL=σH=0. The quantitiesRP andRT represent the control reproduction

numbers associated with the prophylactic (RP ) or therapeutic (RT ) use of antiviralsin the community only.

3.3 Endemic equilibrium and backward bifurcation

In order to find endemic equilibria of the basic model (1) (that is, equilibria whereatleast one of the infected components of the model (1) are non-zero), the followingsteps are taken. Let

E1 = (S∗∗L , S∗∗H , P∗∗, L∗∗, I ∗∗1 , I ∗∗2 , H∗∗, T ∗∗, R∗∗)

represents an arbitrary endemic equilibrium of the model (1). Further, let

λ∗∗ = β(η1L∗∗ + I ∗∗1 + η3 I ∗∗2 + η4H∗∗1 + η6T ∗∗)

N∗∗,

123


be the associated force of infection at steady-state. Solving the equations of the modelat steady-state yields:

S∗∗L =π(1 − p)

λ∗∗ + σL + μS∗∗H =

πp

θHλ∗∗ + σH + μP∗∗ = σL S

∗∗L + σH S∗∗H

θPλ∗∗ + μ =π(Aλ∗∗ + B)

Q

L∗∗ = λ∗∗(S∗∗L + θH S∗∗H + θP P∗∗)

K1= λ

∗∗(Fλ2∗∗ + Gλ∗∗ + H)K1Q

I ∗∗1 =α

K2L∗∗ = αλ

∗∗(Fλ2∗∗ + Gλ∗∗ + H)K1K2Q

I ∗∗2 =γ

K3I ∗∗1 =

γαλ∗∗(Fλ2∗∗ + Gλ∗∗ + H)K1K2K3Q

H∗∗ = ψK4

I ∗∗2 =ψγαλ∗∗(Fλ2∗∗ + Gλ∗∗ + H)

K1K2K3K4Q

T ∗∗ = τ1 I∗∗1 + τ2 I ∗∗2

K5= τ1αλ

∗∗(Fλ2∗∗ + Gλ∗∗ + H)K1K2K5Q

+ τ2γαλ∗∗(Fλ2∗∗ + Gλ∗∗ + H)K1K2K3K5Q

R∗∗ = φI2 I∗∗2 + φH H∗∗ + φT T ∗∗

μ= φI2γαλ

∗∗(Fλ2∗∗ + Gλ∗∗ + H)μK1K2K3Q

+ φHψγαλ∗∗(Fλ2∗∗ + Gλ∗∗ + H)μK1K2K3K4Q

+ φT τ1αλ∗∗(Fλ2∗∗ + Gλ∗∗ + H)

μK1K2K5Q+ φT τ2γαλ

∗∗(Fλ2∗∗ + Gλ∗∗ + H)μK1K2K3K5Q

.

(4)

Using (8) and the expression for λ∗∗ we get

a0(λ∗∗)3 + b0(λ∗∗)2 + c0(λ∗∗) + d0 = 0 (5)

where,

a0 = A1A8b0 = A6 + A2A8 −

(RcΩ

)A1

= πθH θP(1 − Rc

Ω

)+ πpθp(1 − θH ) + A2A8

c0 = A4 + A3A8 −(Rc

Ω

)A2

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278 M. Imran et al.

= π(1 − p)[θP (σH + μ) + μθH ](1 − Rc

Ω

)

+ πp[μ + θP (σL + μ)](1 − θHRc

Ω

)+ π A7

(1 − θPRc

Ω

)+ A3A8

d0 = A5 −(Rc

Ω

)A3

= πμ(1 − p)(σH + μ)[μ

(1 − Rc

Ω

)+ σL

(1 − θPRc

Ω

)]

+ πpμ(σL + μ)[μ

(1 − θHRc

Ω

)+ σL

(1 − θPRc

Ω

)]

The large number of new coefficients (Ai ) are defined in the Appendix.The positive endemic equilibria of the model (1) are obtained by solving the cubic

(5) for λ∗∗ and substituting the results (positive values of λ∗∗) into (4). The coefficienta0, of (5), is always positive, and the signs of the remaining parameters are dependentupon the value ofRc. Since we have a cubic equation to solve, it might not be possibleto capture the dynamics of the system in general. Therefore, wewill try tomake naturaland reasonable assumptions to study the system. Now we will discuss the solutions of(5) under different conditions.

Theorem 3 IfRcΩ

>1

θP, then the model (1) exhibits a unique endemic equilibrium.

However in case ofRcΩ

<1

θHno endemic equilibrium exists.

Proof IfRcΩ

<1

θH, it is easy to verify that all the coefficients of (5) are positive.

Therefore by Descartes’s sign rule there is no positive root and hence no endemicequilibrium.

However ifRcΩ

>1

θP, we have d0 < 0. Considering the other parameters, there is

exactly one sign change among consecutive parameters. According to Descartes signrule, we have exactly one positive root. Thus we have a unique endemic equilibrium.This completes the argument. ��

Now the question arises what happens when1

θH<

RcΩ

<1

θP. In general, we

cannot make any comment about this scenario. However the numerical simulations(using actual parameter estimates) suggest that there is no endemic equilibrium for1

θH<

RcΩ

< 1 and there exists a unique endemic equilibrium when 1 <RcΩ

<1

θP.

Themost important case arises whenwe assume that vaccination was perfect (θP =0) and that infection rate of high risk susceptibles is unity (θH = 1). These assumptionsreduce (5) to a quadratic equation given below

e0(λ∗∗)2 + f0(λ∗∗) + g0 = 0 (6)

123


0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.90

500

1000

1500

2000

2500

3000

3500

4000

Rc

I 1

Fig. 2 Backward bifurcation

where,

e0 = πμZf0 = π(σL(1 − p) + σH p) + πμ + πμ[(σH + μ)(1 − p) + p(σL + μ)]Z − π Rc

Ω

g0 = μ[(σH + μ)(1 − p) + p(σL + μ)](1 − Rc

Ω

)+ πB

Lemma 3 IfRcΩ

< 1 (along with perfect vaccination, θP = 0 and infection rate ofhigh risk susceptibles, relative to that of low risk susceptibles, unity θH = 1), thenBackward Bifurcation is observed.

Proof Clearly e0 > 0. Now ifRcΩ

< 1, we have g0 > 0. Thus, whenRcΩ

< 1 and

f0 < 0, we have two positive roots and hence we have Backward Bifurcation. Theactual parameter estimates clearly show this phenomenon (Fig. 2). ��

4 Uncertainty and sensitivity analysis

This section attempts to discuss the possible means of uncertainty in the values of ourmodel parameters and to quantify the results. In a deterministic model, uncertaintycan be generated by only input (initial conditions, parameters) variation and modelparameters are among the most important part of the input data. The input data, partic-ularly parameter estimates, can be uncertain due to the lack of sophisticatedmeasuringmethods, error in collecting and interpreting data and natural variations. Therefore, inorder to study this uncertainty we perform global uncertainty and sensitivity analysison our model output.

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280 M. Imran et al.

0.2

0.4

0.6

0.8

10

2000

4000

β: M

ean

= 0

.50

, Std

= 0

.050

0.06

0.08

0.1

0.12

0

2000

4000

σ L: M

ean

= 0

.08

, Std

= 0

.005

0.12

0.14

0.16

0.18

0

2000

4000

σ H: M

ean

= 0

.14

, Std

= 0

.005

0.2

0.3

0.4

0.5

0

2000

4000

α: M

ean

= 0

.35

, Std

= 0

.034

0.4

0.5

0.6

0.7

0

2000

4000

τ 1: M

ean

= 0

.50

, Std

= 0

.049

0.4

0.5

0.6

0.7

0.8

0

2000

4000

τ 2: M

ean

= 0

.60

, Std

= 0

.049

0.1

0.2

0.3

0.4

0

2000

4000

φ I2:

Mea

n =

0.2

0 , S

td =

0.0

19

0.1

0.2

0.3

0.4

0

2000

4000

φ H: M

ean

= 0

.20

, Std

= 0

.019

0.2

0.25

0.3

0.35

0

2000

4000

φ T: M

ean

= 0

.25

, Std

= 0

.019

0.05

0.1

0.15

0.2

0

2000

4000

η 1: M

ean

= 0

.10

, Std

= 0

.010

11.

11.

21.

31.

40

2000

4000

η 2: M

ean

= 1

.20

, Std

= 0

.038

0.08

0.09

0.1

0.11

0.12

0

2000

4000

η 3: M

ean

= 0

.10

, Std

= 0

.003

0.1

0.2

0.3

0.4

0

2000

4000

η 4: M

ean

= 0

.20

, Std

= 0

.019

1.1

1.15

1.2

1.25

1.3

0

1000

2000

θ H: M

ean

= 1

.20

, Std

= 0

.058

0.2

0.25

0.3

0.35

0.4

0

2000

4000

1 −

θP: M

ean

= 0

.30

, Std

= 0

.020

0.4

0.5

0.6

0.7

0

2000

4000

ψ: M

ean

= 0

.50

, Std

= 0

.050

0.02

0.04

0.06

0.08

0.1

0

2000

4000

γ: M

ean

= 0

.06

, Std

= 0

.006

−0.

010

0.01

0.02

0.03

0

2000

4000

δ: M

ean

= 0

.01

, Std

= 0

.005

−0.

010

0.01

0.02

0.03

0

2000

4000

θ 1δ:

Mea

n =

0.0

1 , S

td =

0.0

05

22.

53

3.5

4

x 10

−5

0

2000

4000

μ: M

ean

= 3

e−5

, Std

= 1

.99e

−06

0.5

11.

52

2.5

0

2000

4000

R0:

Mea

n =

1.4

5 , S

td =

0.1

77

Fig.3

Uncertainty

analysis

123


0 2 4 6 8 10 12 14 16 18 20 22−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Val

ues

of S

ensi

tivity

Inde

xes

(PR

CC

)Sensitivity of Rc with respect to model paramters

β, 0.97

σL, 2e−3

σH

, 3e−4

α, −0.33

τ1, −0.91

φI, −0.025

ψ, −.08

γ, −0.15

δ, −0.01 μ, −0.02

θ1δ, −0.011−θ

P, 0.01

θH

, −0.03η

3, 3e−3

η4, 0.68

η2, 0.02

η1, 0.35

φT, −0.60

φH

, −0.01τ2, −0.06

Fig. 4 Sensitivity analysis

Uncertainty analysis gives a qualitative measure of parameters which are critical inthe model output. This analysis also presents the degree of confidence on the modelparameters. The Latin hypercube sampling will be used to perform the uncertaintyanalysis on the model parameters presented in Table 1. On the other hand, sensitivityanalysis presents a qualitative measure of important parameters and quantifies theimpact of each parameter on the model output, keeping the other parameters constant.Sensitivity analysis comprises of calculating the Partial Rank Correlation Coefficient(PRCC). The rank correlation coefficient is a standard method of quantifying thedegree of monotonicity between an input parameter and the model output.

So far the dynamics of the model are determined by the threshold quantity Rc,which determines the prevalence of the disease and therefore we consider it as ourmodel output. Both uncertainty and sensitivity analyses are performed on the modelparameters as the input data andRc as the model output. The parameter estimates andthe assigned distributions are given in the Appendix.

The estimate ofRc from uncertainty analysis is 1.32 with 95%CI (1.13, 1.81). Theprobability that Rc > 1 is 99%. This suggests that H1N1 will get endemic under theassumed conditions. However, the time taken to reach that state could be different.

From the sensitivity analysis, it is apparent that most significant parameters to Rcare (the ones having PRCC value above 0.5 or below −0.5) β, τ1, φT and η4. Thisresult implies that these parameters need to be estimated with utmost precision andaccuracy in order to capture the transmission dynamics of H1N1 (Figs. 3, 4).

5 Optimal control

Optimal control theory has found use in making decisions involving epidemic andbiologicalmodels. The desired results and performance of the control functions dependon the different situations. Lenhart’s HIV models [15,27] used optimal control to

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282 M. Imran et al.

design the treatment strategies. Lenhart [25] provides a very good example of decidinghow to divide the efforts between two treatment strategies (case holding and casefinding) of the two strain TB model. Yan [52] used an optimal isolation strategy tofight the SARS epidemic. In [24] Joshi formulated two control functions as coefficientsof the ODE system representing treatment effects in a two drug regime in an HIVimmunology model. The goal was to maximize the concentration of T cells whileminimizing the toxic effects of the drug. The analytic and numerical results illustratedthe level of two drugs to be used over the chosen time interval. The required balancingeffect between two competing goals was well predicted by optimal control theory.

While formulating anoptimal control problem, decidinghowandwhere to introducethe control (through vaccination, drug treatment etc) in the system of differentialequation is very important. The formulation of the optimal control problem must be areasonable and practical representation of the situation to be considered. The form ofthe optimal control depends heavily on the system being analyzed and the objectivefunctional to be optimized. We will consider a quadratic dependence on the control inthe objective functional, this will be briefly justified when we discuss our formulationof the problem.

The existence and uniqueness of the optimal control in an optimality system canbe established using the Lipchitz properties of the differential equations[16,35]. Fora detailed example, see the work of Lenhart [15,16]. After establishing the existenceand the uniqueness results, we can confidently continue to numerically solving theoptimality system to get the desired optimal control.

Control theory for models defined as a system of differential equations was formu-lated by Pontryagin [35]. Since his time, the theory and its application have grownconsiderably in number. Pontryagin’s maximum principle is a technique to optimizethe performance criterion, (mostly cost and efficiency functional) which depends ondifferent control parameters. The control parameters introduced are mostly functionsof time appearing as coefficients in the model. The technique involves reducing aproblem in which an optimal function over an entire time domain is to be determinedto a standard optimization problem. This is achieved by appending an adjoint systemof differential equations (with terminal boundary conditions) to the original model(state system) of differential equations (with initial conditions). The adjoint functionsbehave very similar to Lagrange multipliers (appending constraints to the functionof several variables to be maximized or minimized). The adjoint variables maximizeor minimize the state variables with respect to the desired objective functional. Thedetails of the necessary conditions for the adjoint and optimal controls are presentedhere [16,26,35]. For the application of these results see [15].

We will attempt to optimize the treatment rates τ1 and τ2 of our model (1). We aimto design an optimal treatment strategy that minimizes an objective functional takinginto account both the cost and the number of infectious individuals. Let the treatmentrates τ1 and τ2 be functions of time. The control set V is

V = {τ1(t), τ2(t) : a1 ≤ τ1(t) ≤ b1, a2 ≤ τ2(t) ≤ b2, 0≤ t ≤ T, τ1(t) are τ2(t) are Lebesgue measurable}. (7)

123


The goal is to minimize the cost function defined as

J [τ1(t), τ2(t)] =∫ T0

[X1P + X2 I1 + X3 I2 + X4H + 1

2W1τ

21 (t) +

1

2W2τ

22 (t)

]dt

(8)

This objective functional (performance criterion) involves the numbers of infectedindividuals as well as the cost of treatment. Since the total cost includes not only theconsumption for every individual but also the cost of organization, management, andcooperation, a non-linear cost function seems to be the right choice. In this paper, aquadratic function is implemented for measuring the control cost by reference to liter-ature in epidemics control [15,25,27,52]. The coefficients W1 and W2 are balancingcost factors due to scales and importance of the cost part of the objective functional, b1and b2 are the maximum attainable values of the control, Xi is the positive constantsto keep balance in the population size in each class. We attempt to find an optimalcontrol pair (τ ∗1 , τ ∗2 ) such that

J [τ ∗1 , τ ∗2 ] = minτ1,τ2∈V

J [τ1, τ2]

The existence of an optimal solution to the optimal control problem can be estab-lished by Theorem III.4.1 and its corresponding Corollary in [16]. The boundedness ofsolutions to the system (3.1) for the finite time interval is needed to establish these con-ditions andD satisfies it. Pontryagin’s maximum principle [35] provides the necessaryconditions to be satisfied by the optimal vaccination v(t). As stated earlier this reducesthe problem to one of minimizing point wise a Hamiltonian, H , with respect to v

H = X1P + X2 I1 + X3 I2 + X4H + 12W1τ

21 (t) +

1

2W2τ

22 (t) +

i=9∑i=1

ξi ki (9)

where ki represents the right hand side of the modified model’s i th differential equa-tion. Using the Pontryagin’smaximumprinciple [35] and the optimal control existenceresult from [16], we have

Theorem 4 There exists a unique optimal pair τ ∗1 (t), τ ∗2 (t) which minimizes J overV. Also, there exists adjoint system of ξi ’s such that

dξ1dt

= (λ + σL + μ)ξ1 − σLξ3 − λξ4dξ2dt

= (θHλ + σH + μ)ξ2 − σH ξ3 − θHλξ4dξ3dt

= (θPλ + μ)ξ3 − θPλξ4 − X1dξ4dt

= βN

η1SLξ1 + βN

η1θH SH ξ2 − βN

η1(SL + θH SH + θp P)ξ4 + C1ξ4 − αξ5

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284 M. Imran et al.

dξ5dt

= βNSLξ1 + β

NθH SH ξ2 − β

N(SL + θH SH + θP P)xi4

+ C2ξ5 − γ ξ6 − τ ∗1 (t)ξ8 − X2dξ6dt

= βN

η2SLξ1 + βN

η2θH SH ξ2 − βN

η2(SL + θH SH + θP P)xi4+ C3ξ6 − ψξ7 − τ ∗2 (t)ξ8 − φI2ξ9 − X3

dξ7dt

= βN

η3SLξ1 + βN

η3θH SH ξ2 − βη 3

N (SL + θH SH + θP P)xi4+ C4ξ7 − φH ξ9 − X4

dξ8dt

= C5ξ8 − φT ξ9dξ9dt

= μξ9

where C1 = α1 +μ, C2 = τ ∗1 (t)+γ +μ, C3 = τ ∗2 (t)+ψ +φI2 +μ+ δ, C4 =ψ2 + φ3 + μ + θ1δ, C5 = φT + μ

The transversality condition gives

ξi (t f ) = 0.

The vaccination control is characterized as

τ ∗1 (t) = min[b1,max

(a1,

I1(ξ5 − ξ8)W1

)](10)

τ ∗2 (t) = min[b2,max

(a2,

I2(ξ6 − ξ8)W2

)]. (11)

Proof It is easy to verify that the integrand of J is convex with respect to τ1 and τ2.

Also the solutions of the (1) are bounded as N (t) ≤ �μ

for all time. Also it is clear

the state system (1) has the Lipschitz property with respect to the state variables. Withthese properties and using Corollary 4.1 of [16], we have the existence of the optimalcontrol.

Since we have the existence of the optimal vaccination control. Using the Pontrya-gin’s maximum principle, we obtain

dξ1dt

= − ∂H∂SL

, ξ1(t f ) = 0dξ2dt

= − ∂H∂SH

, ξ2(t f ) = 0...

dξ9dt

= −∂H∂R

, ξ9(t f ) = 0

123


evaluated at the optimal control, which results in the stated adjoint system . Theoptimality condition is

∂H

∂τ1

∣∣∣∣τ∗1

= 0

∂H

∂τ2

∣∣∣∣τ∗2

= 0.

Therefore on the set {t : a1 < τ ∗1 (t) < b1, a2 < τ ∗2 (t) < b2}, we obtain

τ ∗1 (t) =I1(ξ5 − ξ8)

W1

τ ∗2 (t) =I2(ξ6 − ξ8)

W2.

Considering the bounds on τ1 and τ2, we have the characterizations of the optimalcontrol as in (14) and (15). Clearly the state and the adjoint functions are bounded.Also it is easily verifiable that state system and adjoint system has Lipschitz structurewith respect to the corresponding variables, from here we obtain the uniqueness of theoptimal control for sufficiently small time T [16,35]. The uniqueness of the optimalcontrol pair follows from the uniqueness of the optimality system, which consists of(3.1) and (3.5), with characterizations (3.6). Note here the uniqueness of the optimalcontrol is only for a certain length of time. This restriction on the length of the timeinterval arises due to the opposite time orientations of (3.1), and (3.5); the state systemhas initial conditions and the adjoint system has final conditions. For example see [15],this restriction is very common in control problems (see [27]). ��

The optimality system consisting of 9 state system of differential equations andan adjoint system differential equations with SL(0) = SL0, SH (0) = SH0, P(0) =P0, L(0) = L0, I1(0) = I10, I2(0) = I20, H(0) = H0, T (0) = T0, R(0) = R0.(�)

Next, we discuss the numerical solutions of the optimality system and the corre-sponding optimal control pair. The optimal treatment strategy is obtained by solvingthe optimality system. An iterative scheme is used for solving the optimality sys-tem. We start to solve the state equations with a guess for the control pair (τ1, τ2),using a forward Runge–Kutta method. The adjoint functions have final time condi-tions. Because of this transversality conditions on the adjoint functions , the adjointequations are solved by a backward Runge–Kutta method using the current iterationsolution of the state equations. Then, the controls are updated by using a convex com-bination of the previous control and the value from the characterizations. This processis repeated and iteration is stopped if the values of unknowns at the previous iterationare very close to the ones at the present iteration.

The parameter values used have Rc > 1 when the model without control is con-sidered. Thus the disease is not expected to die out without intervention strategies.

Figure 5 represents the control strategy to be employed for the optimal results. Thiscontrol strategyminimizes both the cost and the infected population (I1+ I2). It is well

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286 M. Imran et al.

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (Days)

Pro

port

ion

of P

opul

atio

n T

reat

ed

τ1τ2

Fig. 5 Simulations show the optimal treatment control τ1 and τ2

0 10 20 30 40 500

50

100

150

200

250

300

350

400

450

500

Time (Days)

Infe

cted

Pop

ulat

ion

I 1+

I 2

Optimal ControlConstant Control

Fig. 6 Infected population I1 + I2 under different treatment control strategies

understood that in order to eradicate an epidemic, a large fraction of infected populationneeds to be treated. Therefore, an upper bound of 0.7 was chosen for treatment control.The optimal controls remain at higher values initially and then steadily decreasing to 0.In fact, at the beginningof simulated time, the optimal control is staying at higher valuesin order to treat as many infected as possible to prevent the susceptible individualsfrom getting infected and epidemic to break out. The steadily decreasing of the v isdetermined by the balance between the cost of the infected individuals and the cost ofthe controls. Figure 6 shows the infected (I1 + I2)population for the optimal controland constant control. It is easy to see that the optimal control is much more effectivefor reducing the number of infected individuals and decreasing the time-span of theepidemic. As normally expected, in the early phase of the epidemic breakouts, keepingthe control at their upper bound will directly lead to the decreasing of the number of

123


0 10 20 30 40 50 60 70 80 900

1

2

3

4

5

6

7

8

9

10x 10

4

Time (Days)

Cos

t of D

isea

se C

ontr

ol

Optimal Control

τ1 = 0.18, τ

2 = 0.15

τ1 = 0.25, τ

2 = 0.2

τ1 = 0.3, τ

2 = 0.3

τ1 = 0.45, τ

2 = 0.25

τ1 = 0.5, τ

2 = 0.4

Fig. 7 Simulation of accumulated cost of different control strategies

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (Days)

Pro

port

ion

of P

opul

atio

n T

reat

ed (τ

1* )

β = 0.3β = 0.35β = 0.4β = 0.45

(a)

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (Days)

Pro

port

ion

of P

opul

atio

n T

reat

ed (τ

2* )

β = 0.3β = 0.35β = 0.4β = 0.45

(b)

Fig. 8 Optimal control strategies for different β, a optimal control τ1, b optimal control τ2

the infected people. Figure 7 shows the cost associated with the optimal and constantcontrol strategy. It is clear the cost of optimal strategy is much less than the cost ofconstant strategy. Moreover Fig. 8 represents control strategies for different values ofthe effective contact rate β. It can be observed that increased effective contact raterequires us to implement control strategy for a longer duration of time.

6 Estimating the basic reproduction number for the 2009 H1N1influenza outbreak in Manitoba, Canada

Our purpose in this section is to estimate the transmission ability of the H1N1 virusduring the two waves of the 2009 H1N1 epidemic in Manitoba, Canada, using epi-demic data in the form of the daily confirmed cases of influenza. We will make use ofthe previous work of Cintron-Arias et al. [12], who have developed a standard algo-

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rithm involving the use of a generalized least squares (GLS) scheme to fit epidemicdata to a proposed deterministic ODE model. The GLS scheme is executed in thecontext of a statistical model and using the assumption that the epidemic data con-tains random noise. We then use statistical asymptotic theory to obtain the limitingprobability distribution of the unknown model parameters. A detailed description ofthe methodology can be found in [12].

A simple but effective measure of the transmission ability of an infectious diseaseis given by the basic reproduction number, defined as the total number of secondaryinfections produced by introducing a single infective in a completely susceptible pop-ulation. We will use the methods developed by Cintron-Arias et al. [12] to estimatethe parameters of model (1) and thereby estimate the effective reproduction numberRc for the H1N1 epidemic.

Estimates for several of the model parameters used in model (1) can be obtainedfrom existing studies on H1N1 influenza. Table 1 lists these parameters along withreasonable estimates of their values.

The effective contact rate for transmitting H1N1 influenza β, which is a measureof the rate at which contact between an infective and a susceptible individual occursand the probability that such contact will lead to an infection, is extremely difficult todetermine directly. Consequently, we adopt an indirect approach, similar to previousstudies such as [12], byfirst finding the value of the parameterβ forwhich themodel (1)has the best agreement with the epidemic data, and then using the resultant parametervalues to estimateRc.

As mentioned before, for the purpose of simulating model (1) we require knowl-edge of the initial conditions. It is possible to consider the initial conditions as modelparameters along with the effective contact rate β and estimate values for all parame-ters.

Such an approach produces slightly unreliable results. This is explained by the factthat the available epidemic data is restricted to the daily confirmed cases of H1N1,while the optimization schemes that we will employ produce estimates for ten vari-ables. There are thus too many degrees of freedom and the ’best-fit’ may result inunrealistic estimates for the initial conditions. We will therefore use reasonable esti-mates for the initial conditions given in Table 2 and restrict ourselves to optimizingonly the effective contact rate β.

The epidemic data to be used in this study is in the form of daily confirmed casesof H1N1 influenza in Manitoba, Canada, reported over approximately eight months,from the 24th of April 2009 to the 3rd of January 2010. The data is displayed in Fig. 9.

The best-fit trajectory obtained by applying the GLS methodology in [12] to model(1) is displayed in the figure below. We have assumed that the first wave began on the24th of April 2009 and control measures involving antiviral and therapeutic agents wasstarted 48 days later. Similarly, we assume that the second wave began on the 3rd ofOctober and controlmeasures for the secondwavebegan50days later. Furthermore,weassume that the antiviral and therapeutic control interventions that were implementedduring the first wave also continue to be implemented before, during and after thesecond wave. Figure 10 shows data fitting of daily confirmed cases of H1N1 influenzain Manitoba, Canada.

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Table 2 Estimates of the initialconditions for model (1)

Initial condition Value

SL (0) 1,119,583 ∗ 0.6SH (0) 1,119,583 ∗ 0.4P(0) 0

L(0) 0

I1(0) 1

I2(0) 0

H(0) 0

T (0) 0

R(0) 0

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

80

90

100

Time / Days

Num

ber

of C

ases

Daily Confirmed Cases of H1N1 Influenza in Manitoba, Canda

Fig. 9 The daily confirmed cases of H1N1 influenza in Manitoba, Canada

• For the pre-control 1st wave, we obtain Rc ∼ N (1.9, 0.10);• For the post-control 1st wave, we obtain Rc ∼ N (0.52, 0.05);• For the pre-control 2nd wave, we obtain Rc ∼ N (2.23, 0.11);• For the post-control 2nd wave, we obtain Rc ∼ N (0.66, 0.04).

7 Summary

A deterministic model is designed and rigorously analyzed to assess the transmissiondynamics and impact of antiviral drugs in curtailing the spread of disease of the 2009swine influenza pandemic. The analysis of themodel, which consists of ninemutually-exclusive epidemiological compartments, shows the following:

(i) The disease-free equilibriumof themodel is shown to be globally-asymptotically

stable when Rc ≤ ΩθH

.

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290 M. Imran et al.

Fig. 10 The best-fit trajectory of model (1) along with the daily confirmed cases of H1N1 influenza inManitoba, Canada.Dashed lines represent post-antiviral and therapeutic interventions for 1st and 2nd wave

(ii) The disease persists in the community ifRc > 1.(iii) The model has a unique endemic equilibrium when the associated reproduction

threshold

(Rc > Ω

θP

).

(iv) Backward bifurcation is observed when all susceptible individuals are equallylikely to acquire infection (θH = 1) and the vaccination is perfect (θP = 0).

(v) Uncertainty analysis suggests that with mean value of Rc = 1.32 and 95%confidence interval (1.13, 1.81), the disease will get endemic under the givenconditions and data.

(vi) Sensitivity analysis recommends that we must estimate the parameter values ofβ, τ1, φT and η4 with great precision as they greatly influence the magnitude ofRc, which in turn controls the spread of the disease.

(vii) So far all the analysis is dependent on Rc for the spread of disease. Therefore,we have designed an optimal treatment strategy (using Pontryagin’s maximumprinciple) to prevent the epidemic. It is observed that the optimal strategy ismuch more efficient in combating infection with minimal cost and resources.

Appendix

Proof of Lemma 1

Proof Adding the equations of the model (1) gives

dN

dt= π − μN − δ(I2 + θ1H) ≤ π − μN . (12)

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Since N (t) ≥ 0, it follows using a standard comparison theorem [29] that

N (t) ≤ N (0)e−μt + πμ

(1 − e−μt).

Therefore, N (t) ≤ π/μ if N (0) ≤ π/μ.This proves the positive invariance of D.To prove that D is attracting, from (12), it is clear that dNdt < 0, whenever N (t) >

π/μ. Thus, either the solution enters D in finite time, or N (t) approaches π/μ, andthe variables denoting the infected classes approach zero. Hence, D is attracting andall solutions in R9+ eventually enter D. ��

DFE

Rc calculation

The matrices F (for the new infection terms) and V (of the transition terms) are given,

respectively, by F =⎡⎢⎣

βη1Ω βΩ βη2Ω βη3Ω βη4Ω0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

⎤⎥⎦ V =

⎡⎢⎣

K1 0 0 0 0−α f K2 0 0 00 −γ K3 0 00 0 −ψ K4 00 −τ1 −τ2 0 K5

⎤⎥⎦, where,

Ω = S∗L + θH S∗H + θP P∗

N∗, K1 = α+μ, K2 = τ1+γ +μ, K3 = τ2+ψ+φ2+μ+δ,

K4 = ψ2 + φ3 + μ + θ1δ, K5 = φT + μ.Reproduction number Rc is given as

Rc = ρ(FV−1)

where, ρ denotes the spectral radius.

Proof of Theorem 1

Proof Consider the model (1) with θH = 1 and θP = 0. Further, consider the Lya-punov function

F = g1L + g2 I1 + g3 I2 + g4H + g5T,

where,

g1 = η1K2K3K4K5 + αK3K4K5 + ατ1η4K3K4 + αγ η2K4K5 + αγψη3K5+ αγ τ2η4K4

g2 = K1(γ τ2η4K5 + γψη3K5 + γ η2K4K5 + τ1η4K3K4 + K3K4K5)g3 = K1K2K5(η2K4 + η3 + τ2η4)g4 = η3K1K2K3K5g5 = η4K1K2K3K4

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292 M. Imran et al.

TheLyapunovderivative is givenby (where a dot represents differentiationwith respectto t)

Ḟ = g1 L̇ + g2 İ1 + g3 İ2 + g4 Ḣ1 + g5Ṫ ,= g1[λ(SL + SH + θP P) − (α + μ)L] + g2[αL − (τ1 + γ + μ)I1]

+ g3[γ I1 − (τ2 + ψ + φ2 + μ + δ)I2] + g4[ψ I2 − (ψ2 + φ3 + μ + θ1δ)H ]+ g5[τ1 I1 + τ2 I2 − (φT + μ)T ],

so that,

Ḟ = g1λ[SL(t)+ θH SH (t)+ θP P]− K1K2K3K4K5(η1L+ I1+η2 I2+ η3H+η4T ),≤ g1λθH [SL(t) + SH (t) + P(t)] − K1K2K3K4K5 λN

β, since θP < 1 < θH

≤ g1λθH N − K1K2K3K4K5 λNβ

, since (SL(t) + SH (t) + P(t)) ≤ N (t) in D.

It can be shown that g1 = RcβΩ K1K2K3K4K5. Hence,

Ḟ ≤ RcβR∗

K1K2K3K4K5λN − K1K2K3K4K5 λNβ

,

= K1K2K3K4K5 λNβ

(RcR∗

− 1)

.

Thus, Ḟ ≤ 0 if Rc ≤ R∗ with Ḟ = 0 if and only if L = I1 = I2 = H = T =0. Further, the largest compact invariant set in {(SL , SH , P, L , I1, I2, H, T, R) ∈D : Ḟ = 0} is the singleton {E0}. It follows from the LaSalle invariance principle(Chapter 2, Theorem 6.4 of [30]) that every solution to the equations in (1) withθH = 1 and θP = 0 and with initial conditions in D converges to DFE, E0, ast → ∞. That is, [L(t), A(t), I1(t), I2(t), H(t), H2(t), T (t)] → (0, 0, 0, 0, 0, 0, 0)as t → ∞. Substituting L = A = I1 = I2 = H = H2 = T = 0 into the firstthree equations of the model (1) gives SL(t) → S∗L , SH (t) → S∗H and P(t) →P∗ as t → ∞. Thus, [SL(t), SH (t), P(t), L(t), I1(t), I2(t), H(t), T (t), R(t)] →(S∗L , S∗H , P∗, 0, 0, 0, 0, 0, 0) as t → ∞ for R̃c ≤ R∗, so that the DFE, E0, is GAS inD ifRc ≤ R∗. ��

Proof of Theorem 2

Proof Let X = {(SL , SH , P, L , I1, I2, H, T, R) ∈ R9+ : L = I1 = I2 = H = T =0}. Thus X is the set of all disease free states of (1) and it can be easily verified that Xis positively invariant. Let M = D ∩ X . Since both D and X are positively invariant,M is also positively invariant. Also note that E0 ∈ M and E0 attracts all the solutionsin X . So, Ω(M) = {E0}.

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By setting x(t) = (L(t), I1(t), I2(t), H(t), T (t))T , equations for the infected com-ponents of (1) can be written as

x ′(t) = Y (x)x(t) (13)

where Y (x) =[(

SL+θH SH+θP PNΩ

)F − V

]and it is clear that Y (E0) = F − V . Also

it is easy to check that Y (E0) is irreducible. We will apply the Lemma A.4 in [1] toshow that M is a uniform weak repeller. Since E0 is a steady state solution, we canconsider it to be a periodic orbit of period T = 1. P(t, x), the fundamental matrixof the solutions for (7) is etY . Since the spectral radius of Y (E0) = Rc − 1 > 0,the spectral radius of eY (E0) > 1. So condition 2 of Lemma A.4 is satisfied. Takingx = E0, we get P(T, E0) = eY (E0) which is a primitive matrix, because Y (E0) isirreducible, as mentioned in Theorem A.12(i) [40]. This satisfies the condition 1 ofLemma A.4. Thus, M is a uniform weak repeller and disease is weakly persistent.By definition, M = ∂D. M is trivially closed and bounded relative to D and hence,compact. Therefore by Theorem 1.3 of [42], we have that M is a uniform strongrepeller and disease is uniformly persistent. ��

Endemic equilibrium

The coefficients are defined as follows

A1 = πθH θPA2 = π(1 − p)[θP (σH + μ) + θHμ] + πθH p(θP (σL + μ) + μ) + πθP A1A3 = πμ(1 − p)(σH + μ) + πθH pμ(σL + μ) + πθP BA4 = π(1 − p)[θP (σH + μ) + μθH ] + πp[μ + θP (σL + μ)] + π A1A5 = πμ(1 − p)(σH + μ) + πpμ(σL + μ) + π A2A6 = π(1 − p)θH θP + πpθPA7 = σLθH (1 − p) + σH pA8 = 1

K1+ α

K1K2+ αγ

K1K2K3+ αγψ

K1K2K3K4+ ατ1

K1K2K5+ αγ τ2

K1K2K3K5

+ αγφI2μK1K2K3

+ αγψφHμK1K2K3K4

+(

φT

K1K5

)ατ1

K1K2K5+

(φT

K1K5

)αγ τ2

K1K2K3K5.

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294 M. Imran et al.

Sensitivity analysis

The estimated parameters are presented in Table 3.

Table 3 Mean values of the model parameters with their assigned distributions

Parameter Mean Distribution Parameter Mean Distribution

μ 3e-5 N η1 0.1 G

β 0.5 G η2 1.2 G

σL 0.08 N η3 0.2 G

σH 0.14 N η4 0.2 G

α 0.35 G θH 1.2 U

τ1 0.5 N θP 0.3 N

τ2 0.6 N ψ1 0.5 N

φI2 0.2 N γ1 0.06 G

φH 0.2 N δ 0.01 N

φT 0.25 N θ1δ 0.01 N

N , G and U represents the normal, gamma and uniform distribution respectively

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Mathematical analysis of swine influenza epidemic model with optimal controlAbstract1 Introduction2 Model formulation2.1 Basic properties of the model

3 Existence and stability of equilibria3.1 Local stability of disease-free equilibrium (DFE)3.2 Global stability of DFE3.3 Endemic equilibrium and backward bifurcation

4 Uncertainty and sensitivity analysis5 Optimal control 6 Estimating the basic reproduction number for the 2009 H1N1 influenza outbreak in Manitoba, Canada7 SummaryAppendixProof of Lemma 1DFEmathcalRc calculationProof of Theorem 1Proof of Theorem 2

Endemic equilibriumSensitivity analysis

References

Mathematical analysis of swine influenza epidemic model …...Mathematical analysis of swine...

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