Hepatitis C Virus Epidemic Control Using a Nonlinear Adaptive Strategy
Javad Khodaei-Mehr, Samaneh Tangestanizadeh, Mojtaba Sharifi, Ramin Vatankhah, Mohammad Eghtesad
HHepatitis C Virus Epidemic Control Using a NonlinearAdaptive Strategy
Javad Khodaei-Mehr a , Samaneh Tangestanizadeh b , Mojtaba Sharifi a , RaminVatankhah b , Mohammad Eghtesad b a School of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada b School of Mechanical Engineering, Shiraz University, Shiraz, Iran
Abstract
Hepatitis C is a viral infection that appears as a result of the Hepatitis C Virus(HCV), and it has been recognized as the main reason for liver diseases. HCVincidence is growing as an important issue in the epidemiology of infectious dis-eases. In the present study, a mathematical model is employed for simulatingthe dynamics of HCV outbreak in a population. The total population is dividedinto five compartments, including unaware and aware susceptible, acutely andchronically infected, and treated classes. Then, a Lyapunov-based nonlinearadaptive method is proposed for the first time to control the HCV epidemicconsidering modelling uncertainties. A positive definite Lyapunov candidatefunction is suggested, and adaptation and control laws are attained based onthat. The main goal of the proposed control strategy is to decrease the pop-ulation of unaware susceptible and chronically infected compartments by pur-suing appropriate treatment scenarios. As a consequence of this decrease inthe mentioned compartments, the population of aware susceptible individualsincreases and the population of acutely infected and treated humans decreases.The Lyapunov stability theorem and Barbalat’s lemma are employed in orderto prove the tracking convergence to desired population reduction scenarios.Based on the acquired numerical results, the proposed nonlinear adaptive con-troller can achieve the above-mentioned objective by adjusting the inputs (rates ∗ Corresponding author
Email address:
[email protected] (Mojtaba Sharifi)
Preprint submitted to Journal of L A TEX Templates July 28, 2020 a r X i v : . [ q - b i o . P E ] J u l f informing the susceptible people and treatment of chronically infected ones)and estimating uncertain parameter values based on the designed control andadaptation laws, respectively. Moreover, the proposed strategy is designed tobe robust in the presence of different levels of parametric uncertainties. Keywords:
Nonlinear adaptive control, Hepatitis C epidemic, Infectiousdisease dynamics, Uncertainty and stability, Lyapunov stability theorem,Robust performance.
1. Introduction
The Hepatitis C Virus (HCV) is a blood-borne virus identified as the maincause of the liver disease [1, 2, 3]. Globally, about three percent of the worldpopulation (170 million) are dealing with the HCV and 71 million people havechronic hepatitis C infection [1, 4, 5, 6]. Several studies showed that the chronicstage of HCV will develop the cirrhosis and liver cancer in the case of no treat-ment and approximately 339 thousands people die every year due to these dis-eases[1, 7]. Despite previously mentioned statistics which makes HCV infectionas one of the important health threats, this disease received little attention es-pecially in the regions with higher rate of infectiousness [4].Although fatigue and jaundice were mentioned as symptoms of the HCV, butthis disease often has no considerable symptom, even in the advanced stages.This is the reason that the HCV outbreak is called ”the silent epidemic” [4, 8].Several different ways were reported for HCV prevalence, which include shar-ing injection equipments, unsafe sexual contacts, inadequately sterilization ofsyringes and needles especially for health-care personnel and transfusion of pol-luted blood [1, 9]. Even though these are the main causes of HCV epidemic,but some other reasons may also be critical in some societies based on specialconditions. For instance in developed countries, since there is precise control onthe blood transfusion, the importance of injecting drug use in transmission ofthe disease has increased compared to the transfusion of polluted blood and its2roducts [2, 9].Natural cure at the chronic stage of HCV is not common, but it can happenfor about 10-15% of patients that the RNA of HCV is indistinguishable in theirserum [5, 6]. For the rest of patients (80-85 %) that the HCV could not behealed by their immune system response, some drug therapy regimes should beemployed. Hepatitis C drugs have recently had some developments. Availablesafe, highly effective and endurable combinations of oral antivirals that act di-rectly, have currently developed for this disease [4, 10]. Although vaccinationis the vital way of controlling different viral diseases, but unfortunately thereis no vaccine for the HCV yet [5]. Therefore, preventing this disease has animportant role in stopping the extension of the its epidemic.In the present work, a nonlinear adaptive method is developed for treatmentand control of the HCV epidemic. For this purpose, the recently publishednonlinear HCV epidemiological model in [4] is employed and different parametricuncertainties are taken into account, despite the previous optimal strategies [4].The main goal of the proposed control scheme is the population decrease inthe unaware susceptible and chronically infected compartments in the existenceof parametric uncertainties. Accordingly, two control inputs (efforts to informsusceptible individuals and treatment rate) are employed to track descendingdesired populations of the previously mentioned compartments. The asymptoticstability and tracking convergence of the closed-loop system having uncertaintiesare proven using the Lyapunov stability theorem and the Barbalat’s lemma.Innovations of this research are as follows: • For the first time a nonlinear adaptive method is developed to controlthe HCV epidemic by defining a novel Lyapunov function candidate thatprovides the tracking convergence proof. • Due to the lack of accurate information about HCV model parameters ineach society, parametric uncertainty is taken into account in this research3or the first time, and the defined control objectives are achieved in thepresence of these inaccuracies. • In all of the previous studies that have been conducted on the control ofthe HCV outbreak, populations of some undesired compartments at theend of investigation period were considered as the criterion for designingcontrol inputs [11, 12, 4, 13, 14]. However, in the present work for the firsttime, the populations of two unawared susceptible and chronically infectedclasses during the entire treatment period are considered as the criterionand control inputs are designed in a way to track desired values insteadof focusing on their final populations at the end of process.The rest of this chapter is organized as follows. In Sec. 2, related researchwork will be explained. Description of the dynamic model and the proposedcontrol scheme will be presented in Sec. 3 and Sec. 4, respectively. Results ofsimulations will be depicted and discussed in Sec. 5, and the concluding remarkswill be mentioned in Sec. 6.
2. Related research work
Previous related studies are presented in this section and divided into threeparts, including mathematical modeling, and Optimal control for HCV andadaptive control strategies for different biological systems.Several analytical analyses were conducted on the dynamic modeling of theHCV epidemic which are presented here. Martcheva and Chavez [15] presented asimple mathematical model with three compartmental variables including sus-ceptible, acutely infected and chronically infected. They considered differentepidemiological observations in the model. Yuan and Yang [8] added the ex-posed class to the previous model [15]. They considered that the susceptibleindividuals transmit to the exposed compartment in the case of having contactwith the infected compartments. Zhang and Zhou [5] added a new term in the4odel of Yuan and Yang [8], which denotes the death rate due to the HCV. Huand Sun [16] proposed another epidemiological model for the HCV with fourclasses in which the recovered compartment was taken into account for the firsttime. Naturally the recovered people transmit to this class from the acutelyinfected and chronically infected compartments and become immune againstthis. Ainea et al. [17] extended the previous model [16] by adding the exposedclass. Both of these models [16] and [17] considered the HCV disease-induceddeath rate for both acutely-infected and chronically infected classes. Shen et al.[18] proposed a dynamical model with six classes including susceptible, exposed,acutely infected, chronically infected, treated and recovered populations. Theypropounded treatment influence for the first time and classified treated peoplein the distinct class. Shi and Cui [19] improved the model in [18] and divided thetreated class into two different classes by defining the treatment for chronicalinfection and awareal reinfection.Some researches have been conducted for optimal control of the HCV out-break. Okosun [11] employed a SITV (susceptible, acutely infected, treated andchronically infected) model for the HCV that was an extended form of the dy-namics presented in [8]. This model [11] included the treatment compartmentand considered movement for susceptible, treated, and acutely and chronicallyinfected people among their compartments. Some time dependent optimal con-trol strategies are proposed, in order to control the HCV disease. Cost functionis calculated for these strategies in order to evaluate effectiveness of the controlmethods and select the most efficient one. Okosun and Makinde [12] employed aSEITV (susceptible, exposed, acutely infected, treated and chronically infected)dynamical model for the HCV outbreak considering the screening rate and drugefficacy as control inputs for acutely and chronically infected populations andused the Pontryagin’s Principle to solve the optimal control problem. Anotherepidemiological model was investigated in [4] for the HCV outbreak in whichthe susceptible class was divided into aware and unaware classes. Moreover,they considered two control inputs including screening and treatment rates for5he HCV epidemic model which were determined by an optimal control law.In [4], the dynamics was formulated with the susceptibility reduction due tothe publicity and the treatment process to identify the feasible effect of publicconcerns and treatment on the HCV. An optimal neuro-fuzzy strategy was alsointroduced in [13] in order to control the HCV epidemic. They [13] employedthe mathematical model proposed in [12] as a deterministic model and utilizedthe genetic algorithm to obtain optimal control inputs.As described, all of previous studies on the control of HCV epidemic wereconducted on the optimal strategies. On the other hand, some other researchworks were performed on the adaptive control of different diseases as presentedhere. Moradi et al. [20] suggested a Lyapunov-based adaptive method to con-trol three different hypothetical models of the cancer chemotherapy inside thehuman body and compared results among these models. In the next step of thisresearch [21], a composite adaptive strategy has been developed for online iden-tification of cancer parameters during the chemotherapy process. Boiroux et al.[22] employed a model predictive controller for the type 1 diabetes’ model andused an adaptive controller to balance the blood glucose. They determined themodel parameters based on clinical information of past patients. Aghajanzadehet al. [23] suggested an adaptive control strategy for hepatitis B virus infectioninside the human body by antiviral drugs. They considered model parametersuncertainties on model parameters and employed adaptive controller to controlthe dynamic despite uncertainties of the system. Sharifi and Moradi[24] de-signed a robust scheme with adaptive gains to control the influenza epidemic,considering its dynamic model’s uncertainties. Padmanabhan et al. [25] pro-posed an optimal adaptive method to control the sedative drug in anesthesiaadministration. They employed an integral reinforcement learning method inorder to overcome the uncertainty of parameter values.6 . Dynamic model of hepatitis C virus epidemic
Mathematical modeling is an useful way of analysis for epidemiology of adisease. These models have two important capabilities: 1. finding out mech-anistic understanding of the disease, and 2. exploring potential outcomes ofthe epidemic under different conditions [26]. For assessment of the proposedmethod for the HCV prevalence control in a population, a nonlinear compart-mental model is used with five different classes including unaware susceptible( S u ), aware susceptible ( S a ), acutely infected ( I ), chronically infected ( C ) andthe treated ( T ) humans [4]. The susceptible compartment is divided into twoclasses, including aware and unaware people. Note that aware people have in-formation about the HCV transmission ways and preventing methods despiteunaware population. Since there is no available vaccine for the HCV, informingpeople about preventing methods is a very important way to reduce the risk ofinfection for susceptible people [1]. Therefore, the unaware susceptible individ-uals ( S u ) will be infected in contact with the infected population ( I, C and T )with a higher rate in comparison with the aware susceptible individuals ( S a )[4]. Thus, the transmission rate for unaware susceptible humans ( S u ) shouldbe considered larger than this rate for aware susceptible humans ( S a ) in thedynamic model [4]. The nonlinear mathematical model of HCV epidemic is asfollows: ˙ S u = b − λ S u S u N − ( µ + u ( t )) S u + (1 − q ) γI ˙ S a = u ( t ) S u − λ S a S a N − µS a + (1 − p ) ξT ˙ I = λ S u S u N + λ S a S a N − ( µ + γ ) I (1)˙ C = qγI − ( µ + u ( t ) + θ ) C + pξT ˙ T = u ( t ) C − ( µ + ξ ) T where λ S u = β ( I + K C + K T ) and λ S a = αλ S u . u and u are controlinputs and defined as effort rate to inform unaware susceptible individuals andtreatment rate for chronically infected class, respectively. N denotes the total7opulation and will be calculated as: N = S u + S a + I + C + T (2)The population of unaware susceptible ( S u ) increases with the rate of b .Unaware and aware susceptible individuals are also infected in contact withacutely and chronically infected and treated individuals at the rates of λ S u and λ S a , respectively. Infectiousness rate for acutely infected people is higher thanchronically infected individuals, and the treated people have the lowest rate;thus, it is assumed that K > K [4, 5]. The total population ( N ) decreaseswith two different rates µ and θ , where µ denotes the rate of natural deaththat decreases populations of different compartments. However, θ is the rateof HCV induced death and decreases the population of the chronically infectedcompartment ( C ).During the acute stage ( I ), the HCV could have different behaviors for eachpatient based on his/her immune system response. For 15 to 25% of cases in thisstage, the RNA of HCV becomes indistinguishable in their blood serum and theALT level returns to the normal range. This observation is defined by the term(1 − q ) γI in the proposed HCV dynamics [4, 6]. Approximately, the immunesystem in 75-85% of the patients could not remove the hepatitis C virus in theacute stage and their disease becomes advanced to the chronic stage. Note thatif the HCV RNA reamins in the patient’s blood for at least six months afteronset of acute infection, the chronic level of the disease will appear which isdefined by the term qγI in Eq. (1) [5, 6]. Finally, the defeat in the treatmentprocess is defined by the term p . The treated population decreases by the rateof ξT and join the chronic class by the rate of pξT in the case of treatment de-feat and the rest of this population (1 − p ) ξT will join aware susceptible class ifthe treatment be successful. The schematic diagaram of the proposed nonlineardynamics of the HCV epidemic is depicted in the Fig. 1 and descriptions of theparameters are presented in Table 1 [4].8 igure 1: Schematic diagram of transition among different classes of HCV epidemicTable 1: Parameters of the mathematical model of the HCV (1) [4] Parameter Descriptionb Rate of birth µ Rate of death β Transmission coefficient K Chronic stage infectiousness relative to acute stage K Treated individuals’ infectiousness relative to acute ones α Rate of being infected for aware people relative to unaware ones γ Leaving rate of acutely infected class q Progressing proportion from acute stage to chronic one ξ Transferring rate from treated class to other ones p Moving back proportion from treated class to chronic one θ HCV induced death rate9 . Nonlinear adaptive controller formulation for epidemiology of HCV
In the present section, a new nonlinear adaptive controller is formulated foruncertain hepatitis C virus epidemic. The main purpose of the control methodis to minimize the populations of unaware susceptible ( S u ) and chronically in-fected ( C ) classes. Two control inputs u ( t ) and u ( t ) are considered in orderto reach this objective. u ( t ) denotes the effort rate to inform the susceptibleindividuals from the HCV by media publicity, educational campaigns, publicservice advertising and so on, and u ( t ) is employed to reflect the rate of treat-ment on chronically infected individuals [4].Using the above-mentioned control inputs, the populations of unaware sus-ceptible ( S u ) and chronically infected ( C ) classes will decrease by tracking somedesired values. Moreover, due to decrease of the mentioned components, thenumber of aware susceptible ( S a ) and treated ( T ) individuals will increase anddecrease, respectively. The Lyapunov theorem is employed to prove stabilityof the closed-loop system. In addition, some adaptation laws are defined inorder to update estimated parameters of the system to guarantee the stabilityand robustness of the system against uncertainties of the dynamic model. Aconceptual diagram of the proposed nonlinear feedback controller with adaptivescheme is illustrated in Fig. 2. Control inputs ( u ( t ), u ( t )) could be calculated using dynamics of the un-aware susceptible and chronically infected compartments from Eq. (1) as: u = − ˙ S u S u + bS u − βN ( I + K C + K T ) − µ + (1 − q ) γ IS u (3) u = − ˙ CC + qγ IC − ( µ + θ ) + pξ TC (4) Property.
The right-hand sides of Eqs. (3) and (4) can be linearly parameter-10 igure 2: Conceptual diagram of nonlinear adaptive method developed to control the HCVepidemic in the existence of uncertainties on parameters of the model ized in terms of their available parameters. φ and φ are considered to be thearbitrary variables instead of ˙ S u and ˙ C . Now the equations could be rewrittenas: − ˙ S u S u + bS u − βN ( I + K C + K T ) − µ + (1 − q ) γ IS u = − φ S u + Y θ (5) − ˙ CC + qγ IC − ( µ + θ ) + pξ TC = − φ C + Y θ (6)where Y and Y are the regressor matrices contain known functions of HCVepidemic variables. θ and θ are parameter vectors which contain unknownparameters of the dynamic. Eqs. (7) and (8). Accordingly, these matrices andvectors are defined as Y = (cid:20) S u − IN − CN − TN IS u − (cid:21) ; θ = (cid:104) b β βK βK (1 − q ) γ µ (cid:105) T (7) Y = (cid:20) IC TC − (cid:21) ; θ = (cid:104) qγ pξ ( µ + θ ) (cid:105) T (8)11his regressor presentation is used to summarize the equations and definethe adaptation and control laws. In order to design nonlinear control laws, twonew variables φ and φ are defined as follows: φ = ˙ S u d − λ ˜ S u (9) φ = ˙ C d − λ ˜ C (10)where λ and λ are the controller gains and considered to be positive andconstant. Now, the nonlinear adaptive control laws are defined as u = − ˙ S u d − λ ˜ S u S u + Y ˆ θ (11) u = − ˙ C d − λ ˜ CC + Y ˆ θ (12)where ˆ θ and ˆ θ are the vectors of estimated parameters.In the next section, taking advantages of the Lyapunov stability theorem, itwill be proven that the control laws (11) and (12) together with some adapta-tion laws, guarantee the tracking convergence, stability and robustness for thetreatment of HCV outbreak. The closed-loop dynamics of the system is achieved firstly by substitutingthe control laws (11) and (12) into the dynamics of HCV epidemic (1):˙˜ S u + λ ˜ S u S u = − Y ˜ θ (13)˙˜ C + λ ˜ CC = − Y ˜ θ (14)where ˜ θ i (for i =1, 2) is defined as ˆ θ i − θ i .The adaptation laws are designed to update parameters’ estimation to keepthe system’s robustness against uncertainties, as˙ˆ θ T = S u ˜ S u Y Γ (15)˙ˆ θ T = C ˜ CY Γ (16)12here Γ and Γ are the adaptation gain matrices and considered to be positivedefinite.Now, employing the Lyapunov stability theorem [27] and based on the pre-viously derived close-loop dynamics (13)-(14) and the designed adaptation laws(15)-(16), the tracking convergence, stability and robustness for aware suscep-tible and chronically infected classes will be proven. With this aim, a positivedefinite Lyapunov-candidate-function is selected as V = 12 [ ˜ S u + ˜ C + ˜ θ T Γ − ˜ θ + ˜ θ T Γ − ˜ θ ] (17)The Lyapunov function’s time derivative is determined:˙ V = ˜ S u ˙˜ S u + ˜ C ˙˜ C + ˙ˆ θ T Γ − ˜ θ + ˙ˆ θ T Γ − ˜ θ (18)It should be mentioned that ˙˜ θ = ˙ˆ θ because θ is constant ( ˙ θ is zero). Bysubstituting the adaptation laws (15) and (16) into (18), the time derivative ofV is simplified to: ˙ V = − λ ˜ S u − λ ˜ C (19)As mentioned in the previous descriptions, λ and λ are considered to bepositive; thus, the Lyapunov function’s time derivative is negative-semi-definite.Thus, based on the Barbalat’s Lemma (described in the Appendix A) and theLyapunov stability theorem [27], it is proven that ˜ S u and ˜ C converge to the zero.In other words, employing the suggested adaptive feedback control strategy en-sures the tracking convergence and stability ( ˜ S u → C → t → ∞ ) inthe presence of uncertainties. Thus, the numbers of unaware susceptible ( S u )and chronically infected ( C ) converge to the desired values ( S u → S u d and C → C d ).
5. Results and discussion
For effectiveness evaluation of the proposed method, some simulations arepresented in this section. Note that computer simulations have proven to be use-13 able 2: Values of the HCV parameters in its mathematical model (1) [4]
Parameter Valueb 0.012 µ β K K α γ q ξ p θ S u d ) and treatment of chronically infectedpeople ( C d ). S u d = ( S u − S u f ) exp ( − a t ) + S u f (20) C d = ( C − C f ) exp ( − a t ) + C f (21)where a and a are the desired population reduction rates. S u and S u f arethe initial and final (steady state) populations of unaware susceptible class, re-spectively. C and C f are the initial and final (steady state) populations ofchronically infected compartment, respectively.14 able 3: Values of parameters in the desired HCV population reduction scenarios (20) and(21) Parameter Value S u C S u f C f a a I and C compartments and will jointhe acutely infected class ( I ). Since there is no treatment for acutely infectedindividuals (as seen in Eq. (1)), the HCV disease will progress and reach thechronic stage. Thus, the population of the chronically infected compartment( C ) will increase and the populations of all other compartments will decrease.Figure 4 depicts the above-mentioned points about the HCV outbreak in thecase of no control input. 15 Time (year)
Desired Populations S ud : Desired unaware susceptible populationC d : Desired chronically infected population Figure 3: Desired scenarios for reduction of unaware susceptible and chronically infectedcompartments in the HCV epidemic
However, applying the proposed strategy based on the designed nonlinearcontrol laws (11) and (12) with the obtained adaptation laws (15) and (16), thepopulation changes in different compartments in the presence of 20% parametricuncertainty are depicted in Fig. 5.As seen, due to the employment of the first control input ( u ), the unawaresusceptible individuals ( S u ) reduce and join the aware susceptible class ( S a ).Since the aware susceptible people become less infectious than unaware onesbecause of the control input u , extension of the HCV infection decreases com-pared with the no-control-input case (shown in Fig. 4). Moreover, using thetreatment rate as the second control input ( u ), the population of chronicallyinfected compartment ( C ) decreases (Fig. 5) based on the described scenarios( C d in Fig. 3). Thus, the populations of unaware susceptible and chronically in-fected classes reduce and the population of aware susceptible increases in Fig. 5,which are in accordance with the HCV dynamics (1). Although 20% parametric16 Time (year) P o pu l a t i o n o f C o m p a r t m e n t s S u : Unaware susceptible populationS a : Aware susceptible population (a) Time (year) P o pu l a t i o n o f C o m p a r t m e n t s I: Acutely infected populationC: Chronically infected populationT: Treated population (b)
Figure 4: Populations of (a) unaware and aware susceptible, and (b) acutely infected, chron-ically infected and treated classes in the absence of control inputs Time (year) P o pu l a t i o n o f C o m p a r t m e n t s S u : Unaware susceptible populationS a : Aware susceptible population (a) Time (year) P o pu l a t i o n o f C o m p a r t m e n t s I: Acutely infected populationC: Chronically infected populationT: Treated population (b)
Figure 5: Populations of (a) unaware and aware susceptible, and (b) acutely infected, chron-ically infected and treated classes in the presence of control inputs ( u and u ) based on theproposed laws (11) and (12) Time (year) P o pu l a t i o n o f C o m p a r t m e n t s S u : Actual unaware susceptible populationS ud : Desired uaware susceptible populationC: Actual chronically infected populationC d : Desired chronically infected population Figure 6: Convergence of unaware susceptible and chronically infected populations ( S u and C ) to their desired values ( S u d and C d ) uncertainty is taken to account for the nonlinear model, simulation results showthat the proposed control strategy satisfied its objective which is convegence todesired population reduction and treatment scenarios ( S u → S u d and C → C d ).Figure 6 depicts the desired and real populations of unaware susceptible andchronically infected classes,which implies the appropriate convergence perfor-mance using the nonlinear controller. The corresponding tracking errors arepresented in Fig. 7.As described, two control inputs are adjusted according to the proposednonlinear adaptive strategy in order to prevent the HCV outbreak. The firstcontrol input u ( t ) denotes the effort rate to inform the susceptible individualsfrom the HCV and the second one u ( t ) is the treatment rate for chronically in-fected individuals. These control inputs are considered to be normalized in Eq.(1) to be in the range of [0,1]. The obtained values for these inputs using theproposed control strategy are shown in Fig. 8, which satisfy the physiological19 Time (year) -0.6-0.5-0.4-0.3-0.2-0.100.10.20.3 P o pu l a t i o n T r a c k i n g E rr o r s S u - S ud C - C d Figure 7: Tracking errors between the desired and real values of unaware susceptible andacutely infected compartments constraint ( u ∈ [0 , θ and ˆ θ ) based on the designed adaptation laws (15)and (16) in the presence of 20% uncertainty. In this section, effects of different uncertainty levels are investigated for theHCV epidemic dynamics. For this purpose, 50%, 70% and 90% uncertaintiesare considered on the initial guess of parameters in θ and θ (defined in Eqs.(7) and (8)). Performance of the adaptation laws (15) and (16) on tuning ofestimated model parameters is investigated in Fig. 10. As discussed and provenin Sec. 4, these adaptation laws guarantee that the estimation errors of theHCV dynamic parameters remain bounded against different uncertainty levels.20 Time (year) C o n t r o l I npu t s u : Effort rate to inform the susceptible individualsu : Treatment rate for chronically infected individuals Figure 8: Control inputs ( u and u ) during the treatment period of HCV epidemic Figure 11 shows the population errors of unaware susceptible and chronicallyinfected classes in tracking their desired value ( ˜ S u = S u − S u d and ˜ C = C − C d ).As observed in Fig. 11 the increment of parametric uncertainties, increasesthe magnitude of errors ( ˜ S u and ˜ C ) and their initial variations. However, aftera period of time (about 0.2 year), the errors magnitudes has reached zero, whichmeans that the tracking convergence has been achieved for different values ofuncertainties. In other words, the population of unaware susceptible and chron-ically infected compartments converged to their desired values ( S u → S u d and C → C d ) in the existence of different levels of uncertainty.21 Time (year) E s t i m a t e d P a r a m e t er s Estimation of (2)Estimation of (4)Estimation of (6) (a) Time (year) E s t i m a t e d P a r a m e t er s Estimation of (1)Estimation of (2)Estimation of (3) (b) Figure 9: Estimation of parameters in (a) θ and (b) θ during the treatment period basedon the adaptation laws (15) and (16), respectively Time (year) -0.0100.010.020.030.040.05 E s t i m a t i o n o f M o d e l P a r a m e t er s Estimation of (1) for 50% uncertaintyEstimation of (1) for 70% uncertaintyEstimation of (1) for 90% uncertaintyActual value of (1) (a) Time (year) E s t i m a t i o n o f M o d e l P a r a m e t er s Estimation of (1) for 50% uncertaintyEstimation of (1) for 70% uncertaintyEstimation of (1) for 90% uncertaintyActual value of (1) (b) Figure 10: Adaptation of (a) θ (1) and (b) θ (1) using Eqs. (15) and (16), respectively, fordifferent uncertainty levels Time (year) -2.5-2-1.5-1-0.500.511.5 T r a c k i n g E rr o r f o r U n a w a re Su s ce p t i b l e P o pu l a t i o n (S u - S ud ) for 50% of uncertainty(S u - S ud ) for 70% of uncertainty(S u - S ud ) for 90% of uncertainty (a) Time (year) -0.8-0.6-0.4-0.200.20.4 E s t i m a t i o n E rr o r f o r C h r o n i c a ll y I n f ec t e d P o pu l a t i o n (C - C d ) for 50% of uncertainty(C - C d ) for 70% of uncertainty(C - C d ) for 90% of uncertainty (b) Figure 11: The difference between (a) unaware susceptible population and its desired value( ˜ S u = S u − S u d ) and (b) chronically infected population and its desired value ( ˜ C = C − C d )for different parametric uncertainty levels . Conclusion In the present study, a new nonlinear adaptive strategy was developed tocontrol the hepatitis C virus epidemic based on a mathematical model havinguncertainties. For the first time, an adaptive feedback controller was employedto decrease the populations of unaware susceptible and chronically infected com-partments based on the desired scenarios. Two control inputs were employedfor this goal. The first one u ( t ) is the effort rate to inform the susceptibleindividuals from the HCV and the second one u ( t ) is the rate of treatmentfor chronically infected people. The Lyapunov stability theorem and the Bar-balat’s lemma were used to prove the tracking convergence to desired treatmentscenarios. The proposed control laws and adaptation laws provided the sta-bility of the closed-loop HCV epidemic system in the presence of parametricuncertainties. Results of numerical simulations showed that by adjusting thecontrol inputs and the estimated parameters based on this strategy, number ofthe unaware susceptible and chronically infected individuals are decreased. As aresult, population of the aware susceptible was increased and population of theacutely infected and treated classes reached out to zero at the end of process.Moreover, the obtained results implied that the tracking convergence is achievedfor a wide range of uncertainties. Designing optimal trajectories and employingunstructured uncertainties can be considered as the next steps of this researchin the future. Appendix A. Barbalat’s lemma
The Lyapunov function V ( ˜ S u , ˜ C , ˜ θ , ˜ θ ) in (17) is positive definite and itstime derivative ( ˙ V ( ˜ S u , ˜ C )) in (19) is negative semi definite. Thus, based on theLyapunov stability theorem [27], V is bounded and it is concluded that ˜ S u , ˜ C ,˜ θ and ˜ θ remain bounded. 25 arbalat’s lemma: if g is a uniformly continues function and lim t →∞ (cid:82) t g ( η ) dη exists and has a finite value, it is guaranteed that [27]:lim t →∞ g ( t ) = 0 (A.1)In order to use this lemma for the HCV controlled system, g(t) is consideredto be - ˙ V : g ( t ) = − ˙ V = λ ˜ S u + λ ˜ C (A.2)By integrating both sides of (A.2), one can write: V (0) − V ( ∞ ) = lim t →∞ (cid:90) t g ( η ) dη (A.3)Since ˙ V is negative, V (0) is larger than V ( ∞ ) and V (0) − V ( ∞ ) ≥
0. More-over, as mentioned previously, V is bounded based on the Lyapunov stabilitytheorem . Thus, lim t →∞ (cid:82) t g ( η ) dη in (A.3) exists and has a bounded value.Therefore, it concluded using the Barbalat’s lemma that:lim t →∞ ( λ ˜ S u + λ ˜ C ) = 0 (A.4) References [1] World Health Organization (WHO). Hepatitis c [online] (January 2017).[2] D. Prati, Transmission of hepatitis C virus by blood transfusions and othermedical procedures: A global review, Journal of Hepatology 45 (2006) 607–616 (2006).[3] A. Wasley, M. J. Alter, Epidemiology of Hepatitis C: Geographic Differ-ences and Temporal Trends, Seminars in Liver Disease 20 (1) (2000) 1–16(2000). 264] S. Zhang, X. Xu, Dynamic analysis and optimal control for a model ofhepatitis c with treatment, Communications on Nonlinear Science and Nu-merical Simulation 46 (2017) 14–25 (2017). doi:https://doi.org/10.1016/j.cnsns.2016.10.017 .[5] S. Zhang, Y. Zhou, Dynamics and application of an epidemiological modelfor hepatitis C, Mathematical and Computer Modelling 56 (2012) 36–42(2012). doi:https://doi.org/10.1016/j.mcm.2011.11.0817doi:https://doi.org/10.1016/j.mcm.2011.11.0817