Analysis of a model for hepatitis C virus transmission that includes the effects of vaccination and waning immunity

Transcription

1 Analysis of a model for hepatitis C virus transmission that includes the effects of vaccination and waning immunity Daniah Tahir Uppsala University Department of Mathematics 7516 Uppsala Sweden daniahtahir@gmailcom Abid Ali Lashari Stockholm University Department of Mathematics Stockholm Sweden abidlshr@yahoocom Kazeem Oare Okosun Vaal University of Technology Department of Mathematics Private Bag X21 Vanderbijlpark South Africa kazeemoare@gmailcom Abstract: This paper considers a mathematical model based on the transmission dynamics of Hepatitis C virus HCV infection In addition to the usual compartments for susceptible exposed and infected individuals this model includes compartments for individuals who are under treatment and those who have had vaccination for HCV It is assumed that the immunity provided by the vaccine fades with time The basic reproduction number R and the equilibrium solutions of the model are determined The model exhibits the phenomenon of backward bifurcation where a stable disease-free equilibrium co-exists with a stable endemic equilibrium whenever R is less than unity It is shown that only the use of a perfect vaccine can eliminate backward bifurcation completely Furthermore a unique endemic equilibrium of the model is proved to be globally asymptotically stable under certain restrictions on the parameter values Numerical simulation results are given to support the theoretical predictions Key Words: epidemiological model; equilibrium solutions; backward bifurcation; global asymptotic stability; Lyapunov function 1 Introduction The liver of hepatitis patient is one of the most frequently damaged organs in the body and it is indeed fortunate that it has a very large functional reserve In the experimental animal it has been shown that only 1 % of the hepatic parenchyma the functional part of the liver is required to maintain normal liver function [1 The liver can be infected due to a variety of infectious agents such as parasites viruses and bacteria and the diseases of liver have a variety of causes such as obstructive vascular metabolic and toxic involvements Hepatitis C is the inflammation of the liver caused by hepatitis C virus HCV and spreads through contact with contaminated blood Hepatitis C may be an acute infection which spans over a period of weeks to a few months or chronic infection in which the virus persists for a long time [2 Acute hepatitis is characterized by moderate liver injury and if symptoms appear they include fatigue loss of appetite abdominal pain fever and jaundice However most of the times acute hepatitis is asymptomatic A large percentage of patients with HCV infection recover completely but some develop the long term chronic hepatitis or massive necrosis of the liver Chronic HCV infection may damage the liver permanently it can cause cirrhosis hepatic failure and sometimes liver cancer [1 Today HCV infects an estimated 17 million people worldwide [4 Around 15 million people are chronically infected with HCV HCV infection is a major cause of death of more than 5 people every year Countries with the highest prevalence of chronic liver infection are Egypt 15 % Pakistan 48 % and China 2 % [5 Although treatment for HCV infection does exist the current drug therapies are ineffective in completely eliminating the virus and patients suffering from chronic illness may require a E-ISSN: Volume

2 liver transplant [4 Unfortunately there is no effective vaccine yet developed that may help prevent the spread of the disease At present various attempts are being made to create such a vaccine [6 Thus it is crucial to assess the potential impact of HCV vaccine on the population Some mathematical models on HCV infection have been formulated recently but much work has not been done since it is a relatively new disease discovered in 1989 and data is not available on account of the high variability of the HCV In contrast more research has been carried out on Hepatitis B virus H- BV infection Several epidemiological models have focused on the effects of preventive measures as well as control of HBV infection [7 This has helped in creating cost effective disease prevention techniques The modes of transmission of both HCV and HB- V are same ie through blood thus mathematical models on both infections are somewhat inter related Some mathematical models were formed on HCV infection that considered infected cells uninfected cells and viral cells in the human host The basic aim of these models was to study the effects of liver transplant in patients with HCV infection But in major cases HCV infection is not completely eliminated even after the transplant Thus these models were extended to include more infected compartments [8 Martcheva and Castillo-Chavez [9 introduced an epidemiologic model of HCV infection with chronic infectious stage in a varying population Their model does not include a recovered or immune class and it falls within the susceptible-infected- susceptible SIS category of models A susceptible-infected-recovered SIR model was used by Kretzschmar and Wiessing [1 to study the transmission of HCV among injecting drug users while susceptible-infected-removedsusceptible SIRS type models that allow waning immunity are presented in Zeiler et al [11 Also a deterministic model for HCV transmission is used by Elbasha with the objective of assessing the impact of therapy on public health [12 Our aim is to meticulously analyze the model and examine various parameters to explore their effect on transmission of HCV and its control The model focuses on studying the effects of imperfect vaccines on the control of HCV infection The model shows that an imperfect vaccine reduces the number of individuals who are exposed to HCV while a perfect vaccine completely removes them We have subdivided the total population into six mutually-exclusive compartments: susceptible exposed acutely infectious chronically infectious treated and vaccinated individuals Ordinary differential equations are used to model the HCV infection This model can help provide insight into the spread of HCV infection and the assessment of the effectiveness of immunization techniques This paper is organized as follows: The mathematical model is developed in Section 2 The model is analyzed in Section ie stability of disease free and endemic equilibrium is discussed along with the effects of vaccination on backward bifurcation phenomenon Numerical simulations are provided in the same section Section 4 summarizes the final results of the paper 2 Model Formulation The total population at time t denoted by Nt is divided into sub-populations of susceptible individuals St exposed individuals with hepatitis C symptoms Et individuals with acute infection It individuals undergoing treatment T t individuals with chronic infection C h t and vaccinated individuals V t so that Nt = St + Et + It + T t + C h t + V t It is assumed that the mode of transmission of HCV infection is horizontal We further assume that mixing of individual hosts is homogeneous every person in the population Nt has an equal chance of getting infected The following system of ordinary differential equations describes the dynamics of HCV infection: ds = 1 bλ + ρt + αv β 1 I + β 2 C h + β T S + σc h S de = β 1I + β 2 C h + β T S + 1 ψβ 1 I + β 2 C h + β T V ϵ + E di = ϵe κ + I dt = π 1κI + π 2 C h ρ + T dc h = 1 π 1 κi π 2 + σ + C h dv = bλ α + V 1 ψβ 1 I + β 2 C h + β T V 1 The recruitment rate of susceptible humans is Λ A proportion b of these susceptible individuals is vaccinated The death rate is denoted by The rate of progression from acute infected class to both treated and chronic infected class is given by κ The a- cutely infected proportion of individuals who enter the treated class is π 1 The remaining infected proportion 1 π 1 progresses to chronic infectious stage The rate of progression for treatment from chronic hepatitis is given by π 2 The term ϵ is the rate of progression E-ISSN: Volume

3 from exposed class to acute infected class The recovery rates due to treatment and naturally from the chronic group are ρ and σ respectively The transmission coefficients of HCV infection by individuals with acute hepatitis upper case C It chronic hepatitis upper case C C h t and individuals undergoing treatment but not yet cured T t are β 1 β 2 and β respectively Following effective contact with It C t and T t susceptible individuals can acquire HCV at a rate β 1 I +β 2 C h +β T Here ψ < ψ 1 represents the vaccine efficacy with ψ = 1 representing a perfect vaccine and ψ 1 corresponding to an imperfect vaccine which will wane with time The term 1 ψ corresponds to the decrease in disease transmission in vaccinated individuals in contrast to susceptible individuals who are not vaccinated Hence vaccinated individuals acquire HCV at a reduced rate 1 ψβ 1 I + β 2 C h + β T The rate at which the vaccine wanes is denoted by α The parameter description is described in Table 1 Para Table 1: Description of parameters Description Λ recruitment rate death rate α waning rate of vaccine ψ vaccine efficacy β i transmission rate i=1 2 b proportion of vaccinated individuals κ rate of progression from acute state to treated and chronic state ϵ rate of transfer from exposed class to acute infected class π 1 proportion of individuals who enter the treated class from acutely infected class π 2 rate of progression for treatment from chronic hepatitis ρ rate of recovery due to treatment σ rate of recovery from the chronic class In the proposed model 1 the total population is S + E + I + T + C h + V = Λ t provided that S + E + I + T + C h + V = Λ Thus the biologically feasible region for system 1 given by = { S E I T C h V R 6 : S + E + I + T + C h + V = Λ } is positively invariant with respect to the system 1 21 Local stability of disease-free equilibrium DFE For mathematical model 1 the disease free equilibrium DFE P is given by S E I T C h V 1 bλ = + αbλ bλ α + α + 2 The local stability of P is determined by the next generation operator method [1 on system 1 For this purpose the basic reproduction number the average number of secondary infections produced by an infected individual in a completely susceptible population denoted by R is first calculated below Using the same notation as in [1 the matrices F and V evaluated at P the basic reproduction number R is given by R = ϵ 1 bλ + αbλ + 1 ψ bλ K 1 K 2 K 5 K 5 κ1 π 1 [β 1 + β 2 π 1 κ + π 2 κ1 π 1 +β K where K 1 = ϵ + K 2 = κ + K = ρ + = π 2 + σ + and K 5 = α + Using Theorem 2 in [1 the following result is established Theorem 1 The DFE of the model 1 is locally asymptotically stable LAS if R < 1 and unstable if R > 1 22 Endemic equilibria and backward bifurcation To calculate the endemic equilibrium we consider the following reduced system of differential equations: E-ISSN: Volume

4 de = β 1I + β 2 C h + β T Λ E I T C h ψv K 1 E di = ϵe K 2I dt = π 1κI + π 2 C h K T dc h = 1 π 1 κi C h dv = bλ K 5V 1 ψβ 1 I + β 2 C h + β T V We will consider the dynamics of the flow generated by in the invariant region Ω = {E + I + T + C h + V Λ } The endemic equilibrium for system is P E I T C h V where E = K 2I ϵ T = π 1κ +π 2 1 π 1 κi K C h = 1 π 1κI V = bλ K 5 +1 ψ[β 1 +β 2 κ1 π 1 +β π 1 κ +π 2 κ1 π 1 K I 4 and I is the root of the following quadratic equation with a 1 I 2 + a 2 I + a = 5 a 1 = 1 ψb 2 [K 2 K + ϵ K + π 1 κ + π 2 κ1 π 1 ϵ + ϵκ1 π 1 K a 2 = B[K 2 K K 5 + ϵk K 5 +ϵk 5 π 1 κ + π 2 1 π 1 κ +1 π 1 ϵκk K ψk 1 K 2 K 1 ψλϵbk a = K 1 K 2 K K 5 1 R 6 where B = [ κ1 π 1 π 1 κ + π 2 κ1 π 1 β 1 +β 2 +β K 7 The endemic equilibria of the model can then be obtained by solving for I from 5 and substituting the positive values of I into the expressions in 4 Hence S can be determined from Λ E I T C h V From 6 it can be seen that a 1 is always positive for an imperfect vaccine and a is positive negative if R is less than greater than unity Thus the following result is established: Theorem 2 The HCV model has: i a unique endemic equilibrium if a < R > 1; ii a unique endemic equilibrium if a 2 < and a = or a 2 2 4a 1a = ; iii two endemic equilibria if a > a 2 < and a 2 2 4a 1a > ; iv no endemic equilibrium otherwise Acute Infectious Individuals R Figure 1: Backward Bifurcation diagram with parameter values β 2 = 9 β = 19 = 4 α = 1 ρ = 152 π 1 = 1 π 2 = 2 ϵ = 22 κ = 2 Λ = 52 σ = 2 ψ = 95 Hence the model has a unique endemic equilibrium P whenever R > 1 as evident from case i of the above theorem Also case iii indicates a possible chance of backward bifurcation where a locally asymptotically stable DFE exists along with a locally asymptotically stable endemic equilibrium when R < 1 Since for a > R < 1 the model will have a disease-free equilibrium and two endemic equilibria To check for this the discriminant a 2 2 4a 1a is set to zero and is solved for the critical value of R denoted by R c given by R c = 1 a 2 2 4a 1 K 1 K 2 K 5 K Backward bifurcation occurs for those values of R such that R c < R < 1 This is illustrated by simulating the model with these parameter values: β 1 = E-ISSN: Volume

5 β = 19 = 4 α = 1 ρ = 152 π 1 = 1 π 2 = 2 ϵ = 22 κ = 2 Λ = 52 σ = 2 ψ = 95 These values are used merely for illustration purposes and may not be realistic from epidemiological point of view The result is shown in Figure 1 It can be seen that a locally asymptotically stable disease free equilibrium a locally asymptotically stable endemic equilibrium and an unstable endemic equilibrium coexist when R < Proof of backward bifurcation phenomenon The phenomenon of backward bifurcation can be proved by using center manifold theory on system 1 A theorem Castillo-Chavez and Song in [14 will be used here To apply this method the following change of variables is made in the model Let = S = E = I = T = C h and x 6 = V Let X = x 6 T Thus system 1 can now be written as dx = f 1 f 2 f f 4 f 5 f 5 f 6 T given below d = f 1 = 1 bλ + ρ + αx 6 β 1 + β 2 + β + σ d = f 2 = β 1 + β 2 + β +1 ψβ 1 + β 2 + β x 6 K 1 d = f = ϵ K 2 d = f 4 = π 1 k + π 2 K d = f 5 = 1 π 1 k dx 6 = f 6 = bλ K 5 x 6 1 ψβ 1 + β 2 + β x 6 8 Choose β 1 as the bifurcation parameter and let R =1 Solving for β 1 = β 1 from R =1 gives where β 1 = β 1 = K 1K 2 ϵa β 21 π 1 κ β π 1 k + π 2 1 π 1 κ K A = 1 bλ + αbλ K ψ bλ K 5 9 The Jacobian matrix J of system 8 calculated at the DFE P with β 1 = β 1 is given as follows β 1 K h ρ β K h σ β 2 K h α K 1 β 1 A β A β 2 A ϵ K 2 J = π 1 κ K π 2 1 π 1 κ β 1 K m β K m β 2 K m K 5 1 where K h = Λ bλ K 5 and K m = 1 ψλb K 5 The characteristic equation in λ of the Jacobian matrix J is given as λ K 5 λλ 4 +D 1 λ +D 2 λ 2 +D λ+d 4 = 11 where D 1 = K 1 + K 2 + K + D 2 = K + K 1 K + K 2 K + K 1 + K 2 +K 1 K 2 β 1 ϵa D = K 1 K + K 2 K + K 1 K 2 K + K 1 K 2 β 1 ϵak + β κπ 1 ϵa 1 π 1 β 2 κϵa D 4 = K 1 K 2 K 1 R 12 For R =1 the characteristic equation 11 becomes λ λ K 5 λλ + D 1 λ 2 + D 2 λ + D = 1 Hence the equation 1 has a zero eigenvalue and two negative eigenvalues and K 5 The remaining three eigenvalues are given by the following cubic equation in λ λ + D 1 λ 2 + D 2 λ + D = 14 D 1 is clearly positive and D 2 and D can easily be shown to be positive when β 1 is replaced with β 1 Similarly D 1 D 2 D > Hence using Routh- Hurwitz criterion [15 all roots of the characteristic equation 14 have negative real parts Therefore the Jacobian matrix of the linearized system has a simple zero eigenvalue with all other eigenvalues having negative real parts Hence the Center Manifold Theory [14 can be used to analyze the dynamics of system 8 Corresponding to the zero eigenvalue Jacobian matrix J β1 = β 1 can be shown to have a right eigenvector given by w = w 1 w 2 w w 4 w 5 w 6 T E-ISSN: Volume

6 where w 1 = 1 ρ π1 κ +π 2 1 π 1 κ K + σ 1 π 1κ K 1K 2 ϵa w 1 bλ + αbλ K 5 + α1 ψbλ K 2 5 w 2 = K 2 ϵ w w = w w 4 = π 1κ K w + π 21 π 1 κ K w w 5 = 1 ψbλk 1K 2 K 2 5 ϵa w 15 Similarly corresponding to the zero eigenvalue J β1 = β 1 has a left eigenvector given by v = v 1 v 2 v v 4 v 5 v 6 where v 1 = v 2 = ϵ K 1 v v = v v 4 = ϵβ A K 1 K v v 5 = ϵβ 2A K 1 v + ϵβ π 2 A K 1 K v v 6 = Calculation of a For the system 8 the corresponding non-zero partial derivatives of f i i = calculated at the disease free equilibrium P are given by 2 f 1 = β 1 2 f 1 = β 2 f 2 = β 2 2 f 2 = 1 ψβ 1 x 6 2 f 2 = 1 ψβ x 6 2 f 6 = 1 ψβ 2 x 6 2 f 1 = β 2 2 f 2 = β 1 2 f 2 = β 2 f 2 x 6 = 1 ψβ 2 2 f 6 x 6 = 1 ψβ 1 2 f 6 x 6 = 1 ψβ Consequently the associated bifurcation coefficient a is given by 6 2 f k a = u k w i w j x kij=1 i x j = ϵu w 2 κ1 π 1 β 1 +β 2 K 1 π 1 κ +π 2 κ1 π 1 +β ρ K π1 κ + π 2 1 π 1 κ K 1K 2 ϵa K 1 bλ + αbλ K σ 1 π 1κ α1 ψbλ K ψ2 bλk 1 K 2 K5 2ϵA 16 Calculation of b The required partial derivative for the computation of b is calculated at the disease 2 f 2 free equilibrium ie β 1 = A Hence the associated bifurcation coefficient b is given as b = 6 ki=1 u k w i 2 f k x i ϕ = Aϵu w K 1 > 17 Since the coefficient b is always positive it follows from Theorem that the system will undergo backward bifurcation if the coefficient a is positive The phenomenon of backward bifurcation poses a lot of problems since it jeopardizes the possibility of total disease eradication from the population when the basic reproduction number is less than unity Hence it is instructive to try to eliminate the backward bifurcation effect Since this effect requires the existence of at least two endemic equilibria when R < 1 [16 17 it may be removed by considering such a model in which positive endemic equilibria cease to exist 222 Use of perfect vaccine to eliminate backward bifurcation The backward bifurcation behavior of the proposed HCV infection model 1 can be eliminated by using a perfect vaccine ie when ψ=1 For ψ=1 the original model now becomes ds = 1 bλ + ρt + αv β 1I + β 2 C h +β T S + σc h S de = β 1I + β 2 C h + β T S K 1 E di = ϵe K 2I dt = π 1κI + π 2 C h K T dc h = 1 π 1 κi C h dv = bλ K 5V 18 The system 18 has a DFE P S V which is the same as the original model given in equation 1 The corresponding vaccinated reproduction number R for model 18 is given as R = R ψ=1 = ϵ 1 bλ + αbλ K 1 K 2 K 5 κ1 π 1 π 1 κ + π 2 κ1 π 1 β 1 +β 2 +β K Consider the quadratic equation 5 rewritten below for convenience a 1 I 2 + a 2 I + a = 19 E-ISSN: Volume

7 For ψ = 1 using the values given in equation 6 the coefficients a 1 a 2 and a of the above quadratic equation reduce to a 1 = a 2 > and a whenever R = R ψ=1 1 In this case the quadratic equation 5 will have just a single non positive solution I = a a 2 Hence whenever R 1 the model 18 with perfect vaccine has no positive endemic equilibrium This clearly suggests the impossibility of backward b- ifurcation because for backward bifurcation to occur there must exist at least two endemic equilibria whenever R Efficacy of vaccine ψ Proportion of vaccinated humans b Figure 2: Simulation of the model 18 showing a contour plot of R as a function of proportion of vaccinated humans b and vaccine efficacy ψ A contour plot of vaccinated reproduction number R as a function of proportion of vaccinated humans b and vaccine efficacy ψ is shown in Fig 2 The parameter values used to generate this diagram are as given in Table 1 The contours illustrate a significant decrease in the vaccinated reproduction number R with increasing vaccine efficacy ψ and proportion of vaccinated humans b It can be seen that very high vaccine efficacy and vaccine coverage is required to control HCV infection effectively in the population Almost all of the susceptible individuals should have had vaccination and vaccine efficacy must be % for R to be less than one so that the spread of HCV infection is controlled effectively The global stability of the disease free equilibrium can be proved in the region as follows Theorem The disease-free equilibrium DFE is globally asymptotically stable in whenever R 1 5 Proof: Let the Lyapunov function be where Then V = A 1 E + A 2 I + A T + A 4 C h 2 A 1 = S Λ A 2 = S K 1 ϵλ A = β S K A 4 = β 2S + β S π 2 K 21 V = A 1 E + A 2 I + A T + A 4 C h = A 1 β 1 I + β 2 C h + β T S K 1 E +A 2 ϵe K 2 I + A π 1 κi + π 2 C h K T +A 4 1 π 1 κi C h 22 Since S + E + I + T + C h + V Λ implies that S Λ Therefore V becomes V A 1 β 1 I + β 2 C h + β T Λ K 1E +A 2 ϵe K 2 I + A π 1 κi + π 2 C h K T +A 4 1 π 1 κi C h = E K 1 A 1 + ϵa 2 Λ +I β 1 A 1 K 2A 2 + π 1 κa + A 4 κ1 π 1 Λ +T β A 1 K a Λ +C h β 2 A 1 + π 2A A 4 κ1 π 1 S β 1 + β 2 = I +β π 1 κ + π 2 κ1 π 1 = IK 1K 2 ϵ whenever S K 2 K 1 K ϵλ 2 R S Λ R S Λ < 1 24 Hence V for R S Λ It should also be noted that S Λ = Λ bλ α+ Λ < 1 and V = only when E + I + T + C h = System 18 then becomes ds de dv = 1 bλ + αv S = di = dt = dc h = = bλ α + V 25 When t the solution of the last equation in the system 25 becomes V = bλ α + 26 E-ISSN: Volume

8 Putting this value back into the equation ds and letting t gives S = = 1 bλ + αv S 27 1 bλ + αbλ α + 28 The solution to the remaining equations in 25 is obviously zero Clearly when t the solution to system 25 approaches the DFE P S V By using LaSalle s invariance principle [18 P is found to be globally asymptotically stable in This result is illustrated by simulating the model 18 using a reasonable set of parameter values given in Table 1 The plot shows that the disease is eliminated from the population Fig 2 Global stability of the endemic equilibrium Theorem 4 The endemic equilibrium P S E I T C h V of the system 1 with ρ = and σ = is globally asymptotically stable whenever it exists In order to prove the above theorem we have used the method given in [19 2 At the endemic equilibrium P with ρ = and σ = the following equations are satisfied: = 1 bλ + αv β 1 I + β 2 C h + β T S S = β 1 I + β 2 C h + β T S + 1 ψβ 1 I + β 2 C h + β T V ϵ + E = ϵe κ + I = π 1 κi + π 2 C h T = 1 π 1 κi π 2 + C h = Λ α + V 1 ψβ 1 I + β 2 C h +β T V Let 29 = S S = E E = I I = T T = C h Ch x 6 = V V 1 Number of Susceptible individuals Number of Exposed individuals Number of Acutely Infected individuals Then 1 can be rewritten as [ x 1 = bλ 1 S αv S x 6 1 β 1 I 1 β 2 Ch 1 β T 1 x 2 = [ β1 I S E 1 + β 2C h S E ψ β 1I V E x ψ β 2C h V E x ψ β T V E x β T S E E-ISSN: Volume

9 Number of Treated individuals Number of Chronically infected individuals Number of Vaccinated individuals Figure : Simulation of system 18 showing the total number of susceptible exposed acutely infected chronically infected treated and vaccinated individuals as a function of time years Parameter values are given in Table 1 with ψ = 1 for perfect vaccine β 1 = 9 β 2 = 6 β = 1 and R = 654 < 1 The numerical simulation shows that the disease is eliminated when R < 1 It is assumed that the acute phase is more infectious than the chronic stage which is in turn more infectious than the treatment phase So β 1 > β 2 > β [ x = x ϵe I x 2 1 [ x 4 = x π1 κi 4 T x 1 + π 2Ch T x 5 1 [ x 5 = 1 π 1 κi 1 C h x 6 = [ x bλ 6 V 1 x ψβ 1 I 1 1 ψβ 2 Ch 1 1 ψβ T 1 2 The endemic equilibrium P S E I T Ch V corresponds to the positive equilibrium of 2 Since the global stability of P is the same as that of P the global stability of P is described below instead of P We define the Lyapunov function as follows L = a 1 S 1 ln + a 2 E 1 ln +a I 1 ln + a 4 T 1 ln +a 5 C h 1 ln + a 6 V x 6 1 lnx 6 where a 1 a 2 a a 4 a 5 and a 6 are positive numbers which are to be determined Using 29 the time derivative of L along the solutions of system 1 is given as dl = a 1 21 bλ+αv β 1 I S β 2 ChS β T S + a 2 β 1 I S + β 2 ChS +β T S + 1 ψβ 1 I V + 1 ψβ 2 ChV +1 ψβ T V + a ϵe + a 4 π 1 κi +a 5 1 π 1 κi + a 6 2bΛ 1 ψβ 1 I V 1 ψβ 2 C hv 1 ψβ T V a 1 1 bλ + a 1 αv a 1 β 1 I S a 1 β 2 ChS a 1 β T S + a 2 β 1 I S a 2 β 2 ChS a 2 β T S a 2 1 ψβ 1 I V a 2 1 ψβ 2 ChS a 2 1 ψβ T V + a ϵe + a 1 β 1 I S a ϵe + a 4 π 1 κi +a 5 1 π 1 κi + a 6 1 ψβ 1 I V + a 1 β T S a 4 π 1 κi a 4 π 2 C h +a 6 1 ψβ T V + a 1 β 2 C hs + a 4 π 2 C h a 5 1 π 1 κi +a 6 1 ψβ 2 C hv E-ISSN: Volume

10 x 6 a 1 αv + a 6 bλ a 6 1 ψβ 1 I V a 6 1 ψβ 2 ChV a 6 1 ψβ T V + x 6 a 2 1 ψβ 2 ChV a 6 1 ψβ 2 ChV + a 1 β 1 I S + a 2 β 1 I S + a 1 β 2 ChS + a 2 β 2 ChS + a 1 β T S + a 2 β T S + x 6 a 2 1 ψβ 1 I V a 6 1 ψβ 1 I V + x 6 a 2 1 ψβ T V a 6 1 ψβ T V + 1 a 1 1 bλ + 1 a 6 bλ + x 6 a 1 αv 6 + a 5 1 π 1 κi + a 4 π 2 C 5 x h 4 + a 4 π 1 κi + a ϵe 4 + x 6 a 2 1 πβ 1 I V + a 2 β 1 I S + a 2 β 2 C x hs 2 + a 2 β T S + x 6 a 2 1 πβ T V + x 6 a 2 1 πβ 2 C x hv 2 =: F x 6 We define the function H = is given as P 1 = b P 2 = b 2 2 x 6 1 x 6 i=1 P i where P i i = P = b 1 x 6 x 6 P 4 = b 4 1 P 5 = b P 6 = b P 7 = b P 8 = b 8 1 x 6 x 6 P 9 = b x 6 x 6 P 1 = b x 6 x 6 P 11 = b x 6 x 6 P 12 = b x 6 x 6 P 1 = b x 6 x 6 P 14 = b x 6 x 6 4 To determine all the coefficients a i > i = b i i = we let F x 6 = H Comparing coefficients of F and H we see that the terms x 6 x 6 and x 6 of F do not appear in H Hence their coefficients will be equal to zero ie a 2 β 1 I S a 2 β 2 C hs a 2 β T S a 2 1 ψβ 1 I V a 2 1 ψβ 2 C hs a 2 1 ψβ T V + a ϵe = 5 a 1 β 1 I S a ϵe + a 4 π 1 κi +a 5 1 π 1 κi + a 6 1 ψβ 1 I V = 6 a 1 ρt + a 1 β T S a 4 π 1 κi a 4 π 2 C h + a 6 1 ψβ T V = 7 a 1 β 2 C hs + a 1 σc h + a 4 π 2 C h a 5 1 π 1 κi + a 6 1 ψβ 2 C hv = 8 a 2 1 ψβ 2 C hv a 6 1 ψβ 2 C hv = 9 a 2 1 ψβ 1 I V a 6 1 ψβ 1 I V = 4 a 2 1 ψβ T V a 6 1 ψβ T V = 41 a 1 β 1 I S + a 2 β 1 I S = 42 a 1 β 2 C hs + a 2 β 2 C hs = 4 a 1 β T S + a 2 β T S = 44 The above equations have the following solution a 1 = 1 a 2 = 1 a 6 = 1 a = ϵ + ϵ a 4 = β S + 1 ψβ V a 5 = S + 1 ψv β 2 + β π 2 π 2 + Substituting these values into L = F x 6 and using equations 29 gives F x 1 x 6 = 2Λ + αv + ϵ + E +β S + 1 ψv T + S + 1 ψv β 2 + β π 2 Ch S x 6 V 1 1 bλ 1 bλ x 6 αv x 6 x S + 1 ψv β 2 + β π 2 C h E-ISSN: Volume

11 x 5 β S + 1 ψv π 2 Ch x β 4 S + 1 ψv π 1 κi x 2 ϵ + E x x 6 1 ψβ 1 I V β 1 I S β 2 C x hs 2 β T S x 4x 6 1 ψβ T V x 5x 6 1 ψβ 2 C x hv 45 2 Comparing the remaining coefficients of F and H gives b 1 = S αv + b 12 + b 1 + b 14 b 2 = V b = αv b 12 b 1 b 14 b 4 = β 1 I S b 12 b 5 = β T S β S + 1 ψv π 2 Ch +b 1 b 14 b 6 = β 2 Ch S b 1 b 7 = β S + 1 ψv π 2 Ch b 1 b 8 = 1 ψβ 1 I V b 9 = 1 ψβ T V b 1 b 11 = 1 ψβ 2 Ch V 46 To assure that b 1 b b 4 b 5 b 6 b 7 and b 9 are non negative b 1 b 12 b 1 b 14 must satisfy the following inequalities : αv S b 12 + b 1 + b 14 αv { 1 ψβ T V } b 1 min β S + 1 ψv π 2 Ch b 14 b 1 β T S β S + 1 ψv π 2 C h b 12 β 1 I S b 1 β 2 C h S 47 Finally using equations 29 the equality for the constant terms between F x 6 and H is verified as follows 2b 1 + 2b 2 + b + b 4 + 4b 5 + 4b 6 + 5b 7 + b 8 +4b 9 + 5b 1 + 4b b b 1 + 5b 14 = 2 [S αv + b 12 + b 1 + b V + [αv b 12 b 1 b 14 + [β 1 I S b 12 [ +4 β T S β S + 1 ψv π 2 Ch +4 [b 1 b 14 β 2 ChS b 1 [ β +5 S + 1 ψv π 2 Ch b 1 +1 ψβ 1 I V + 4 [1 ψβ T V b 1 +5b ψβ 2 ChV + 4b 12 +5b 1 + 5b 14 = 2S + 2V + αv + β 1 I S + 4β 2 C hs +4β T S + 1 ψβ 1 I V +41 ψβ 2 C hv + 41 ψβ T V + β S + 1 ψv π 2 C h = 41 bλ + 4αV 4β 1 I + β 2 C h + β T S +4V + αv + 4β 1 I S + 4β 2 C hs +4β T S + 41 ψβ 1 I V +41 ψβ 2 C hv + 41 ψβ T V + β S + 1 ψv π 2 C h 2S β 1 I S 1 ψβ 1 I V 2V = αv + β S + 1 ψv π 2 Ch 2S 2V + 4Λ ϵ + E +β 2 Ch + β T S + 1 ψv = αv + 2Λ + ϵ + E +β S + 1 ψv T +S + 1 ψv β 2 + β π 2 Ch +2 [Λ S V ϵ + E = αv + 2Λ + ϵ + E +β S + 1 ψv T + S + 1 ψv β 2 + β π 2 Ch 48 which is the same as the constant term of F x 6 The constrained conditions in 47 show that the available values of b 1 b 12 b 1 and b 14 are not u- nique Since b 1 b b 4 b 5 b 6 b 7 and b 9 depend on b 1 b 12 b 1 and b 14 their values will also be non u- nique Using inequalities 47 we can assign different values to b i i = 1 14 i and hence H can have different forms in following three subregions: Case 1: S > αv and β S + 1 ψv π 2 C h 1 ψβ T V For Case 1 using equations 46 and 47 choose b 1 = S αv b = αv b 4 = β 1 I S E-ISSN: Volume

12 b 5 = β T S b 6 = β 2 Ch S b 7 = b 9 = 1 ψβ T V β S + 1 ψv π 2 Ch b 1 = β S + 1 ψv π 2 Ch b 12 = b 1 = and b 14 = Using these values and the values of b 2 b 8 and b 11 the function F x 6 becomes F x 6 = S αv 2 1 +V 2 x 6 1 x 6 +αv 1 x 6 x 6 +β 1 I S 1 x 1 +β T S 4 1 +β 2 C hs ψβ 1 I V 1 x x 6 x 6 x ψβ T V β S + 1 ψv π 2 Ch 4 1 x 6 x x 6 + β S + 1 ψv π 2 C h x x 6 +1 ψβ 2 C hv 4 1 x 6 x 6 β Case 2: S = αv S + 1 ψv π 2 Ch 1 ψβ T V For Case 2 using equations 46 and 47 choose b 1 = b = αv b 4 = β 1 I S b 5 = β S + 1 ψv π 1κI b 6 = β 2 C h S b 7 = β S + 1 ψv π 2 C h 1 ψβ T V b 9 = b 1 = 1 ψβ T V b 12 = b 1 = and b 14 = Using the above values and the values of b 2 b 8 and b 11 the function F x 6 becomes F x 6 = V 2 x 6 1 x 6 +αv 1 x 6 x 6 +β 1 I S 1 x 1 +β S + 1 ψv π 1κI 4 1 x +β 2 ChS 4 1 x β + S + 1 ψv π 2 Ch 1 ψβ T V 5 1 x 5 +1 ψβ 1 I V 1 x x 6 x 6 +1 ψβ T V 5 1 x 6 x 6 x 5 +1 ψβ 2 C hv 4 1 x 6 x 6 Case : S < αv 1 ψβ T V β S + 1 ψv π 2 C h For Case using equations 46 and 47 we assume that αv β S + 1 ψv π 1κI and choose b 1 = S b = b 4 = β 1 I S b 5 = β S + 1 ψv π 1κI αv b 6 = β 2 C h S b 7 = β S + 1 ψv π 2 C h 1 ψβ T V b 9 = b 1 = 1 ψβ T V b 12 = b 1 = and b 14 = αv Using the above values and the values of b 2 b 8 and b 11 the function F x 6 becomes F x 6 = S V 2 x 6 1 x 6 +β 1 I S 1 x 1 + β S + 1 ψv π 1κI αv 4 1 x +β 2 C hs β S + 1 ψv π 2 Ch 1 ψβ T V 5 1 x E-ISSN: Volume

13 +1 ψβ 1 I V 1 x x 6 x 6 +1 ψβ T V 5 1 x 6 x 5 x 6 +1 ψβ 2 ChV 4 1 x 6 x x 6 +αv 5 1 x 6 x 6 Since the arithmetic mean is greater than or equal to the geometric mean F x 6 in each of the above three cases The equality holds only when = 1 x 6 = 1 and = = = ie { x 6 : F x 6 = } = x 6 = 1 = x 6 = = = This corresponds to the set = {S E I T C h V : S = S V = V E/E = I/I = T/T = C h /Ch } Hence the maximum invariant set of 1 on the set is the singleton {P } Therefore by LaSalle s Invariance principle the endemic equilibrium P is globally stable in when ρ = and σ = This result is illustrated by simulating the model 1 using a reasonable set of parameter values given in Table 1 The plot shows that the disease persists in the population Fig 4 Conclusion This paper presents a deterministic model for the transmission dynamics of Hepatitis C virus infection The formulated model realistically allows HCV transmission by acutely and chronically infected individuals Most importantly the model includes a compartment of vaccinated individuals and considers the effect of a waning vaccine on the transfer of individuals from one compartment to another The model was rigorously analyzed to gain insights into its qualitative dynamics We obtained the following results: 1 The model has a locally stable disease free e- quilibrium whenever the associated reproduction number is less than unity Number of Susceptible individuals Number of Exposed individuals Number of Acutely Infected individuals The model exhibits the phenomenon of backward bifurcation suggesting a case where stable disease-free equilibrium co-exists with a stable endemic equilibrium whenever the basic reproductive number is less than unity Using an imperfect vaccine would have no positive epidemiological impact to reduce disease burden in the community 4 Using a perfect vaccine can result in effective e- limination of HCV infection in a community that is the efficacy of the vaccine should be % for complete removal of the disease E-ISSN: Volume

14 Number of Treated individuals Number of Chronically infected individuals Number of Vaccinated individuals Figure 4: Simulation of system 1 showing the total number of susceptible exposed acutely infected chronically infected treated and vaccinated individuals as a function of time years when R > 1 Parameter values are given in Table 1 with ψ = 6 ρ = σ = β 1 = 9 β 2 = 6 and β = 1 The numerical simulation shows that the disease persists when R > 1 References: [1 S L Robbins Pathologic Basis of Disease W B Saunders Philadelphia [2 A M Di Bisceglie Natural history of hepatitis C: Its impact on clinical management Hepatology [ P Das D Mukherjee and A K Sarkar Analysis of a disease transmission model of hepatitis C J Biol Sys [4 R Qesmi Jun Wu J Wu J M Heffernan Influence of backward bifurcation in a model of hepatitis B and C viruses Math Biosc [5 World Health Organization Hepatitis C Factsheet no /en/ Accessed December 212 [6 J Y Chen F Li Development of hepatitis C virus vaccine using hepatitis B core antigen as immuno-carrier World J Gastroenterol [7 S Zhang Y Zhou The analysis and application of an HBV model Appl Math Model [8 H Dahari A Feliu M Garcia-Retortillo X Forns A U Neumann Second hepatitis C replication compartment indicated by viral dynamics during liver transplantation J Hepatol [9 M Martcheva and C Castillo-Chavez Disease with chronic stage in a population with varying size Math Biosci [1 M Kretzschmar and L Wiessing Modelling the transmission of hepatitis C in injecting drug users in Hepatitis C and Injecting Drug Use: Impact Costs and Policy Option eds J C Jager W Limburg M Kretzschmar M J Postma and L Wiessing Lisbon: European Monitoring Centre For Drugs And Drug Addiction 4 [11 I Zeiler T Langlands J M Murray and A Ritter Optimal targeting of Hepatitis C virus treatment among injecting drug users to those not enrolled in methadone maintenance pro- grams Drug Alcohol Depend [12 E H Elbasha Model for hepatitis C virus transmission Math Biosci Eng E-ISSN: Volume

15 [1 P van den Driessche J Watmough Reproduction numbers and sub-threshold endemic e- quilibria for compartmental models of disease transmission Math Biosci [14 C Castillo-Chavez B Song Dynamical model of tuberclosis and their applications Math Biosci Eng [15 L J S Allen An Introduction to Mathematical Biology Pearson Education Inc 7 [16 M A Safi A B Gumel Mathematical analysis of a disease transmission model with quarantine isolation and an imperfect vaccine Comp Math Appl [17 S M Garba A B Gumel M R Abu Bakar Backward bifurcations in dengue transmission dynamics Math Biosci [18 J P LaSalle The Stability of Dynamical Systems Regional Conference Series in Applied Mathematics SIAM Philadelphia PA 1976 [19 J Li Y Yang Y Zhou Global stability of an epidemic model with latent stage and vaccination Non Anal RWA [2 J Li Y Xiao F Zhang Y Yang An algebraic approach to proving the global stability of a class of epidemic models Non Anal RWA [21 N K Martin P Vickerman G R Foster S J Hutchinson D J Goldberg M Hickman Can antiviral therapy for hepatitis C reduce the prevalence of HCV among injecting drug user populations? A modeling analysis of its prevention utility J Hepatol [22 N K Martin P Vickerman M Hickman Mathematical modelling of hepatitis C treatment for injecting drug users J Theor Biol E-ISSN: Volume