Mathematical Analysis of Control Strategies of HCV in a Community with Inflow of Infected Immigrants

In this paper, we derive and analyse rigorously a mathematical model of control strategies (screening, education, health care and immunization) of HCV in a community with inflow of infected immigrants. Both qualitative and quantitative analysis of the model is performed with respect to stability of the disease free and endemic equilibria. The results show that the disease free equilibrium is locally stable at threshold parameter less than unity and unstable at threshold parameter greater than unity. Using Lyapunov method, endemic equilibrium is globally stable under certain conditions. Numerical simulation of the model is implemented to investigate the sensitivity of certain key parameters on the HCV model in a community with inflow of infected immigrants. However, analysis shows that screening, education, health care and immunization have the effect of reducing the transmission of the disease in the community.


Introduction
Hepatitis C is a blood borne liver disease, caused by the Hepatitis C Virus (HCV), first identified by [1].Moreover, the link between infectious diseases and screening must be understood in relation to infectives on the spread of HCV infections.[2] analysed the screening of HCV in a health maintenance Organization.Mathematical modelling of the spread of infectious diseases continues to become an important tool in understanding the dynamics of diseases and in decision making processes regarding diseases intervention programs for disease in many countries.For instance, [3] formulated and analysed a mathematical model on the effect of Treatment and Infected Immigrants on the spread of Hepatitis C Virus disease at Acute and Chronic stages.[4] considered SEI (Susceptible-Exposed-Infective) epidemic model with acute and chronic stages.[5] investigated the effects of a HCV educational intervention or a motivational intervention on alcohol use and sexual risk behaviours among injection drug users.[6] studied the potential impact of vaccination on the hep C virus epidemic in injection drug users.[7] presented the study on immunization strategies in chronic HCV infection.[8] reported that HCV patient education is associated with positive outcomes in various models of HCV care.However, in all the above studies, none of them incorporated the HCV infectiology and control strategies (screening, education, health care and immunization) in a community with inflow of infected immigrants.The aim of the paper is to have a deeper understanding of the effects of screening, education, health care and immunization in controlling the spread of HCV.

Model Formulation
A mathematical model is proposed and analysed to study the effect of screening, education, health care and immunization on the spread of HCV disease in the community.The model has five epidemiological classes: The susceptible S , exposed individuals E , the acute Infectives A , the chronic infectives C and recovered group R .Total population at time t is given by: ( ) ( ) ( ) ( ) ( ) ( ).

N t S t E t A t C t R t
The interaction between the classes is being assumed as follows: Exposed individuals, acute infected and chronic infected immigrants enter into the population with the rates 1 2 3 , , π π π respectively.Susceptible indi- viduals are infected with the HCV virus at a rate υ , where are effective contact rate of individuals with acute and chronic hepatitis C respectively.It is assumed that the rate of contact of susceptibles with chronic individuals is much less than that of acute infectives ( ) β β ≤ because at chronic stage, people become aware of their infection and may choose to use control measures and change their behaviour and thus may contribute little in spreading the infection.The control variable based on screening programme aimed at, reduces the inflow of infected immigrants into the community at the rate τ and η is the control variable based on education, health care and immunization to decrease the infection contact rate.
Taking into account the above considerations, we then have the following schematic flow diagram (Figure 1): From the above flow chart, and with ( ) ( ) the model will be governed by the following system of equations: with nonnegative initial conditions and ( ) are the effective contact rates of individuals with acute and HCV respectively, ( ) are the rates at which exposed, acute and Chronic infected immigrants enter into the population respectively, Q is the recruitment rate, θ is the rate of progression to acute infected class from exposed class, ( ) are the rates at which acute and exposed infective develop chronic respectively, ( ) are the rates at which acute and chronic individuals recovered respectively, a is the death rate of acute infected group due to the disease, σ is the rate at which infectious humans after recovery become immediately susceptible again, µ is the natural death rate, d is the death rate of chronic infected group due to the disease, τ is the screening rate of infected immigrants.

Model Analysis
The model system of Equations (2) will be analysed qualitatively to get insight into its dynamical features which will give a better understanding of the effects of screening, education, health care and immunization on the transmission of HCV infection in the population with inflow of infected immigrants.The threshold which governs elimination or persistence of HCV will be determined and studied.We begin by finding the invariant region and show that all solutions of system (2) are positive 0 t ∀ > .

Invariant Region
In this section, a region in which solutions of the model system (2) are uniformly bounded is the proper subset 5 .
, , , , S E A C R R + ∈ be any solution with positive initial conditions.Then from Equation (2) it is noted that in the absence of the disease related mortality (i.e.0, 0 d a = = ), the rate of change of the population is given by, Using Birkhoff and Rota's theorem [9] on the differential inequality (3), the following expression is obtained; ( ) where ( ) is the value evaluated at the initial conditions of the respective variables.
Thus, as t → ∞ in (4), the population size, In respectof this, all the feasible solutions of system (2) enter the region , , , , : Hence, Ω is positively invariant and it issufficient to consider solutions in Ω .Furthermore, existence, uniqueness and continuation of results for system (2) hold in this region.Lemma 1: The region 5 R + Ω ⊂ is positively invariant for the model system (2) with initial conditions in 5 R + .

Positivity of Solutions
Lemma 2: Let the initial data be From the first equation of the model system (2), we have ( ) ( ) The Integration factor is ( ) , multiplying both sides by the integration factor and integrating leads to )

The Disease Free Equilibrium Point (DFE)
In the absence of the disease, which implies that ( ) the disease free equilibrium points is given by ( )

The Effective Reproductive Number Re
In this section, the threshold parameter that governs the spread of a disease which is called the effective reproduction number is determined.Mathematically, it is the spectral radius of the next generation matrix [10].This definition is given for the models that represent spread of infection in a population.It is obtained by taking the largest (dominant) Eigen value, (spectral radius) of ( ) ( ) where i F is the rate of appearance of new infection in compartment i , i V is the transfer of individuals out of the compartment i by all other means and 0 ∆ is the disease free equilibrium.Therefore, ( ) ( ) and The partial derivatives if ( 6) and ( 7) with respect to , and In the absence of the disease and when Q N µ = , the matrix (8) becomes ( ) ( ) Now, taking the inverse of matrix (9) leads to where The spectral radius (dominant eigenvalue) of the matrix Hence, the effective reproduction number of the model system ( 2) is given by The effective reproduction number e R measures the average number of new infections generated by a typi- cal infectious individual in a community with inflow of infected immigrants when screening, education, health care and immunization strategies are in place.
Theorem 1: The disease free equilibrium of the model system (2) is locally asymptotically stable if 1 e R < and unstable if 1 e R > .Theorem 1 implies that HCV can be eliminated from the community when 1 e R < if the initial size of the sub population of the model are in the basin of attraction of the disease free equilibrium.That means if 1 e R < , then on average an infected individual produce less than one new infected individual over the course of its infectious period and the infection cannot grow.
From Equation (12), for e R to be less than 1 this will only be possible when η (which implies HCV educa- tion, health care and immunization) are increased without bound in collaboration with other intervention strategies to all people including immigrants in a given locality which may result into the decreasing effect on 1 2 , In the absence of interventions (screening, education, health care and immunization) that is ( ) < .Hence the presence of screening, education, health care and immunization can eradicate the HCV infection if e R can be reduced to below unity.

Local Stability of Disease Free Equilibrium (DFE)
Local stability of disease free equilibrium 01 ∆ , can be determined by the variational matrix 01 J of the model system (2).The Jacobian matrix at the steady states is given by; The local stability analysis of the Jacobian matrix (13) of the system (2) can be done by the trace/determinant method.Where by matrix ( )

J
is locally asymptotically stable if and only if the trace of matrix ( ) 01 J is strictly negative and its determinant is strictly positive.Whose trace and determinant are given by and ( )

The Endemic Equilibrium Point ∆
Endemic equilibrium point ∆ is a steady state solution in which the disease persists in the population (i.e.0, 0, 0, 0 where, ( )( ) The equation, ( ) 0 f A * = corresponds to a situation when the disease persists (endemic).In case of back- ward bifurcation, multiple endemic equilibrium must exist.
However it is important to note that K is always positive if

Global Stability of the Endemic Equilibrium Point ∆
The global stability of the endemic equilibrium ∆ is analysed using the following constructed Lyapunov func- tion by [11] Theorem 4: If 1 e R > , the endemic equilibrium ∆ of the model (2) is globally asymptotically stable.Proof: To establish the global stability of the endemic equilibrium ∆ , we construct the following Lyapunov function: ( ) , , , , log log log log log .
By direct calculating the derivative of V along the solution of (2) we have; where, ∆ where ∆ is the endemic equilibrium of the system (2).By LaSalle's invariant principle, it implies that ∆ is globally asymptotically stable in Ω if P Q < .

Numerical Sensitivity Analysis
In determining how best to reduce human mortality and morbidity due to HCV, we calculate the sensitivity indices of the basic reproduction number, e R to the parameters in the model using approach of [12].These in- dices are crucial in determining the importance of each individual parameter in transmission dynamics and prevalence of the disease.Sensitivity analysis determines parameters that have a high impact on e R and should be targeted by intervention strategies.Sensitivity indices allow us to measure the relative change in a state variablewhen a parameter changes [12].When a variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives.
Numerical values of sensitivity indices of e R to parameter values for the HCV model, evaluated using the following estimated parameter values: 1 0.35 . Definition 1: The normalised forward sensitivity index of a variable " p " that depends differentiable on a parameter " q " is defined as: .
Having an explicit formula for e R in Equation ( 18), we derive an analytical expression for the sensitivity of to each of parameters involved in e R .For example the sensitivity indices of e R with respect to 1 β and θ are given by; are obtained following the same method and tabulated as follows: From Table 1, it can be observed that when the parameters 2 k and 1 k are increased keeping the other parameters constant they increase the value of e R implying that they increase the endemicity of the dis- ease as they have positive indices.While the parameters η , a , 2 ρ , 1 ρ , d , θ and µ decrease the value of e R when they are increased while keeping the other parameters constant, implying that they decrease the The specific interpretation of each parameter shows that, the most sensitive parameter is the control based on education, health care and immunization η , followed by effective contact rate of individuals with chronic dis- ease 2 β , then recovered rate of chronic individuals due to treatment 2 ρ , effective contact rate of individuals with acute 1 β , followed by recovery rate naturally from acute a , death rate of chronic infected d , recovered rate of acute individuals due to treatment 1 ρ , natural mortality rate µ , rate at which exposed develop chronic 2 k , the rate which acute infective are detected by a screening method from exposed group θ , the rate at which screened develop to chronic 1 k , which is the least sensitive parameter.

Numerical Simulations
In this section, we illustrate the analytical results of the study by carrying out numerical simulations of the model system (2) using the following estimated parameter values: 1 2 0.38, 0.001  The phase portrait in Figures 2(a)-(d) shows that for any initial starting point or initial value, the solution curves tend to the endemic equilibrium point ∆ .Hence, we infer that the system (2) is globally stable about the endemic equilibrium point ∆ for the set of parameters above.
In Figures 3(a)-(d), the variation of proportions of exposed, recovered, acute and chronic infective populations for different rates of education, health care and immunization ( ) Figures 3(a)-(d), shows that the infected population decreases as the control strategies (education, health care and immunization), η increased.This confirms that, if η is not effective the disease will invade the population.
Figures 4(a)-(d) shows the variation of proportions of exposed, acute and chronic infective populations and recovered population for different rates of screening.
From Figures 4(a)-(d) we vary the screened rate of infected immigrants, and it is seen that as the degree of screening increases, the infected population decreases.The results further show that increasing the screening rate, decreases the severity of the epidemic.Once again this confirms that, screening can reduce the inflow of infected immigrants into the community.

Discussions and Conclusion
In this paper, a mathematical model of control strategies of HCV in a community with inflow of infected immigrants been established.Both qualitative and numerical analysis of the model was done.The model incorporates the assumption that infected immigrants enter in the community.It is shown that there exists a feasible region where the model is well posed in which a unique disease free equilibrium point exists.The disease free and endemic equilibrium points were obtained and their stabilities investigated.The model showed that the disease free equilibrium is locally stable at threshold parameter less than unity and unstable at threshold parameter greater than unity.Using Lyapunov method, endemic equilibrium is globally stable under certain conditions.A sensitivity analysis shows that the control based on education, health care and immunization η is the most sensitive parameter on e R and the least is 1 k .A numerical study of the model has been conducted to see the effect of certain key parameters on the spread of the disease.It was observed that the spread of the disease decreases due to the presence of control strategies (screening, education, health care and immunization).As the control strategies increase, the exposed, acute and chronic infective individuals also decrease in the population.Finally, from the analysis, it may be hypothesized that preventive measures, through reducing rates of transmission of HCV are therefore necessary to the community.Since reduced transmission leads to lower prevalence of the disease in ∆ is locally asymptotically stable if and only if 1 e R < .These results are summarized with the N. Ainea et al.

Figure 2 .
Figure 2. Phase portrait of the dynamics of susceptibles and the infected and recovered population.
endemic equilibrium of the model system (2) using the estimated parameter values above.

Figure 3 .
Figure 3. Variation population under different values of η (education, health care and immunization).

Figure 4 .
Figure 4. Variation of population under different values of τ (screening).

Table 1 .
Numerical values of sensitivity indices of e R .