Sensitivity Analysis of the Parameters of a Mathematical Model of Hepatitis B Virus Transmission

In this paper, we developed a new mathematical model for the dynamics of hepatitis B virus (HBV) transmission in a population with vital dynamics, incorporating vertical transmission and sexual maturity. We obtained the basic reproduction number, 0 R , proof the local and global stability of the disease-free equilibrium of the model. Sensitivity analysis of 0 R with respect to the model parameters were carried out. Our result shows that birth rate, death removal rate, HBV sexual transmission probability per contact rate, and the average total sexual contacts rate are highly sensitive parameters that affect the transmission dynamics of HBV in any population. Thus, vaccination, condom usage and reduced-average sexual partner(s) are good strategies that can lead to controlling HBV transmission.


Introduction
Hepatitis (plural Hepatitides) is a general term that means injury to the liver characterized by the presence of inflammatory cells in the tissue of the organ (liver). Hepatitis B is a disease caused by hepatitis B virus (HBV). This disease reduces the liver's ability to perform life-preserving functions, including filtering harmful infectious agents from the blood, storing blood sugar and converting it to usable energy forms, and producing many proteins necessary for life.
Hepatitis B is fifty to one hundred times more infectious than HIV [1,2]. It has caused epidemics in part of Asia and Africa, and it is endemic in China [3] and Nigeria [2]. About a third of the world's population, more than two billion peoples have been infected with hepatitis B virus at some stage in their life time. Of these, about 360 million people remain chronically infected carriers of the disease, most of whom are unaware of their HBV status [4,5].
Transmission of hepatitis B virus results from exposure to infectious blood or body fluids containing blood. Possible forms of transmissions include (but are not limited to) unprotected sexual contact, blood transfusions, re-use of contaminated needles and syringes, and vertical transmission from mother to child during child birth [6].
Infection with the HBV has been a major public health problem. This has two phases: Acute and Chronic. The Acute phase causes liver inflammation, vomiting, and jaundice in which the individual is infectious. Chronic hepatitis B is an infection with hepatitis B virus that last longer than six months. Once the infection becomes chronic, it may never be cured completely, and may eventually cause liver cirrhosis and hepatocellular carcinoma (HCC) [5,7]. HBV causes approximately 600,000 deaths each year world-wide. Moreover, 10% of people infected with HIV (approximately four million people world-wide) are co-infected with HBV [8].
During the last two and a half decades, [9][10][11][12][13][14][15] have designed mathematical models to evaluate the effect of public health programs and provided long-term predictions regarding HBV prevalence and control in various region. These models are defined by a series of equations, input factors (variables and parameters) aimed at characterizing the process being investigated. The input factors are subject to change and errors which will likely affect the output of the model. Sensitivity Analysis which is define by [16] as the study of how the variation (uncertainty) in the output of a model (numerical or otherwise) can be apportioned (attributed) to different variations in the input of the model therefore becomes not only important but necessary to carry out in order to determine the relative importance of the different factors responsible for the transmission and prevalence of the disease. It may shed light into issues not anticipated at the beginning of a study. This in turn, may dramatically improve the effectiveness of the initial study and assist in the successful implementation of the final solution. In this paper, we developed a new HBV mathematical model incorporating vital dynamics (birth and death removal rates are not equal), vertical transmission, standard incidence function, disease induced death due to both acute and chronic infection and sexual maturity. We carried out stability analysis of the disease-free equilibrium as well as the sensitivity analysis of the basic reproduction number, 0 R with respect to the model parameters so as to know the strength and relevance of the input factors in determining the variation in the output.

Model Description and Formulation
We developed a model for the spread of HBV in the human population with the total population size at time, t given by N(t) with the following assumptions: (  The corresponding mathematical equations of the schematic diagram can be described by a system of ordinary differential equations given in (1)- (7). where: and (14) in the biological-feasible region: , , , , , , :

Existence and Stability Analysis of Disease-free
Equilibrium, 0

E
The model has a disease-free equilibrium (DEF), obtained by setting the right-hand side of (1)-(7) to zero, given by ( ) 0, 0, 0, 0 Using the next generation operator technique described by [20] and subsequently analyzed by [21], we obtained the basic reproduction number, 0 R of the model (1)-(7) which is the spectral radius ( ) ρ of the next generation matrix, Thus, the basic reproduction number is then given as: The epidemiological implication of the theorem is that HBV can be eliminated (control) from the population when 0 1 R < , if the initial size of the sub-populations of the model are in the basin of attraction of the DFE. Furthermore, using Castillo-Chavez et al. [22] global stability theorem, the following global stability result can be established (see Appendix B for proof).

Theorem 2:
The disease-free equilibrium 0 E of the model equations (1)- (7) is globally asymptotically stable (GAS) in Global stability of equilibrium removes the restrictions on the initial conditions of the model variables. In global asymptotic stability, solutions approach the equilibrium for all initial conditions. And similarly, the existence and local stability of the endemic equilibrium can be proved.

Sensitivity Analysis (SA) of the Basic Reproduction Number with Respect to the Model Parameters
One of the most important concerns about any infectious disease is its ability to invade a population. The basic reproduction number, 0 R is a measure of the potential for disease spread in a population, and is inarguably 'one of the foremost and most valuable ideas that mathematical thinking has brought to epidemic theory' [23]. It represents the average number of secondary cases generated by an infected individual if introduced into a susceptible population with no immunity to the disease in the absence of interventions to control the infection. If 1 0 < R , then on average, an infected individual produces less than one newly infected individual over the course of its infection period. In this case, the infection may die out in the long run. Conversely, if 1 0 > R , each infected individual produces, on average more than one new infection, the infection will be able to spread in a population. A large value of 0 R may indicate the possibility of a major epidemic. We thus, carried out sensitivity analysis of the basic reproduction number, 0 R with respect to the model parameters in order to determine the relative importance of the different factors responsible for the transmission and prevalence of the disease. This will assist in curtailing the transmission of the disease by using appropriate intervention strategies.
There are more than a dozen ways of conducting SA, all resulting in a slightly different sensitivity ranking [24]. Following [25][26][27][28][29], we used the normalized forward sensitivity index also called elasticity as it is the backbone of nearly all other SA techniques [24] and are computationally efficient [25]. The normalized forward sensitivity index of the basic reproduction number, 0 R with respect to a parameter value, P is given by: It is important to stress that with the exception of sensitivity indices of HBV-sexual transmission probability per contact, p and average total sexual contacts rate, c , i.e. 0 0 and R R p c S S respectively, the expressions for the sensitivity indices of other parameters are complex with little obvious structure. We therefore, evaluate the sensitivity indices at the baseline parameter values given in Table 1, using Maple software.

Results and Discussion
It is important to stress that the sensitivity indices of all the parameters remain unchanged regardless of the baseline value of p and c . But changes with different baseline values of b and µ . We, thus considered three (3) different set values of µ . Table 2 shows that all the parameters have either positive or negative effects on the basic reproduction number. The death removal rate, µ have the highest sensitivity index followed by the birth rate, b . The HBV-sexual transmission probability per contact rate, p and the average total sexual contacts, c have high sensitivity index of +1 each in all cases. This means that 0 R is an increasing function of both c p and . Thus, decreasing (or increasing) c p or by 10% decreases (or increases)  Table 2. Sensitivity indices of the basic reproduction number to model parameters. The parameters are ordered from the most sensitivity to the least. Parameter values used are as in Table 1 S/N SA value for μ= 0.011 SA value forμ = 0.016 SA value for μ = 0.021 10 is the assumed average number of sexual contacts per year and y the average number of sexual partner(s); we show in Figure 2 the linear relationship between 0 R and the average number of sexual partner(s), y .
Similarly, for b we expected 0 R to be an increasing function of b . This is because increasing b increases the number of susceptible individuals that are likely going to be infected and infect many others more. This is evident on Table 2 were b has the second highest sensitivity index in all the 3 set values of µ . Furthermore, for µ , the higher the Universal Journal of Applied Mathematics 1(4): 230-241, 2013 235 death removal rate the lower the basic reproduction number. This is because as more people die there will be less to be infected. We illustrated on Figure 3 the relationship between the per capital birth rate, the natural death rate and the basic reproduction number, 0 R .  Table 1 Table 1 Figure 6 shows the effects of different levels of the average total sexual contacts, birth and death removal rates on human population. The figures clearly illustrated that 0 R is an increasing function of the average total sexual contacts and the birth rates and a decreasing function of death removal rate. The initial population data used are: , giving a total of 12, 000, 000 N = .

Conclusion
We developed a new mathematical model for hepatitis B virus (HBV) transmission dynamics in a population with vital dynamics, incorporating vertical transmission and sexual maturity. Sensitivity indices of the basic reproduction number with respect to the model parameters were computed. These sensitivity indices allowed us to determine the most influential parameters in controlling disease transmission and prevalence.
Our analysis shows that all the parameters are sensitive to the transmission and prevalence of HBV either positively or negatively. The most influential been the natural death rate, µ and birth rate, b . Next, are the HBV-sexual transmission probability contact rate, p and the total sexual contact rate, c, each with +1 sensitivity index. For optimum control, intervention strategies should be target towards those parameters with high sensitivity index. Nevertheless, even the low sensitive parameters should be included in model formulation so as to determine the number of new HBV-positive birth, morbidity, as well as mortality due to both acute and chronic infection which undermines the social, economic and political systems of the human population concern.
Though, intervention strategies cannot directly target most of the highly sensitive parameters, they can be indirectly targeted through vaccination, condom usage and reduced-average sexual partner(s) by individuals in different classes.
Vaccination will reduce the number of susceptible individuals to be infected as those that are vaccinated are immune for at least 25 years [30]. Thus Finally, there is need to quantify the relationship between the parameters in our model and the possible intervention strategies such as vaccination (at birth, infant and adult) and condom usage in a human population. This would enable us determine the efficiency and cost-effectiveness of different intervention strategies on curtailing morbidity and mortality of hepatitis B virus in the population.

Appendix A. Proof for local stability of the disease-free equilibrium
In this appendix, we proof the local stability of the disease-free equilibrium for the model (1)- (7). We used the qualitative matrix stability technique of determining the local stability of a system. Now, we observed that the variable R does not appear in the first six (6) equations of the model, i.e.

Appendix B. Proof for global stability of the disease-free equilibrium
To establish the global stability of the disease-free equilibrium, the two conditions (H1) and (H2) as in [22] must be satisfied for 0 1 R < . We rewrite the model (1)- (7) in the form: with the components of 3 1 X ∈  denoting the uninfected population and the components of 4 2 X ∈  denoting the infected population. Thus, the disease-free equilibrium is now denoted as: Now, for the first condition, that is globally asymptotically stability of * 1 a linear differential equations. Solving, we have ( ) Thus, is globally asymptotically stable.
Next, for the second condition, that is This is clearly an M-matrix (the off-diagonal elements of A are non-negative). It is thus obvious that ( ) 1 2 , 0 G X X = . Hence, the proof is complete.

Appendix C. Estimation of model parameters values
Population parameters can be divided into two, namely: population-dependent and population-independent parameters. The population-dependent parameters values usually have to be estimated based on HBV epidemiology and the demographic profile of the population (country) concern. While the population-independent parameters value usually has to be estimated on HBV epidemiology and published data. Below we estimated the model parameters values given reasons in details correct to 3 decimal places value accuracy.

C.1. Birth rate, b
The estimated birth rate of countries varies from 6.85 to 47.60 births per year per 1000 people for the year 2012 [33]. As the birth rate in any country may be different, we used 3 levels of birth rates as our hypothetical values. These are low, moderate and high levels with 6.85, 27.225 and 47.60 births per year per 1000 people respectively. Thus, approximating correct to 3 decimal places value, we have: low level birth rate

C.4. Average total sexual contacts, c
This is the average total number of sexual contacts, effective or not, per year. As some individual may have more than one sexual partner it is necessary to take into consideration the mean number of sexual partners since sexual partnerships and disease spread are co-evolving dynamic process and the number of sexual partners is important for the determination of epidemic thresholds. The rate of acquisition of new partners depends largely on social and environmental factors that determine the living conditions, resources and social opportunities [34]. Cultural and religious beliefs have an influence on the number of new partners one can acquire. In some cultural settings, men are allowed to have as many partners as they wish and this has a significant impact on the value of c . Many people indulge in risky behaviours due to poverty, need to get financial support, revenge for having been infected unjustifiably and lack of knowledge on disease dynamics [35]. Thus, a direct estimation of the average total sexual contacts can be given by

C.5. Modification parameter associated with reduce infection by chronic infected individual, η
The chronic infected class comprises of both carriers with HBeAg-positive and HBeAg-negative. The later are less infectious, which make chronic carriers less infectious to acute individuals by a rate η . This is in consistent with [10,36] The average time for a chronically infected child to become sexually active is 14 years 7.5 months. That is: About 90% of children born infected are expected to become carriers [4,38]. That is, the maximum proportion of acute infected infants becoming carriers is 0.9, but the average rate can be less as some infants will progress to immune class due to one reason or another such as influence of nutrition (nutrient-sufficient). We therefore accept [10] value. That is, 885 . 0 = U ϕ as the proportion of acute infected infants becoming carriers. And therefore, the proportion of acute infants who become recovered is About 10% of acutely infected adults are expected to become carriers [4,38]. i.e. 1 . 0 = F ϕ , this is also in consistence with [10]. And therefore, the proportion of acute individual who become recovered is ( ) 9 . 0 1 = − F ϕ C.13. Average rate of recovery from F C to R , C γ A few chronic carriers naturally clear HBV and become removed later at adult age. The age-related annual rate of the viral clearance has been reported to be as low as 1-2% (i.e. 0.01 -0.02) on average [11,39]. We adopted [10]  People with chronic HBV infection are called chronic carriers. About two-third of these people do not themselves get sick or die of the virus but can transmit it to others. The remaining one-third develops chronic hepatitis B, a disease of the liver that can be very serious. People with chronic hepatitis B have a chance of 15-25% of dying prematurely from hepatitis B related cirrhosis or liver cancer-often during the most productive adult years [4]. Assuming an average age of 51years as the usual age of patient at the time of diagnosis (with chronic hepatitis B), we have: