Global Dynamics and Applications of an Epidemiological Model for Hepatitis C Virus Transmission in China

An epidemiological model is proposed and studied to understand the transmission dynamics and prevalence of HCV infection in China. Theoretical analysis indicates that the basic reproduction number R 0 provides a threshold value determining whether the disease dies out or not. Two Lyapunov functions are constructed to prove the global asymptotic stability of the disease-free and the endemic equilibria, respectively. Based on data reported by the National Health and Family Planning Commission of China, the basic reproduction number is estimated as approximatelyR 0 = 1.9897, which is much less than that for themodel when a treatment strategy is not considered. An ever-increasing HCV infection is predicted in the near future. Numerical simulations, performed to investigate the potential effect of antiviral treatment, show that increasing the treatment cure rate and enlarging the treatment rate for patients at the chronic stage remain effective in reducing the number of new infections and the equilibrium prevalence. The finding suggests that treatment measures are significantly beneficial for disease control in terms of reducing new infections and, in particular, more attention should be paid to treatment for patients at the chronic stage.


Introduction
Hepatitis C virus (HCV) has been considered as a leading cause of liver cirrhosis and hepatocellular carcinoma and is becoming a major and growing global health problem [1,2].HCV is an enveloped single-stranded RNA virus in the Flaviviridae family and mutates so rapidly that no vaccine is currently available [3].The spread of HCV mainly results from blood-to-blood contact through blood transfusions, intravenous drug use (IDU), and the use of inadequately sterilized or unsterilized medical equipment.According to the World Health Organization (WHO) estimates [4], nearly 3% of the world's population (more than 170 million) has been infected with HCV.In a recent cross-sectional study [5], for which 8,762 Chinese subjects from six areas of China were randomly selected, the overall average prevalence of anti-HCV was estimated to be 0.58% in China, which is much lower than the WHO estimates.Surveillance data show that the annual numbers of newly reported HCV cases increased sharply from 21,145 in 2003 to 223,094 in 2013 [6], which indicates that HCV infection is becoming a serious threat to public health in China.Therefore, it is urgently necessary to understand the present epidemic situation and to provide suggestions on how to control HCV infections.
Mathematical models have been used to analyze the spread and control of HCV infection in [7][8][9][10][11][12][13][14][15][16][17] and provide some insights into the disease's transmission.Martcheva and Castillo-Chavez [7] considered an epidemiological model of hepatitis C in a varying population, divided into susceptible and infected individuals with acute and chronic hepatitis C, and studied the role of the chronic infectious stage on the long-term dynamics of HCV.This model was extended by Das et al. [8] by incorporating the immune class and was also extended by Yuan and Yang [9] by incorporating the latent period.The impacts of HCV treatment on prevalence among active injecting drug users (IDUs) have been studied in [10][11][12] where the treated populations are assumed not to infect the susceptible populations.Imran et al. [14] and Khan et al. [17] formulated epidemic models of HCV containing an isolation class and analyzed the effects of the isolation class on

Model Formulation
We propose a mathematical model to understand the transmission dynamics and prevalence of HCV in mainland China using a system of ordinary differential equations.The population are divided into six classes:  (susceptible),  (exposed),  (infected with acute hepatitis C),  (infected with chronic hepatitis C),  (treated population), and  (recovered population with partial immunity).Let  denote the total population; that is,  =  +  +  +  +  + .New susceptible individuals enter into the  class at a fixed rate Λ.Let  be the natural death rate of the population.Susceptible individuals are infected by patients in the , , and  classes at rates of  1 ,  2 , and  3 , respectively.It is generally thought that the acute phase is more infectious than the the chronic one and that the treated individuals have the lowest infectiousness [15,18], so it is reasonable to assume that  1 >  2 >  3 .Once infected, the individuals move into the exposed class  and then progress to the acute stage at a rate of .An individual at the acute stage who can spontaneously clear the virus recovers at rate ; otherwise he/she will progress to the chronic stage at rate .Acutely and chronically infected individuals are treated at rates  and , respectively.After the treatment, some patients succeed in clearing HCV and move to the class  at rate , while the others who have not responded to the treatment move back to the chronic state  at rate .The disease-induced death rate in the  class is denoted as .Individuals in the class  lose their immunity and eventually return to the susceptible class  at rate .A flow diagram for the model is shown in Figure 1 and the variables and parameters are described in Table 1.The complete dynamical model is as follows: Since / = Λ −  −  ≤ Λ − , we can study the dynamical behavior of system (1) in the positively invariant set: By using the next generation matrix approach given in [21], we obtain the basic reproduction number: which can be interpreted as the average number of new infections generated by a single infectious individual in the acute (), chronic (), and treated () classes.
To investigate the effect of therapy, we study variation in the basic reproduction number  0 with treatment rates , If ( 3 / 1 ) <  < (/), then  0 / < 0. Thus, we get a sufficient condition to make sure that  0 is inversely related to , which means that increasing treatment uptake for the patients at the acute stage is beneficial to control the disease for certain conditions of the treatment failure rate.Model (1) always has a disease-free equilibrium  = (Λ/, 0, 0, 0, 0, 0).There is an endemic equilibrium  * = ( * ,

Global Stability of the Disease-Free Equilibrium
Theorem 1. (i) If  0 ≤ 1, then the unique disease-free equilibrium  of system ( 1) is globally asymptotically stable in D.
There is only one equilibrium  in   , so ∪  0 ∈  ( 0 ) = .Therefore,  is a compact and isolated invariant set for  0 in  0 .
While  >  0 , we have Consider the auxiliary system Define Let ( 1 ( 0 )) be the maximum real part of the eigenvalues of  1 ( 0 ).Since  1 ( 0 ) is irreducible and has nonnegative off-diagonal elements, ( 1 ( 0 )) is a simple eigenvalue of  1 ( 0 ) with a positive eigenvector.It follows from Lemma 2.1 in [22] that ( 1 (0)) > 0 when  0 > 1.Since ( 1 ( 0 )) is continuous for small  0 , there exists a positive constant  0 small enough such that ( 1 ( 0 )) > 0. Therefore, there is a positive eigenvalue of  1 ( 0 ) with a positive eigenvector.It is easy to see   → ∞ as  → ∞,  = 1, 2, . . ., 5. Then according to the comparison principle we have lim which contradicts our assumption.This completes the proof of the claim, which implies that  is an isolated invariant set in  and   () ∩  0 = 0. Using the uniform persistence theory (Theorem 4.2) in [23], we obtain that system (1) is uniformly persistent if  0 > 1.This completes the proof.

Global Stability of the Endemic Equilibrium
The global stability of the endemic equilibrium is studied under the simplified assumption that the immune loss rate is zero (i.e.,  = 0).Thus, we can omit the decoupled equation for / and the following theorem holds.
Proof.Construct a continuously differentiable and nonnegative Lyapunov function: where It can be verified that the global minimum of  occurs at the endemic equilibrium  * , and the function  takes the value  = 0 at the endemic equilibrium  * .Differentiating  along the solutions of system (1) and using the equilibrium relations, we obtain Obviously,  1 > 0 and  2 > 0. Next we will show that   ≥ 0 ( = 3, . . ., 7).From the equations above, we have At the equilibrium, the equalities  * +  * = ( +  + ) * ,  * +  * = ( +  + ) * hold.Then by substituting the values of  2 and  3 , it is easy to obtain that In order to assure   ≥ 0 ( = 3, 4, 5, 6), from (27) we know that  7 must satisfy the inequalities:  The dashed and dash dot lines show the prediction at higher treatment rates  = 0.12 and  = 0.2 for patients at the chronic stage from 2015, respectively.All the other parameters are as shown in Table 1.
Because the geometric mean is always less than or equal to the arithmetic mean, we have / ≤ 0 and the equality holds if and only if (, , , , ) take the equilibrium values ( * ,  * ,  * ,  * ,  * ).Therefore, by LaSalle's Invariance Principle, it follows that the endemic equilibrium  * is globally asymptotically stable in the feasible region D.

Numerical Results
The annual reported HCV case numbers have been released by the National Health and Family Planning Commission of China [6], shown in Figure 2(a).The birth rate is fixed as 0.0121 in [24] and the total population of China is about 1.3 × 10 9 , so we obtain that the recruitment rate is Λ = 0.0121 × 1.3 × 10 9 = 1.573 × 10 7 per year.Okosun [16] chose the rate of progression for treatment from acute infected individuals as in the interval (0.12, 0.189); thus, we choose the mean treatment rate for the patients at the acute stage as  = 0.1545 per year.By fitting (1) to the annual reported HCV data (Figure 2(a)) we obtain estimates for the transmission rates and the initial population size, which are listed in Table 1 and the goodness of fit is shown in Figure 2(b).Figure 2(b) shows that the estimated reported case numbers will reach 809,970 in 2021 in mainland China if the current surveillance, testing, and interventions are unchanged.
Using the estimated parameter values we calculated the basic reproduction number  0 as 1.9897, which is similar to other estimates in the literature [25,26] but is smaller than the value of 4.0636 estimated in [18].This is because the treated individuals included in our model have relatively low infectiousness, which results in a smaller basic reproduction number.Moreover, it is interesting to note that the prevalence of HCV infection in China (Figure 3) is estimated as 0.51% in 2013 and 0.58% in 2014, which is in good agreement with the cross-sectional study [5].Note that this estimated HCV prevalence is much lower than the WHO estimation.This is partial because strict HCV screening in blood began in the 1990s in China, which reduced the transmission rate of HCV by blood transfusion and other sources of iatrogenic infection.
To examine the impact of treatment on HCV transmission dynamics and prevalence and identify the most effective measures to control the transmission of HCV in mainland China we investigated variation in the basic reproduction number and prevalence (i.e., (() + () + () + ())/()) with parameters associated with treatment.If the treatment rate  for the infected people at the chronic stage increases threefold, then  1.
the predicted reported case number will decrease by 13.21% to 702,960 in 2021, while if  increases fivefold, then the predicted number will decrease by 24.74% to 609,580 in 2021, as shown in Figure 2(b).It follows from Figure 3 that the HCV prevalence in China will continue to rise and reach 1.63% in year 2021 under the current circumstances.Figure 3(a) shows that increasing the treatment rate  for the patients at the chronic stage by 200% and 400% from the baseline value can decrease the prevalence in 2021 by 19.75% and 47.15%, respectively.Figure 3(b) shows that increasing the treatment rate  for the patients at the acute stage by 200% and 400% from the baseline value can decrease the prevalence in 2021 by 32.16% and 61.28%, respectively.Figure 3(c) shows that increasing the cure rate  by 200% from  = 0.67 can decrease the prevalence in 2021 by 11.01%.It follows from Figure 3(d) that reducing the treatment failure rate  by 50% and 25% from the baseline value can decrease the prevalence in 2021 by 8.37% and 14.44%, respectively.These results indicate that, to decrease the prevalence effectively, it will be better to enlarge the treatment rate for the patients at the acute and chronic stage and cure rate and decrease the treatment failure rate in the short term.
To access the effectiveness of treatment interventions in the long term, we examine the effects of the corresponding   4 that increasing the treatment rate  for the patients at the chronic stage and the cure rate  and decreasing the treatment failure rate  are more effective than increasing the treatment rate  for the patients at the acute stage in terms of reducing both the basic reproduction number and the equilibrium prevalence in the long run.
To examine the sensitivity of our results to parameter variation, we used Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCCs) [27,28] to examine the dependence of  0 and the equilibrium prevalence on each parameter.Because of limited information on the distributions of each parameter, we chose a uniform distribution as in [28] for all input parameters with ranges listed in Table 2. To know whether the significance of any parameter varies over an entire time interval during model dynamics, PRCC values were calculated for numerous times and plotted versus time.This enables us to assess whether the significance of one parameter changes over an entire time interval during the progression of the model dynamics.1 and 2. population  3 , the cure rate , and the treatment rate  for the population with chronic infection.This implies that a greater efficacy of treatment, a larger treatment uptake for the population with chronic infection, and lower transmission rates at the chronic stage definitely result in lower new HCV infections.Moreover, these four parameters , ,  2 ,  3 also have significant impact on equilibrium prevalence, as shown in (Figure 5(c)).It is worth mentioning that the rate of waning immunity , although it has no effect on  0 , greatly affects the equilibrium prevalence because quick waning of immunity significantly increases the number of the susceptible individuals and hence the prevalence.It should be noted that the transmission rate  1 and the treatment rate  for the population with acute infection have little effect on both the basic reproduction number  0 and the prevalence at the endemic equilibrium because of the relative short duration of the acute stage.

Discussion
In order to understand the prevalence of HCV infection in China based on the reported data [6] and examine the role that treatment plays in the transmission dynamics, we proposed a mathematical model which includes realistic features of HCV transmission such as treatment and partial immunity.Theoretically, the global dynamics of our model are determined by the basic reproduction number  0 .The disease-free equilibrium is globally asymptotically stable if  0 ≤ 1, which means that hepatitis C can be entirely eliminated from the population.When  0 > 1, hepatitis C will persist in the population and the endemic equilibrium is globally asymptotically stable for a special case.It is worth mentioning that although the constructed Lyapunov function () is deterministic when proving the global stability of the positive steady state, the coefficients   ( = 3, 4, . . ., 7) of function / which appeared in (25) were chosen to be a positive solution of (27) and are nonunique, which is more general than the proof of Theorem 3.3 in [15].
When the model was applied to HCV transmission in China, we estimated the basic reproduction number  0 as 1.9897, which is similar to other estimates in the literature [25,26] but is smaller than the value 4.0636 estimated in [18].This is because the treated population has relatively low infectiousness which is included in our model, resulting in a smaller basic reproduction number.Goodness of fit and prediction of HCV trends (Figure 2(b)) show a more accurate result than the overestimated simulation in [18] and the prediction indicates that newly reported cases will continue to rise rapidly in the near future.Moreover, we estimate that the prevalence of HCV infection in China (Figure 3) was 0.51% in 2013 and 0.58% in 2014, in good agreement with the cross-sectional study [5], and will reach 1.63% in 2021.These findings lead us to believe that the exact HCV prevalence is much lower than what the WHO estimation indicates.Also, HCV screening in blood, a practice that may significantly reduce the transmission rate of HCV by blood transfusion and other sources of iatrogenic infection, began in the 1990s in China, and thus it is reasonable to believe that strict blood screening and other procedural measures are preventing the spread of HCV and are leading to a lower prevalence.
It follows from sensitivity analysis (Figure 5) that the transmission rate  3 of the treated population contributes greatly to the transmission of HCV throughout the period of the disease spread.So the prevalence of HCV may be underestimated in [10][11][12] because in their models the treated population are assumed not to infect the susceptible populations.Figures 4 and 5 show that the basic reproduction number  0 and equilibrium prevalence are not sensitive to the transmission rate  1 and the treatment rate  for the population with acute infection, indicating that the acute stage does not substantially affect the transmission of HCV in the long run due to its relatively short duration, but it may affect the prevalence at the beginning of the epidemic (Figure 3(b)).
It should be acknowledged that one limitation of our results is that the reported national data may not be completely composed of exposed people who enter into the acute stage.The data may contain some cases diagnosed at the acute or chronic stage; though the number of these cases is low, it may still result in a slight overestimate of the prevalence of HCV in China.However, the slightly overestimated results were not caused by our model, but rather by the deficiency of data which did not distinguish which stage the reported cases came from.More realistic models about HCV infection on complex networks [29][30][31] will be studied in the future work.

Figure 2 :
Figure 2: (a) Annual newly reported HCV cases for mainland China; (b) goodness of fit and prediction of HCV trends until 2021.Stars represent the reported number of people with HCV by year.The solid line shows the fit based on the current circumstances ( = 0.04).The dashed and dash dot lines show the prediction at higher treatment rates  = 0.12 and  = 0.2 for patients at the chronic stage from 2015, respectively.All the other parameters are as shown in Table1.

Figure 3 :
Figure 3: Plots of the prevalence against time at varying parameter values: (a) the treatment rate  for patients at the acutely infected stage; (b) the treatment rate  for patients at the chronic stage; (c) the treatment cure rate ; (d) the treatment failure rate .All the other parameters are as shown in Table1.

Figure 4 :
Figure 4: (a) Plots of the basic reproduction number  0 by varying  (the rates of change of ; ; ; and ); (b) plots of the prevalence at the steady state by varying  (the rates of change of ; ; ; and ).Each column of markers denote that  increases 10% per time.All the other parameters are as shown in Table1.

Figure 5 (Figure 5 :
Figure 5: (a) PRCCs of the eight parameters for prevalence from 2003 to 2021.(b) PRCCs for  0 .(c) PRCCs for the equilibrium prevalence.Sample size is set to 2000.* denotes PRCCs are significant ( value < 0.01).Parameter values and ranges are as shown in Tables1 and 2.

Table 1 :
Parameters and initial data chosen for the simulation.

Table 2 :
Each column of markers denote that  increases 10% per time.All the other parameters are as shown in Table1.PRCC values for  0 and prevalence at endemic equilibrium.