Research Article | Open Access
Mingwang Shen, Yanni Xiao, Weike Zhou, Zhen Li, "Global Dynamics and Applications of an Epidemiological Model for Hepatitis C Virus Transmission in China", Discrete Dynamics in Nature and Society, vol. 2015, Article ID 543029, 13 pages, 2015. https://doi.org/10.1155/2015/543029
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 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 approximately , which is much less than that for the model 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.
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 . 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 , nearly 3% of the world’s population (more than 170 million) has been infected with HCV. In a recent cross-sectional study , 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 , 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 [7–17] and provide some insights into the disease’s transmission. Martcheva and Castillo-Chavez  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.  by incorporating the immune class and was also extended by Yuan and Yang  by incorporating the latent period. The impacts of HCV treatment on prevalence among active injecting drug users (IDUs) have been studied in [10–12] where the treated populations are assumed not to infect the susceptible populations. Imran et al.  and Khan et al.  formulated epidemic models of HCV containing an isolation class and analyzed the effects of the isolation class on the transmission dynamics of the disease. Few models have been proposed to understand the transmission dynamics of HCV in mainland China. In , Zhang and Zhou extended the model in  by considering recovery from the acute infected stage and simulated HCV transmission in the near future based on the available HCV epidemic data from 2003 to 2010 in China. Note that their model does not involve the treatment class although a treatment strategy has been available in mainland China for several years . Antiviral treatment, especially at the early stages of HCV recurrence, is an extremely effective strategy. How effective antiviral treatment remains is therefore an issue of great importance for HCV control, and quantifying this impact through a mathematical modeling framework falls within the scope of this study.
The purpose of this study is to formulate a mathematical model which involves almost all possible stages during HCV infection with the aim of accessing the potential impact of antiviral treatment on HCV prevalence of China and then providing reliable quantitative information on controlling the HCV epidemic in China. On the basis of the model in  we incorporate the treated and recovered classes with partial immunity. Note that we consider that the treated individuals can also infect the susceptible individuals , which adds to the complexity of our model and makes it considerably more insightful from an epidemiological perspective than previous models. We analyze the global dynamical behavior of the proposed system by developing suitable Lyapunov functions  and suggest some measures to control HCV infection in China.
The paper is organized as follows. In Section 2, we introduce the model and derive the basic reproduction number. The global stability of the disease-free equilibrium and the endemic equilibrium are studied in Sections 3 and 4, respectively. In Section 5, we apply the model to simulate the HCV data in China and investigate various control strategies with numerical simulations. Finally, we conclude the paper by a discussion in Section 6.
2. 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 . Let be the natural death rate of the population. Susceptible individuals are infected by patients in the , and classes at rates of , , and , 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 . 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:
|LS, least square.|
Since , we can study the dynamical behavior of system (1) in the positively invariant set: By using the next generation matrix approach given in , 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 with treatment rates , , treatment failure rate , and cure rate . Calculating the derivative of with respect to gives Note that ; then if the condition holds. This condition is likely to be true because the disease-induced death rate is always relatively low  (i.e., ). Thus, decreases with increasing treatment uptake for the patients at the chronic stage. Similarly, by finding the derivative of with respect to , we obtain provided , which implies that decreasing the treatment failure rate results in declining and thus is beneficial to disease control.
Regardless, is inversely related to : Therefore, a higher cure rate leads to lower and a lower intensity of the epidemic. Differentiating with respect to yields If , then . Thus, we get a sufficient condition to make sure that 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 . There is an endemic equilibrium if , where with
3. Global Stability of the Disease-Free Equilibrium
Theorem 1. (i) If , then the unique disease-free equilibrium of system (1) is globally asymptotically stable in .
(ii) If , then is unstable and system (1) is uniformly persistent; that is, there exists a constant such that, for all initial values , the solutions of system (1) satisfy , , , , , and .
Proof. (i) Construct a continuously differentiable and positive definite Lyapunov function
Calculating the derivative of along (1), we obtain
In the domain , it is easy to see that , which gives . Thus, if , then we have for all . Every solution of system (1) converges to , where is the largest invariant set in . When , the equality holds if and only if , which implies that and from (1). When , the equality holds if and only if or . Both of these cases indicate that . By the LaSalle largest invariant set theorem, the disease-free equilibrium is globally asymptotically stable if .
(ii) The Jacobian matrix of system (1) at is For convenience of denotation, let , , , and . Then we obtain the characteristic equation at as follows: where Denote the eigenvalues of the characteristic equation (14) as , . Obviously, , , and , , , satisfy the equation . When , we get which means that . This indicates that at least one of has positive real part. Thus, is unstable if .
Define It then suffices to show that system (1) is uniformly persistent with respect to . First, it is easy to see that both and are positively invariant. Obviously, is relatively closed in . Furthermore, it follows from (2) that system (1) is point dissipative. Next we prove that . Assume, by contradiction, that and there exists a such that at least one of is greater than zero at ; for example, and . Then we get , , . This indicates that there exists a such that , , , and , , for , which means that for , contradicting the assumption that . Other cases can be proved in the same way. Thus we have . Let be an initial value. There is only one equilibrium in , so . Therefore, is a compact and isolated invariant set for in .
Next we claim that there exists a positive constant such that any solution , satisfies Suppose the claim is not true. Then , for any ; namely, there exists a positive constant such that , , , , , and , for any . While , we have Consider the auxiliary system Define Let be the maximum real part of the eigenvalues of . Since is irreducible and has nonnegative off-diagonal elements, is a simple eigenvalue of with a positive eigenvector. It follows from Lemma 2.1 in  that when . Since is continuous for small , there exists a positive constant small enough such that . Therefore, there is a positive eigenvalue of with a positive eigenvector. It is easy to see as , . Then according to the comparison principle we have which contradicts our assumption. This completes the proof of the claim, which implies that is an isolated invariant set in and . Using the uniform persistence theory (Theorem 4.2) in , we obtain that system (1) is uniformly persistent if . This completes the proof.
4. 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., ). Thus, we can omit the decoupled equation for and the following theorem holds.
Theorem 2. When , if , then the unique endemic equilibrium is globally asymptotically stable in .
Proof. Construct a continuously differentiable and nonnegative Lyapunov function:
It can be verified that the global minimum of occurs at the endemic equilibrium , and the function takes the value at the endemic equilibrium .
Differentiating along the solutions of system (1) and using the equilibrium relations, we obtain where and () satisfy the following equations:
Obviously, and . Next we will show that . From the equations above, we have At the equilibrium, the equalities , hold. Then by substituting the values of and , it is easy to obtain that
In order to assure (), from (27) we know that must satisfy the inequalities: Using (28) yields and then inequalities (29) must have a nonnegative solution, and so has (27). This indicates that . Because the geometric mean is always less than or equal to the arithmetic mean, we have 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 .
5. Numerical Results
The annual reported HCV case numbers have been released by the National Health and Family Planning Commission of China , shown in Figure 2(a). The birth rate is fixed as in  and the total population of China is about , so we obtain that the recruitment rate is per year. Okosun  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 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 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 . 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 . 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 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 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 treatment parameters , and on the basic reproduction number and prevalence at the endemic equilibrium (i.e., ). Let () be the rate of change of each parameter; then we could increase the treatment rate for the patients at the chronic (or acute) stage (represented by or ()), the cure rate (), and decrease the treatment failure rate () by increasing parameter . In particular, increasing by 30% or decreasing by 30% from baseline values (while keeping other parameters fixed) can reduce by 16.56% or 11.83% and can reduce the equilibrium prevalence by 27.98% or 19.54%, respectively. However, increasing or by 30% from baseline values can only reduce by 0.36% or 9.42% and reduce the equilibrium prevalence by 0.56% or 15.47%, respectively. It follows from Figure 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 and the equilibrium prevalence on each parameter. Because of limited information on the distributions of each parameter, we chose a uniform distribution as in  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. Figure 5(a) shows PRCC values plotted from 2003 to 2021 and it indicates that there are five PRCC values that are significantly different from zero and that the PRCC values of the eight examined parameters vary little with time and stabilize at fixed values in about 2006. It follows from Figure 5(b) that the first four parameters with the most significant impact on are the transmission rate for the chronically infected individuals and the transmission rate for the treated population , 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 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 , 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 and the treatment rate for the population with acute infection have little effect on both the basic reproduction number and the prevalence at the endemic equilibrium because of the relative short duration of the acute stage.
In order to understand the prevalence of HCV infection in China based on the reported data  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 . The disease-free equilibrium is globally asymptotically stable if , which means that hepatitis C can be entirely eliminated from the population. When , 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 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 .
When the model was applied to HCV transmission in China, we estimated the basic reproduction number as 1.9897, which is similar to other estimates in the literature [25, 26] but is smaller than the value 4.0636 estimated in . 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  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 , 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 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 [10–12] 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 and equilibrium prevalence are not sensitive to the transmission rate 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 [29–31] will be studied in the future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank Professor Robert A. Cheke for his helpful comments on the paper. The authors are supported by the National Natural Science Foundation of China (11171268 (Yanni Xiao)) and by the Fundamental Research Funds for the Central Universities (08143042 (Yanni Xiao)).
- A. Alberti, L. Chemello, and L. Benvegnù, “Natural history of hepatitis C,” Journal of Hepatology, vol. 31, supplement 1, pp. 17–24, 1999.
- A. M. Di Bisceglie, “Natural history of hepatitis C: its impact on clinical management,” Hepatology, vol. 31, no. 4, pp. 1014–1018, 2000.
- R. Colina, C. Azambuja, R. Uriarte, C. Mogdasy, and J. Cristina, “Evidence of increasing diversification of hepatitis C viruses,” Journal of General Virology, vol. 80, no. 6, pp. 1377–1382, 1999.
- World Health Organization, “Hepatitis C,” 2009, http://www.who.int/ mediacentre/factsheets/fs164/en/.
- J. Lu, Y. Zhou, X. Lin et al., “General epidemiological parameters of viral hepatitis A, B, C, and E in six regions of China: a cross-sectional study in 2007,” PLoS ONE, vol. 4, no. 12, Article ID e8467, 2009.
- National Health and Family Planning Commission of the People’s Republic of China, “National Report of Notifiable Diseases,” 2003–2013, http://www.nhfpc.gov.cn/jkj/pgzdt/list.shtml.
- M. Martcheva and C. Castillo-Chavez, “Diseases with chronic stage in a population with varying size,” Mathematical Biosciences, vol. 182, no. 1, pp. 1–25, 2003.
- P. Das, D. Mukherjee, and A. K. Sarkar, “Analysis of a disease transmission model of hepatitis C,” Journal of Biological Systems, vol. 13, no. 4, pp. 331–339, 2005.
- J. Yuan and Z. Yang, “Global dynamics of an SEI model with acute and chronic stages,” Journal of Computational and Applied Mathematics, vol. 213, no. 2, pp. 465–476, 2008.
- 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 programs,” Drug and Alcohol Dependence, vol. 110, no. 3, pp. 228–233, 2010.
- N. K. Martin, P. Vickerman, and M. Hickman, “Mathematical modelling of hepatitis C treatment for injecting drug users,” Journal of Theoretical Biology, vol. 274, no. 1, pp. 58–66, 2011.
- N. K. Martin, P. Vickerman, G. R. Foster, S. J. Hutchinson, D. J. Goldberg, and 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,” Journal of Hepatology, vol. 54, no. 6, pp. 1137–1144, 2011.
- F. Luo and Z. Xiang, “Global analysis of an endemic model with acute and chronic stages,” International Mathematical Forum, vol. 7, no. 1–4, pp. 75–81, 2012.
- M. Imran, M. Hassan, M. Dur-E-Ahmad, and A. Khan, “A comparison of a deterministic andstochastic model for hepatitis C with an isolation stage,” Journal of Biological Dynamics, vol. 7, no. 1, pp. 276–301, 2013.
- E. H. Elbasha, “Model for hepatitis C virus transmissions,” Mathematical Biosciences and Engineering, vol. 10, no. 4, pp. 1045–1065, 2013.
- K. O. Okosun, “Impact and optimal control of movement on a multipatch hepatitis C virus model,” Journal of Pure and Applied Mathematics, vol. 5, no. 1, pp. 80–95, 2014.
- A. Khan, S. Sial, and M. Imran, “Transmission dynamics of hepatitis C with control strategies,” Journal of Computational Medicine, vol. 2014, Article ID 654050, 18 pages, 2014.
- S. Zhang and Y. Zhou, “Dynamics and application of an epidemiological model for hepatitis C,” Mathematical and Computer Modelling, vol. 56, no. 1-2, pp. 36–42, 2012.
- X.-W. Liao, Y. Ling, X.-H. Li et al., “Association of genetic variation in IL28B with hepatitis C treatment-induced viral clearance in the Chinese Han population,” Antiviral Therapy, vol. 16, no. 2, pp. 141–147, 2011.
- J. Li, Y. Xiao, F. Zhang, and Y. Yang, “An algebraic approach to proving the global stability of a class of epidemic models,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2006–2016, 2012.
- P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, no. 1-2, pp. 29–48, 2002.
- W. Wang and X.-Q. Zhao, “An epidemic model in a patchy environment,” Mathematical Biosciences, vol. 190, no. 1, pp. 97–112, 2004.
- J. K. Hale and P. Waltman, “Persistence in infinite-dimensional systems,” SIAM Journal on Mathematical Analysis, vol. 20, no. 2, pp. 388–395, 1989.
- National Bureau of Statistics of China, China Statistical Yearbook 2013, Birth Rate, Death Rate and Natural Growth Rate of Population, 2013, http://www.stats.gov.cn/tjsj/ndsj/2013/indexch.htm.
- O. G. Pybus, A. Cochrane, E. C. Holmes, and P. Simmonds, “The hepatitis C virus epidemic among injecting drug users,” Infection, Genetics and Evolution, vol. 5, no. 2, pp. 131–139, 2005.
- C. Matheï, S. van Dooren, P. Lemey, P. van Damme, F. Buntinx, and A.-M. Vandamme, “The epidemic history of hepatitis C among injecting drug users in Flanders, Belgium,” Journal of Viral Hepatitis, vol. 15, no. 6, pp. 399–408, 2008.
- S. M. Blower and H. Dowlatabadi, “Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example,” International Statistical Review, vol. 62, no. 2, pp. 229–243, 1994.
- S. Marino, I. B. Hogue, C. J. Ray, and D. E. Kirschner, “A methodology for performing global uncertainty and sensitivity analysis in systems biology,” Journal of Theoretical Biology, vol. 254, no. 1, pp. 178–196, 2008.
- Y. Wang, Z. Jin, Z. Yang, Z.-K. Zhang, T. Zhou, and G.-Q. Sun, “Global analysis of an SIS model with an infective vector on complex networks,” Nonlinear Analysis. Real World Applications, vol. 13, no. 2, pp. 543–557, 2012.
- Y. Wang and Z. Jin, “Global analysis of multiple routes of disease transmission on heterogeneous networks,” Physica A: Statistical Mechanics and Its Applications, vol. 392, no. 18, pp. 3869–3880, 2013.
- J. D. Cao, Y. Wang, A. Alofi, A. Al-Mazrooei, and A. Elaiw, “Global stability of an epidemic model with carrier state in heterogeneous networks,” IMA Journal of Applied Mathematics, 2014.
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