Dynamic mathematical models of HIV/AIDS transmission in China

Social, cultural, and political aspects of the disease may alter the dynamics of infection. In the early 1980s, China’s first outbreak of HIV was found in intravenous drug users (IDUs) and in the mid-1990s, there was a concentrated outbreak in individuals who were donors and recipients of illegal blood donations. Recent reports have shown that HIV is now spreading from those engaged in high-risk sexual behavior to the general population.^1–6 In 2003 several studies predicted that there would be an outbreak of HIV in China before 2010.^7–9 Since 2003, however, China has implemented a series effective interventions^10,11 and a generalized epidemic of HIV has not yet occurred in China, despite widespread high-risk factors.^2,12 More than 40 000 HIV/AIDS cases have been reported every year since 2003,¹² and it is estimated that there are approximately 500 000 HIV-positive people who are unaware of their status.^12,13 Since no preventative vaccine is available,¹⁴ China urgently needs to understand the epidemic trend and quantitatively evaluate the effect of intervention strategies and measures.

Transmission dynamic models are based on the characteristics of population growth, disease occurrence, and spread within a population. Classical dynamic models, also known as storage models, including the susceptible infectious recovered (SIR) model (otherwise known as susceptible infectious AIDS (SIA) in HIV/AIDS research), are relatively crude models. For HIV/AIDS, the total population (N) is divided into three states: susceptible (S), infected with HIV (I), and removal with AIDS (R). When the (S) individuals contact (I) persons they may transition to the infected state. Infected persons transition to the (R) state when they develop AIDS. Through this process differential equations can be obtained and through the solutions to these equations, the model’s corresponding outputs can be obtained, such as the incidence of infection, the prevalence of infection and the basic reproductive number (R₀). After the model is constructed, its outputs should be fitted to the reported numbers from the actual epidemic. Transmission dynamic models may also be used to quantitatively analyze how key parameters, such as the transmission probability per contact (β), the contact number of partnerships formed per unit time (C), and the average duration of infectiousness (D), impact HIV epidemic trends. The SIR model should be modified given the variation in factors associated with HIV/AIDS transmission.^15,16 Dynamic models can provide a comprehensive understanding of the epidemic process through simulation and can reflect its transmission by obtaining population-level outcomes from individual-level inputs.¹⁷

Outside of China, studies using dynamic models have provided worthwhile conclusions about HIV transmission.18–20 Among men who have sex with men (MSM), an HIV transmission model was created considering the stages of HIV disease progression and assumed new members to the susceptible group as proportional to the total MSM population.21 Socioeconomic conditions are important factors in determining the dynamics of the HIV epidemic in Cuba by studying transmission dynamics over a fifteen-year period from 1986 to 2000.22 Dynamic analysis has also been used to conclude that the achievement of universal voluntary HIV testing and uptake of anti-retroviral therapy (ART) could greatly impact the HIV epidemic in South Africa.23 A deterministic mathematical model concluded that herpes simplex virus (HSV) type 2 epidemics could lead to a doubling in HIV incidence.24 We reviewed the current literature on dynamic modeling of HIV in China and provides some insight into the direction studies should take in the future.

Previous Chinese studies of HIV/AIDS dynamic models have included two categories: mathematical theory-based and epidemiological application. Theory-based research studies the model based on mathematical theory. Most of these studies are concerned with model structure and parameter selection. Each study includes different components depending on the hypothesis. The majority of these studies established models, obtained mathematical expressions of the reproductive number of the infectious process (R₀), and proved the existence of equilibrium and the local or global asymptotic stability of equilibrium for the model. These theoretical models often do not incorporate actual data. The epidemiological application of dynamic models uses actual data to analyze the transmission dynamics of high-risk groups or the entire population. These studies often forecast and assess the epidemic, as well as evaluate intervention strategies and measures. Articles about applied transmission dynamic models studies of HIV in China were not published until 2003, since parameters such as the probability of transmission per contact, the average number of partnerships formed per unit time, and the size of high-risk populations were not well understood before that time. With a greater understanding of relevant parameters for HIV transmission, better applied dynamic models can be developed. Parameter estimation, model construction, and application are the three most important elements to consider in dynamic model studies.

PARAMETERS

The reproduction number of the infectious process (R₀) is the most important outcome of interest, indicating if an epidemic will occur. R₀ is the product of the transmission probability per contact (β), the average number of partnerships formed per unit time (C), and the average duration of infectiousness of an infected individual (D) in the basic SIA model. The frequent movement of populations can also have an impact on the epidemic. The expression of R₀ will have parameters based on population dynamics such as the input rate, output rate, and mortality of the population. All of these parameters fluctuate based on social, behavioral, and biological factors. Only one Chinese study has analyzed the sensitivity of parameters and it was concluded that in this study population, transmission probability by sharing needles, injecting number per unit time, and sharing probability were the most sensitive parameters.²⁵

Transmission probability per contact (β)

Biological factors such as viral load, CD4 count, co-infection with other STIs, and circumcision can affect β. Interventions such as the 100% condom use program, needle exchange, methadone maintenance treatment, and prevention of mother to child transmission (PMTCT) impact the epidemic by changing factors associated with the model parameters.

The parameter β is set artificially according to different biological considerations. Some studies conducted in other countries have shown that β is related to the stage of HIV infection.^26–28 Some researchers developed models with varying β for different stages of infection; in these models, however, it was assumed that people were not infectious at the AIDS stage.^7,29–31 Some studies have found that the probability of HIV transmission is greater from males than females.³² Other researchers^33–35 used a different β for men and women, whereas others have found that there is no difference in HIV transmission by gender.³⁶ It is, however, considered undesirable to analyze males and females together due to varying biologic and social factors. Some studies set β as a constant based on actual data. Other studies set it as a stochastic variable, they believe that β is impacted by environmental interference factors and developed a stochastic model by assuming β is a Gaussian white noise process.^37,38 Since the value of β is impacted by many factors, it is preferable not to consider it as a constant.

Bridge populations can acquire HIV by more than one transmission mode, such as female sex workers (FSWs) who are also IDUs; these individuals may be infected by unprotected sexual intercourse or contaminated needles and β is, therefore, difficult to set. Some studies using Chinese data considered β in bridge populations as the mixture of two transmission modes.^39,40 Each method of setting β has its limitations and the parameter should be set according to the study hypothesis and determined by fitting available data.

Contact rate (C)

The contact rate, C, is related to the number of susceptible and infected individuals. However, in some Chinese studies the sexual contact number is regarded as a function of only the sexually active portion of the population.^30,41 Other studies in China have set C as a constant.^33,42 Some studies believed that the core group of transmitters is another driving force to sustain HIV transmission since core groups generally have high contact numbers.^43–46 Different core groups can have significantly different contact numbers, so C can vary for different high-risk groups. From this viewpoint, each high risk population should be analyzed separately in different models. Social factors, such as police restrictions of prostitution and drug use, migration, economic factors, and interventions, may also impact C.

Duration of infectiousness (D)

Dynamic models of HIV in China have considered varying times for the duration of infectiousness.⁴⁷ Varying durations have different transmission probabilities and contact numbers. Several studies set the infectious duration as a constant 8–10 years,^7,25,48,49 although different populations have been found to have different durations of infectiousness.^50,51 Yang and Zhang³³ did not consider D in an SI model because the study period was short and no one died from AIDS during the course of the study. Han et al⁵² also developed an SI model, analyzing the effect of D, but considered HIV and AIDS cases as the same.

ART has had a great impact on the duration of infectiousness by increasing life expectancy,⁵³ but there have not yet been any studies on how ART impacts the HIV epidemic using Chinese data. Co-infections can also impact the duration of HIV infection. Tuberculosis (TB) is the most common opportunistic infection in HIV positive individuals and TB is the leading cause of death for approximately one-third of all AIDS deaths.^54–57 Both clinical data and mathematical models indicate that the presence of TB leads to accelerated HIV disease progression.^58,59 Hepatitis C is also a common co-infection in HIV positive individuals, especially among IDUs. Those who are HIV/HCV co-infected have faster liver cirrhosis progression, but the effect on HIV infection is not yet understood.⁶⁰

Population dynamic parameters

Because HIV has a long duration of infection, population changes can have a great impact on the epidemic. The parameters related to population dynamics, such as the population input (initiation of behavior) and output (termination of behavior) rates will also impact the outcomes of equations. The input rate impacts the number of susceptible, while the output rate is impacted by AIDS mortality. Most dynamic modeling studies in China have considered population dynamics. Using three-year data, Yang et al³³ used three functions including linear, exponential, and Lagrange polynomial interpolating to get the average input number of FSWs. In some studies, however, the input number was assumed as constant.^{37,41,49,52,61} In other studies it was assumed that the input of high-risk groups was not constant, but directly proportional to the size of the sexually active population.30 The output rate is often regarded as a constant^7,25,37 and, therefore, the duration is also constant. In practice, these parameters must not be constants or zero; rather, they should be set according to available data.

Floating populations are also at high risk for HIV infection. A dynamic model of HIV transmission characterized by multi-migration was designed based on an SIR model in Shenzhen, China.⁶² Migrants account for 80% of total population in Shenzhen.⁶³ It was concluded that the model including migration characteristics accurately described the HIV transmission dynamics since it fit the data well. At present, China’s mobile population is estimated to be about 420 million⁶⁴ and the transient population could have a great influence on the prevalence of HIV in China.

MODEL STRUCTURE

The mathematical model structure should reflect the nuances of HIV transmission. Several Chinese studies have created models that incorporate the stages of HIV infection, whereas other studies^{25,48,52,61,65} have used the simple SIR or susceptible-infectious (SI) models. HIV has a long infectious duration compared with other infectious diseases and the disease can be defined by at least three stages based on viral load and CD4 cell levels.²⁶ Several studies have developed multi-phase infection models considering stages of HIV infection. Liu divided the HIV infection period into multiple phases.^29,30 The multi-stage model is more mathematically complex, but is likely more representative of the disease process. Chen et al⁴¹ concluded from a theoretical model that if there are more than two stages, the model is not always in a state of stable equilibrium. No actual data were used to simulate HIV transmission to demonstrate an improvement using the multi-phase infection model. Liu et al⁷ however, found that stage of infection was not relevant to predicting HIV in China when similar estimates for the year 2010 were obtained from both SIA and multi-infection phase models.

Several studies have used the SIR model to analyze trends in HIV infection in China.^25,48 These models consider HIV and AIDS as different disease stages. Ye et al⁴² used an SIR model in IDUs, but considered the β parameter as the same regardless of stage of infection. This model illustrates the impact of AIDS mortality on the HIV epidemic, but does not reflect the impact of the distribution of different stages of infection. Some researchers used an SI model to examine HIV in IDUs.^52,61,65 In this model, all stages of infection were considered together. It was assumed the parameters of β, C, and mortality rate were the same in HIV and AIDS stages. From a biological viewpoint, β should vary at different stages of infection because the viral load fluctuates. C should also vary because stages of infection may have an impact on behavior change if time of diagnosis is not independent of the stage of infection; for example those at a later stage of infection may have fewer contacts because of their deteriorating condition.⁶⁴ Mortality is also differs between HIV and AIDS.

Liu et al⁹ developed a model to predict HIV in China, which considered age structure of the population. In this model, C was different at different age stages, but the β parameter was held constant. This model used available data to forecast the number of people living with HIV/AIDS in the Chinese population. Without any interventions, the number of HIV infections in 2010 was predicted to be approximately 10 million people. The largest proportion of HIV cases would be in the 31–40 years of age group. The number of AIDS cases is expected to reach 650 000 with the 41–50 years of age group comprising the largest share. The results from this model suggest that HIV prevention interventions should target Chinese in their 30 years of age. Using available data, a French study also found that including the age-structure in dynamic models can provide guidance for determining the most appropriate age group to target interventions.⁶⁶ Zhou et al⁶⁷ showed that dynamic models incorporating age of infection could provide information on transmission trends and the effectiveness of interventions. Another study constructed a model, which considered age stages for children, youth, and middle-aged adults.⁶⁸ Different age groups were considered to have different contact rates, as well as different β. It was assumed that when a person was infected by HIV in youth, they would not develop AIDS until middle-age and since they would be sexually active during this interval, they would most likely have more sex partners during these years and would have a greater probability of transmitting the infection. This research indicates that efforts should be made to prevent HIV among China’s youth. This model is less mathematically complex, but provides less information than the former study.

MODEL APPLICATION

Using study data, dynamic models can be applied to estimate and forecast the HIV epidemic in China. Such applications have been used to predict the HIV epidemic in IDUs, FSWs, bridge populations, and the general Chinese population. These estimates can provide information for the development of interventions and allocation of health resources.

Application in IDUs

IDUs were the first population affected by the HIV epidemic and IDU is still the main means of HIV transmission in China.^11,69,70 Han et al⁵² and Ye et al⁴² forecasted the HIV/AIDS epidemic trend in IDUs in two cities in Southwest China; they both concluded that without interventions, HIV prevalence in IDUs would escalate; from 11.4% in 2002 to 18.3% in Xichang City in 2010 and from 2.8% in 2001 to 13.2% in 2005 in Ruili City. The parameters were derived from cross-sectional data, cohort studies, and surveillance data. Ye et al⁴² directly selected the transmission probability as 0.003 for needle sharing among IDUs. However, Han et al⁵² did not directly select a transmission probability, but determined the infectious rate to be 0.3 (infectious rate is the product of transmission probability and contact number) from prevalence data derived from two cross-sectional studies in Xichang City.⁷¹ HIV incidence of IDUs was estimated using a dynamic model based on data from these two studies. The calculated incidence using a dynamic model was similar to those found in a cohort study.⁷² Jia et al⁴⁰ estimated the HIV prevalence in IDUs by three different statistical procedures including a dynamic model with all three procedures yielded similar results. Reliable estimates of HIV prevalence can be made by synthesizing several procedures. These two studies provide a methodology to estimate HIV incidence and prevalence in IDUs in China with accuracy, reliability, and at minimal cost.

Application in FSW and clients

Unlike IDU models, which consider this high-risk population as a separate entity, models that study FSWs and male clients often combine these high-risk populations together. FSW and male clients, however, have distinctly different risks for infection. Often these two populations should be considered separately to reflect their differences in the probability of HIV acquisition and transmission. A study in Yuncheng City in Shanxi Province applied dynamic mathematical models to forecast the HIV epidemic in FSWs and their male clients, through separate equations in the same model.³³ It was concluded that entertainment venues are the main HIV transmission sites in the city. In the model including FSWs and male clients, each group’s total number of sexual contacts is the same although individual clients are thought to have lower contacts than individual FSWs. Heterosexual sex is becoming the main means of transmission in China. It was estimated in 2007 that there are 1.8–3.8 millions FSWs and 17.7–37.1 millions male clients in mainland China.¹³ FSWs may transmit HIV to their husbands and children and male clients are a potential bridge to the general Chinese population.

Application in the bridge population-IDU FSWs

FSW IDUs could transmit HIV to their male clients who could then transmit the infection to other FSWs and to the general population. A theoretical dynamic model was used to demonstrate how FSW IDUs could influence the prevalence of HIV among male clients. Since the probability of transmission from sexual intercourse is much less than from IDU, sexual transmission was not considered in this study. Bacaёr et al³⁹ simultaneously analyzed the HIV epidemic in male IDUs who are also clients and in female IDUs who are also FSWs in Kunming, Yunnan Province. This study confirmed that R0 is greater in IDU than from sexual transmission, but found that the β of the dual-risk bridge population was the sum of needle sharing and commercial sex. Jia et al⁴⁰ also considered the two modes of transmission simultaneously by setting β as the function of needle sharing and commercial sex in Dehong, Yunnan Province.

Application in other high risk populations

Little literature is currently available on dynamic models of MSM or MTCT in China. According to 2007 estimates, there are 3.1–6.3 million MSM in mainland China.¹³ Chinese MSM are highly vulnerable to HIV transmission.^74,75 About 70% of MSM get married to women, which may result in transmission to the family.⁷⁶

HIV prevalence in females has risen in recent years.^12,64,69,77 It is estimated that more than 50% of HIV positive women are 15–30 years old.⁷⁰ The risk of MTCT is, therefore, of serious concern.⁷⁸ There is evidence that infant HIV prevalence has increased sharply in recent years, although there have been relatively few cases.⁷⁹

Applying existing data from these populations to dynamic models could help to better predict HIV transmission trends. There are also no dynamic model studies about the spread of HIV from illegal blood collection in the Chinese population, but control measures by the government have, for the most part, reduced this means of HIV transmission.^2,80

Application in the entire population

Some dynamic models have attempted to forecast HIV transmission for the entire Chinese population. Liu et al⁷ used a dynamic model to predict the trend of HIV transmission for all of China in 2002. This model estimated that in 2007 there would be 6 million HIV cases and 400 000 AIDS cases in China if interventions were not implemented. In 2007, the Chinese government, in collaboration with UNAIDS estimated that there were only 700 000 cases of HIV and 85 000 cases of AIDS in China,¹² which is much lower than Liu’s estimate. It is possible that interventions that were conducted between 2002 and 2007 reduced the incidence of HIV.

Zhao et al⁴⁹ forecasted the epidemic of HIV in Henan Province using a dynamic model. In this study, all high-risk populations were included in one model and the characteristics of the high-risk populations were assumed to have the same parameter values. There are several studies using a dynamic model to describe the epidemic for the entire population of a country, but these studies considered either a small number of HIV-positive people and/or a single route of transmission.^22,23 As a chronic infectious disease with many high-risk groups and a variety of transmission modes, it is difficult to study HIV in the entire population through only one set of dynamic systems. These components increase the dimensions and complexity of the model. It has been proposed that mixed dynamic models are appropriate for China.⁸¹

Other methods have been used in the forecasting and evaluation of the HIV epidemic such as Asian Epidemic Model (AEM), Estimation and Projection Package (EPP), Workbook, and Spectrum.⁸² There is, however, no literature available on the comparison of these particular models with dynamic models in China. Dynamic models have strengths over other mathematical models,^25,40 but the availability of high-quality data sources is essential for obtaining reliable predictions and estimates.

Application for evaluating the impact of interventions

Dynamic models can also be uesd to evaluate the impact of interventions, which requires a change in the values of the parameters of β, C and D. Studies have adjusted β to determine how many people the intervention should reach in order to control the HIV/AIDS epidemic.^37,48 Results from these studies could provide a basis for the allocation of health resources.

Han et al⁵² concluded from research based on a dynamic model that the most cost-effective interventions to control the HIV epidemic in China would be for IDUs. If interventions can reduce the infectious coefficient by 50%, then HIV infection rates would decline by more than 50% in IDUSs. The cost-effectiveness varies for different interventions for IDUs in China, such as needle exchange and methadone maintenance treatment, and the cost-effectiveness also differs for infected and susceptible populations. Studies should examine each intervention’s independent and comprehensive role in infected and susceptible populations to achieve the best economic value and impact on health. Zhao et al⁴⁹ analyzed a dynamic model and came to the conclusion that in Henan Province, interventions for AIDS cases were less cost-effective than interventions to prevent HIV transmission. The difference in cost-effectiveness between susceptible and infectious populations is due to the high number of AIDS cases from illegal blood donation from the early years of the epidemic.

Health education can change behavior of high-risk populations and reduce the risk of HIV infection. Researchers determined, using a dynamic model, that when more people receive health education, R0 is smaller, and a disease-free equilibrium is easier to achieve.^61,83 Yang et al³³ believe that health education at entertainment venues can decrease HIV infection rates, given an increase in the availability of condoms. Although health education is a relatively inexpensive intervention measure, its effectiveness is controversial and health education measures can only be effective if health care services are also available.

ART became widely available in China in 2003, however only one group has attempted to model the effect of ART on China’s HIV epidemic: Li et al⁴⁷ developed a model including ART use with the assumption that with treatment, viral loads will decrease and those in the high infectious stage would transfer to a low infectious stage. Since ART can reduce the viral load and the probability of transmission, as well as extend the asymptomatic stage and the duration of infection, the use of ART could positively or negatively affect HIV transmission, but no actual data has been used to simulate the infectious process in China. Transmission-related behavior may increase if patients taking ART perceive that treatment reduces their infectiousness or if the general population no longer takes precautions to prevent HIV infection, which could impact the contact number. Foreign studies have examined how ART can impact the HIV epidemic using dynamic models. Blower et al⁸⁴ forecasted the impact of increasing risky behavior and drug resistance due to increasing ART utilization in the MSM population in San Francisco. Under scenarios evaluating changing to both high risk and low risk behavior in this population, ART use is predicted to decrease mortality. This study’s prediction of the low level of transmitted drug resistance has been validated when compared with empirical data. Mathematical models have also been used in more resource-limited settings in which there are low-to-moderate rates of ART use.^85,86 Since ART use is relatively new in China,^87,88 dynamic models could be used to predict their impact on the HIV epidemic.

Studies from other countries have examined the effect of interventions on the HIV epidemic such as the treatment of STIs^89–91 and male circumcision.^92–94 Mathematical models have also verified these conclusions.^95–98 Research in China to evaluate the effect of interventions, such as the treatment of STIs and male circumcision, is relatively weak. The prevalence of STIs was high in some populations,^99–101 but circumcision is uncommon in China.^102,103 An HIV vaccine is not yet available, but if one is discovered, it could also serve as an effective intervention measure. An experimental vaccine has been found to reduce the risk of HIV infection by approximately 30% in a community-based population in Thailand with largely heterosexual risk.¹⁰⁴ Dynamic models could be used to predict the effect of this vaccine or a hypothetical vaccine on HIV infection in China.

CONCLUSION

Dynamic models of HIV integrate the fields of medicine and mathematics. The models are comprised of mathematical symbols and equations and HIV transmission is a biological process of interaction between the source of infection and the susceptible. Dynamic models to predict HIV infection in China should reflect the biological, social, and behavioral nuances of the population.