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Conceived and designed the experiments: MX BBR JH JW. Performed the experiments: MX. Analyzed the data: MX JW. Contributed reagents/materials/analysis tools: JH. Wrote the paper: MX BBR JW.

Current address: Julius Center for Health Research and Primary Care, University Medical Centre Utrecht, Utrecht, The Netherlands

The authors have declared that no competing interests exist.

Hepatitis D (or hepatitis delta) virus is a defective virus that relies on hepatitis B virus (HBV) for transmission; infection with hepatitis D can occur only as coinfection with HBV or superinfection of an existing HBV infection. Because of the bond between the two viruses, control measures for HBV may have also affected the spread of hepatitis D, as evidenced by the decline of hepatitis D in recent years. Since the presence of hepatitis D is associated with suppressed HBV replication and possibly infectivity, it is reasonable to speculate that hepatitis D may facilitate the control of HBV.

We introduced a mathematical model for the transmission of HBV and hepatitis D, where individuals with dual HBV and hepatitis D infection transmit both viruses. We calculated the reproduction numbers of single HBV infections and dual HBV and hepatitis D infections and examined the endemic prevalences of the two viruses. The results show that hepatitis D virus modulates not only the severity of the HBV epidemic, but also the impact of interventions for HBV. Surprisingly we find that the presence of hepatitis D virus may hamper the eradication of HBV. Interventions that aim to reduce the basic reproduction number of HBV below one may not be sufficient to eradicate the virus, as control of HBV depends also on the reproduction numbers of dual infections.

For populations where hepatitis D is endemic, plans for control programs ignoring the presence of hepatitis D may underestimate the HBV epidemic and produce overoptimistic results. The current HBV surveillance should be augmented with monitoring of hepatitis D, in order to improve accuracy of the monitoring and the efficacy of control measures.

Hepatitis D (or hepatitis delta) virus is a defective virus that requires helper functions from hepatitis B virus (HBV) for virion assembly and propagation

High prevalences of hepatitis D have been reported among individuals infected with HBV, but recent reports indicate that hepatitis D prevalence is on the decline. For instance, in 1986, dual infection with both viruses was found in 91% of Taiwanese drug users infected with HBV

To investigate how hepatitis D affects the HBV epidemic, we use a mathematical model describing the spread of the two viruses in a population. We show that hepatitis D prevalence is very sensitive to changes in the infectivity of HBV. We also show that if hepatitis D virus is highly transmissible, the presence of hepatitis D can result in more severe HBV epidemic.

For the transmission of HBV and hepatitis D, we use a mathematical model, shown in

infected only with HBV from an individual with single HBV:

infected only with HBV from an individual with dual infection:

superinfected with hepatitis D:

coinfected with both HBV and hepatitis D:

The definitions of the parameters are summarised in

Symbol | Definition | Value | Source |

Progression rate from acute to carrier for HBV | 0.4/person/year | ||

Recovery rate from acute infection with HBV | 3.6/person/year | ||

Recovery rate from chronic infection with HBV | 0.02/person/year | ||

Transmission risk of HBV from person with HBV only, stage 1 | 0.46 | ||

Transmission risk of HBV from person with HBV only, stage 2 | 0.65 |
||

Progression rate from acute to carrier for those dually infected | 2/person/year | ||

Recovery rate from acute infection for those dually infected | 2/person/year | ||

Recovery rate from chronic infection for those dually infected | 0.02/person/year | ||

Transmission risk of HBV from person with dual infection, stage |
0.71 |
||

Transmission risk of hepatitis D from person with dual infection, stage |
|||

Initial total population size | 26000 | ||

Rate of departing from the population | 0.018/year | ||

Rate at which individuals enter the uninfected population | |||

Rate of partner change | 1.64 partners/year |

See details in the Supporting Information,

To investigate the long-term dynamics of the two viruses, we performed an equilibrium analysis of the system equations. This analysis allows us to find the steady state of the system and the conditions for the eradication or persistence of the viruses. Solving the model equations (1) with the left-hand side equal to zero, we find all the possible steady states. The model has three types of steady states: one where both viruses are eradicated (the disease-free equilibrium), one where only HBV remains endemic but hepatitis D is eradicated, and one where the prevalences of both viruses are non-zero (in the following it will be shown that there may be more than one point of this third type, with negative or complex entries).

Further, we found conditions for the stability of the disease-free equilibrium and of the endemic equilibria (see Supporting Information, _{b}

If both basic reproduction numbers

The transmission probability of chronic HBV from individuals with single HBV infection (_{b}_{bd}

A. Bifurcation diagram. The two positive equilibria are shown with black, the disease-free equilibrium with grey; solid lines correspond to locally asymptotically stable equilibria, dotted lines to unstable equilibria. Results shown here are with _{bd}

The condition that all reproduction numbers are below one is necessary, but not sufficient for the eradication of the viruses, as shown in

The endemic prevalences of HBV and hepatitis D are shown in

If hepatitis D infectivity is not very high, the presence of hepatitis D results in lower endemic HBV prevalence. This can be understood intuitively, because now many individuals have dual infection and transmit HBV less than individuals with single HBV, resulting in less prevalent HBV infections.

If hepatitis D infectivity is high, then hepatitis D spreads very fast and, hence, despite the lower transmissibility of HBV in those dually infected, those with single HBV infection are superinfected sooner rather than later with hepatitis D and, hence, frequently during acute infection. Since the probability of entering the chronic phase for those with acute dual infection is five times higher than for those with acute single infections, this results in higher endemic HBV prevalence and, thus, in a more severe HBV epidemic.

The plot shows the prevalences of HBV (solid line) and hepatitis D (dashed line) in a population where both viruses are circulating and the prevalence of HBV in a population where only HBV is circulating (grey dotted line). A. The transmission probability of acute hepatitis D (

The above results indicate that the suppression of HBV replication and transmissibility due to hepatitis D infection in those dually infected has an important role in the spread of the two viruses. To investigate this further, the endemic prevalences of HBV and hepatitis D were again calculated, but now for different levels of suppression of HBV replication (

Further we investigated the assumption that HBV carriers who are superinfected with hepatitis D go again through acute HBV infection.

Here we examine the effect of control measures, such as treatment, reducing the transmissibility of chronic HBV equally in those with single (

A, B. Treatment reduces only the infectivity of HBV. It is introduced at time 0, after the epidemic had stabilised at the endemic equilibrium with both viruses present, which is shown with a straight grey line. The infectivity of chronic HBV is reduced by 20% (dotted lines), 30% (dashed-dotted lines), 40% (dashed lines), or 50% (solid lines). Prevalences are shown as percentages of total population. C, D. As in plots A, B, but assuming that treatment reduces also the infectivity of hepatitis D, by the same percentage as that of HBV.

In many countries, surveillance of hepatitis D is limited and therefore the actual prevalence of hepatitis D in the population is unknown. For that reason, we examined the impact of a specific intervention for HBV under different assumptions about the “unknown” prevalence of hepatitis D when the intervention was introduced, but keeping the total prevalence of HBV constant (

Antiviral treatment reduces infectivity of chronic HBV by 20%. The total prevalence of HBV is shown as percentage of the population. When the intervention is introduced (year 0), 36% of the population is infected with HBV; among them, the percentage coinfected with hepatitis D is 30% (solid line), 20% (dashed line), 10% (dashed-dotted line), or 0% (dotted line).

The results presented in this study indicate that the presence of hepatitis D may have a strong impact on the spread of HBV. Hepatitis D virus modulates both the severity of the HBV epidemic and the impact of interventions that are aimed at reducing HBV incidence. The presence of hepatitis D virus may hamper the eradication of HBV. Interventions that aim to reduce the basic reproduction number of HBV below one may not be sufficient to eradicate the virus, as control of HBV depends also on the reproduction numbers of dual infections. This implies that for populations where hepatitis D is endemic, plans for control programs ignoring the presence of hepatitis D may underestimate the HBV epidemic and produce overoptimistic results.

The impact of hepatitis D can be explained as follows. At the individual level, hepatitis D affects HBV infection in two ways: HBV transmission rate is lower and the chance to progress from acute to chronic infection (and not recover) is higher for those with dual infection than for those with single HBV infection. Because of this, hepatitis D superinfection affects the prevalence of HBV at the population level. If hepatitis D infectivity is high, then hepatitis D superinfection will usually occur early during acute HBV infection; that increases the chance to progress to chronic infection and hence the prevalence of HBV. If hepatitis D infectivity is low, then hepatitis D superinfection will mostly occur during chronic HBV infection; that reduces HBV infectivity and hence HBV prevalence. The precise mechanism depends on how much HBV infectivity is reduced by hepatitis D, how much the progression rate to chronic infection is increased by hepatitis D, and by other properties of the two viruses. Our results, for instance, show that if hepatitis D superinfection does not result in re-entering the acute stage, higher HBV prevalence will be observed even with low hepatitis D transmission rates. Unfortunately, knowledge about the properties of dual infection is limited, as there are few studies on hepatitis D. The use of a mathematical model allows us to incorporate existing information and obtain realistic parameter values from the literature; the uncertainty analysis of the model further shows how the outcomes depend on the specific parameter values.

This study follows a considerable amount of research on the epidemic dynamics of interacting pathogen strains (see, for instance,

The work presented here can be broadened towards several interesting research directions. For instance, the model can be extended to account for the effect of HBV vaccination. As yet, there are no effective treatments specific for the hepatitis D component of concurrent HBV and hepatitis D infections. However, vaccination against HBV provides direct protection against hepatitis D virus infection as well. In the recent years, HBV vaccination has been introduced in many countries, resulting in considerable reductions in HBV prevalence and incidence. Moreover, vaccination reduces HBV transmission and, hence, the number of those infected with HBV. As hepatitis D can be transmitted only in the presence of HBV, the hepatitis D prevalence will also decline. Our results for the treatment of HBV infections lead us to expect that vaccination may also have a large impact on the prevalence of hepatitis D. This expectation is confirmed by observations in countries where HBV vaccination is introduced and where hepatitis D is prevalent. For instance, in Taiwan the national HBV vaccination of infants was introduced in 1984; 15 years later, the HBsAg carrier rate in children had decreased from 9.8% to 0.7%

Further, it would be also interesting to account for stochasticity in this framework. Several studies have shown that including chance in the model can change its long-term behaviour, for instance “fit” pathogens that remain endemic in the deterministic model, may go extinct in the stochastic due to chance fading out

An important direction for further work would be to include spatial structure in the model. The assumption of proportionate mixing in a large population ignores the clustering of individuals at high risk. As both HBV and hepatitis D spread through the same transmission routes, it is likely that there are clusters of individuals with a high prevalence of HBV, where hepatitis D can disappear by chance. To study the dynamics of such a system, we require a meta-population model or a network model. It has been shown that the dynamics of infection in such a network may differ, for instance having less opportunity for persistence of the disease, or greater possibility for extinction and limit cycles (see, e.g.

Another issue that could be investigated in further research is the transmission of the two viruses via other contacts. In several countries injecting drugs and household contacts are also important routes of transmission of both HBV and hepatitis D

Finally, certain limitations of this modeling study have to be mentioned. First, because of the limited knowledge on the biological properties of hepatitis D and on how HBV infection is changed in those dually infected, several assumptions were made in the model structure or the parameter values used in the numerical results. We tried to compensate the lack of data by performing uncertainty analyses and examining different scenarios, covering as much as possible of all realistic possibilities. Another consideration is that we did not account for variation in some progression rates of HBV according to age. This was done because these rates are relatively stable for adults during the years of sexual activity examined in the numerical results

To our knowledge, this is the first attempt to model the dynamics of hepatitis D and to investigate the interplay between HBV and hepatitis D. On the empirical side, there is very little known about the epidemiological properties of hepatitis D, and given its importance in affecting the HBV epidemic, more information is needed on basic epidemiological characteristics of hepatitis D infection and the time course of infection with HBV and a concurrent hepatitis D infection. On the theoretical side, understanding the complex dynamics of the interaction between a defective virus and its helper virus would be much helped by additional modeling approaches that incorporate the role of demographic stochasticity and network models. Our findings indicate that hepatitis D plays an important role in the spread and control of HBV. Investigating the transmission dynamics of HBV should account for the presence of hepatitis D in a population. Augmenting the existing HBV monitoring programs with monitoring of hepatitis D could boost the accuracy of the surveillance of HBV prevalence and of the efficacy of control programs.

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The authors would like to thank the referees for constructive comments and suggestions that improved the manuscript.