MAIN PAPER (wileyonlinelibrary.com) DOI: 10.1002/pst.1685
Published online 30 April 2015 in Wiley Online Library
Influenza vaccine efficacy trials: a simulation approach to understand failures from the past Anne Benoit,a,b * Catherine Legrand,a and Walthère Dewéb The success of a seasonal influenza vaccine efficacy trial depends not only upon the design but also upon the annual epidemic characteristics. In this context, simulation methods are an essential tool in evaluating the performances of study designs under various circumstances. However, traditional methods for simulating time-to-event data are not suitable for the simulation of influenza vaccine efficacy trials because of the seasonality and heterogeneity of influenza epidemics. Instead, we propose a mathematical model parameterized with historical surveillance data, heterogeneous frailty among the subjects, survey-based heterogeneous number of daily contact, and a mixed vaccine protection mechanism. We illustrate our methodology by generating multiple-trial data similar to a large phase III trial that failed to show additional relative vaccine efficacy of an experimental adjuvanted vaccine compared with the reference vaccine. We show that small departures from the designing assumptions, such as a smaller range of strain protection for the experimental vaccine or the chosen endpoint, could lead to smaller probabilities of success in showing significant relative vaccine efficacy. Copyright © 2015 John Wiley & Sons, Ltd. Keywords: seasonal influenza; vaccine efficacy; clinical trials; simulation
1. INTRODUCTION Seasonal flu epidemic is a worldwide phenomenon. Yearly vaccination is recommended by the World Health Organization, especially for the populations at risk of complications such as the elderly. Clinical efficacy of a new seasonal influenza vaccine is usually studied in a large phase III randomized multicentered trial. Such a study is typically conducted during one or two influenza seasons: subjects are enrolled and randomized just before the influenza season to be vaccinated either with the vaccine of interest or with a comparator [1]. The clinical cases and their time of onset are then collected until the end of the season. In the season 2008–2009, a large multi-countries trial in the elderly population (over 65 years old) failed to show improved efficacy against all influenza A and B strains (excluding the H1N1 pandemic strain) of an experimental adjuvanted vaccine versus the reference vaccine [2]. Because of the study population, no placebo group could be included in the trial, and all subjects were vaccinated, either with the experimental vaccine or with the reference one. Possible reasons for the failure of this trial include the following:
294
The endpoint was too ambitious: it had been assumed that the adjuvanted vaccine would give protection against the vaccine strains but also against other strains (excluding the pandemic strain). The two vaccines had lower efficacy against placebo (absolute efficacy) than expected. The absolute efficacy of the experimental vaccine was lower than expected, while the absolute efficacy of the reference vaccine was as expected, resulting in a lower relative efficacy defined as the efficacy of the experimental vaccine compared with the reference vaccine.
Pharmaceut. Statist. 2015, 14 294–301
Details about the computation of absolute and relative vaccine efficacy (VE) can be found in the Appendix. To learn from the past and to make further steps into flu vaccine development, it seems important to understand what happened. Indeed, when designing and analyzing VE trials, complex issues related to the flu particularities have to be dealt with. Influenza viruses are constantly evolving, which makes intensity of the seasonal epidemics and circulating strains not (fully) predictable [3]. As many complex factors must be properly accounted for, sound, detailed, evidence-based planning is required to design and conduct powerful clinical trials able to demonstrate efficacy of an experimental vaccine. Therefore, clinical trials may fail to show significant VE for various reasons other than no real efficacy, for example, because of low virus circulation or because of mismatch between the circulating and vaccine strains. In such a complex context, simulation studies are particularly useful to understand the failure of past trials to prove VE, to investigate the performances of various designs, and ultimately to help design high-quality trials. Simulating flu VE trials data comes back to simulating time-to-event data (time-to-flu episode). Indeed, the VE can be defined as 1 minus the risk reduction between an experimental and a reference group. Classical data generation techniques for time-to-event data are usually based on simple models, such as the Cox proportional hazards (PH) model, assuming a parametric baseline function [4]. However, we will show that such
a
Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université catholique de Louvain, Brussels, Belgium
b
GSK Biologicals, Rixensart, Belgium
*Correspondence to: Anne Benoit, Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université catholique de Louvain, Brussels, Belgium. E-mail: [email protected]
Copyright © 2015 John Wiley & Sons, Ltd.
A. Benoit, C. Legrand and W. Dewé models cannot appropriately take into account the facts that flu yearly epidemics are complex and depend on many factors. The particularities of influenza, such as seasonality, heterogeneity of the individuals and their level of exposure [5], virus circulation, mechanisms of vaccine protection [6,7], and regular mismatches between vaccine and circulating strains are not captured by these models. To be able to use simulations to study the impact of those particularities, more flexible models should therefore be preferred to generate data. We therefore propose a mathematical model inspired from the epidemiological literature [8]. Our objective is to generate data consistent with the predominant characteristics of seasonal influenza to assess trial designs and data analysis methodologies. Section 2.1 presents our approach. We describe the characteristics of influenza, and we discuss why we think that they are not reflected in classical data generation methods. In Section 2.2, we detail our model components and our algorithm. Based on this, a simulation study is performed in Section 3 to show how it can help in better understanding the failure of the trial mentioned earlier. We simulate data for similar trials, but adding a placebo group, and use these to investigate the impact of the choice of the trial main endpoint, the range of protection of the experimental vaccine, the presence of an immune portion in the population, and the real efficacy levels, absolute and relative.
2. DESCRIPTION OF THE METHOD 2.1. Particularities of influenza In this section, we first shortly present the traditional Cox PH model. We then describe the particularities of influenza and its spreading dynamic. We explain why they are not taken into account in this model and how we include those characteristics in our proposed model. We have selected influenza characteristics that appear to be relevant with regard to the outcome of VE trials. Our objective is to build a data generation algorithm suitable to compare designs and statistical models for VE trials. We intend our model to be relatively straightforward to use while remaining flexible enough to reflect various scenarios [9,10]. 2.1.1. Cox PH model. Simulating flu VE trials requires a model: the PH Cox regression [11] is frequently used in presence of data. Any covariate is supposed to have a multiplicative effect with respect to the baseline hazard function (Equation. (1)), and given the covariates, subjects are assumed to have the same risk of event. 0
.tjXi / D 0 .t/ eˇ Xi , i D 1, : : : , n
(1)
where .tjXi / is the hazard at time t for subjects with covariate values Xi , 0 .t/ is the baseline hazard function, and ˇ is a vector of fixed effects. In data analysis, the baseline hazard 0 .t/ shape is often left unspecified. However, for simulating data, parametrical assumptions are often made about this quantity. VE is defined as the reduction in risk: VE D 1 where D cine effect.
eˇvaccine
(2)
the year but exhibit a marked seasonal increase, typically during the winter months [12]. As a result, clinical influenza incidence curve over time is almost flat during most of the year and present one or more peaks between November and March in the Northern Hemisphere. Traditional parametric hazard distributions, such as exponential, Weibull, and lognormal densities, are characterized by one or two parameters [13]. Those distributions are characterized by a constant or unimodal hazard function. However, the choice of the right distribution to obtain the right timing and intensity for the peak is neither flexible nor straightforward. Three-parameter extension of those distributions [14,15] and polyhazards models [16] have been developed to improve the flexibility of the hazard functions, allowing multimodal hazards. The piecewise exponential distribution [17,18] is another flexible distribution for modeling time-to-event data. The difficulty in generating data when using these flexible parametric distributions lies in the selection of the parameters and in the case of the piecewise model, the determination of the hazard change points. Therefore, in the context of a seasonal infectious disease such as influenza, a parametric baseline hazard shape is not recommended to generate data. Instead, we propose to use incidence historical data, available at epidemic surveillance websites such as FluNet [19] and Sentinelles [20]. This methodology will be detailed in Section 3 of this paper. An important advantage of this proposal is that it makes the inclusion of the multiplicity of the strains causing yearly epidemics straightforward. Indeed, influenza is not a single, genetically stable virus: several strains co-circulate, and the viruses mutate continuously. The repartition of the strains and their level of circulation differ between geographical regions resulting in varying times of occurrence and magnitudes of epidemic peaks [1]. Generating data using a parametric baseline hazard function for different influenza subtypes and geographical regions would require the selection of parameters for each strain and for the assumed correlation structure between the regions. Instead, we use country-specific and strain-specific historical incidence data to take into account the geographical aspects of the circulation and the repartition of the strains.
2.1.3. Subject fragility. Not everybody is equally affected by influenza. Individual characteristics, such as age or preexisting condition, influence the probability of infection and the severity of the disease. For example, people who have weaker immune systems, such as the elderly, sick people, or young children, have lower vaccine response than healthy adults [21]. Traditional methods of simulation for time-to-event assume an equal risk for all subjects given the covariates. In the seasonal influenza context, this is not verified as unobserved sources of heterogeneity, such as subject fragility, must be taken into account. We propose to include subject weakness and contact rate through random effects. .tjzi , xi / D zi .tjxi /, i D 1, : : : , n
(3)
is the hazard ratio and ˇvaccine is the vac-
Pharmaceut. Statist. 2015, 14 294–301
where zi is an unobservable realization of a nonnegative random variable with a given probability density function fz . The random variable z is often called a frailty (as people with a higher value will have a higher hazard, and thus be more ‘frail’), and
Copyright © 2015 John Wiley & Sons, Ltd.
295
2.1.2. Seasonality and geographical regions heterogeneity. In temperate climates, influenza infections are characterized by seasonality. The disease is thought to exist at a low level throughout
A. Benoit, C. Legrand and W. Dewé model (3) is then referred to as the frailty model [22]. A classical choice for the frailty component is the continuous strictly positive one-parameter gamma distribution. 2.1.4. Contact. Influenza infection is conditional to exposure to the virus, more specifically to contact with an infected person. Subjects who have higher contact rates are more likely to encounter the virus and thus to develop the infection. The daily contact rate can be modeled from a discrete distribution. Selecting a distribution admitting mass at 0 allows for the inclusion of subjects who have no risk of exposure to the virus. It has been shown that average daily contact rate was best modeled through a negative binomial distribution [23], and this is what we will use in our work. For the purpose of our model, we consider that the number of daily contact for one individual is constant across time. 2.1.5. Vaccine protection mechanisms. The traditional model (Equation (1)) actually assumes that all the vaccinated individuals are equally partially protected against the virus via the factor exp.ˇvaccine / acting multiplicatively on the baseline hazard. However, the existence of two mechanisms of vaccine protection, leaky and all-or-none, has been argued. The leaky mechanism corresponds to the traditional model. In the all-or-none model, vaccination is assumed to provide a fraction of the vaccinated people with complete immunity. Most probably, vaccine protection mechanism is a mixture of partial protection and complete protection. The traditional model, however, only reflects the leaky model. In our model, we introduce the two protection mechanisms (Figure 1). Consequently, individuals from the experimental group have either a complete or a partial protection against the studied viruses. Furthermore, we consider that some subjects could be previously immune to the infection, for example, because of prior vaccination. We consider that a portion 0 (0 > 0) of the subjects from the reference group (placebo or active control) is completely protected. This fraction is increased by an additive efficacy term VE in the experimental group, resulting in a fraction 1 of non-susceptible subjects in this group. We therefore use two efficacy parameters [24]. First, VEs represents the vaccine protection in the experimental versus the reference group in the susceptible population. Second, VE represents the proportion of the vaccinees totally protected that is induced s /.11 / , by the vaccine. Total efficacy is computed as 1 .1VE1 0 where 1 D 0 C VE , the proportion of non-susceptible in the vaccinees. In the case of a leaky vaccine, 1 D 0, and VE resumes to VEs . If the vaccine only acts as an all-or-none protection, VES D 1 0, and VE simplifies to 1 1 10 . 2.2. Simulation model
296
Based on the previous points, we generate times of disease occurrence through a mixture cure model based on the principle that there are three conditions for any subject to acquire influenza: first, he or she must be susceptible; second, he or she must enter in contact with an infected person; and third, transmission of the virus must occur [25]. We consider a sample of n independent individuals participating in a phase III VE trial. The subjects are potentially exposed to the infectious agent of interest over a period of time Œ0, T, where T is the stopping time of the trial. So, we only consider censoring because of the termination of the trial. However, the model
Copyright © 2015 John Wiley & Sons, Ltd.
Figure 1. Illustration of mechanisms of vaccine efficacy (VE) (adapted from Halloran et al. [7]): distribution of susceptibility in the reference (top) and experimental vaccine (bottom) groups of subjects. A fraction 0 of the subjects are completely protected in the reference (placebo or active control) group. This fraction is increased by VE in the experimental group, resulting in a fraction of totally protected subjects of 1 . The level of susceptibility of the remaining 1 1 experimental group subjects is reduced by a factor VEs compared with the reference group ones.
could easily be extended to the case of non-informative random censoring. Subjects are randomized to receive either the experimental or the reference vaccine. In our model, each subject has contact with other persons at a rate of ci contact per unit of time. The probability at time t that a contact is infectious for influenza strain k is defined by the proportion of the population that is infected, that is, the prevalence of the disease caused by strain k at time t, pk .t/. If a susceptible reference group person makes a single contact with a person infected with the pathogen of interest, then he or she becomes infected with probability i , which is the transmission probability to a reference group person. If a person from the experimental makes a single contact with an infected person, then that individual becomes infected with rate .1 VEs / i . The fragility term for subject i is noted zi . It acts as a multiplier of the instantaneous hazard. So, 0,k .t/, 1,k .t/ are the instantaneous hazards for strain k at time t for the reference and experimental group subjects, respectively, and are computed as (
0,i,k .t/ D .1 y0,i /zi ci i pk .t/ 1,i,k .t/ D .1 y1,i / f.1 VEs / zi ci i pk .t/g
(4)
with i D 1, : : : , n, t D 1, : : : , T, y0,i Bernouilli .0 / for the reference group and y1,i Bernouilli .0 C VE / for the experimental group.
Pharmaceut. Statist. 2015, 14 294–301
A. Benoit, C. Legrand and W. Dewé The overall survival function for subject i, Si .t/, for any strain, can be derived as 8 0 0 111 0 t ˆ X Z ˆ ˆ ˆ @ pk .u/ duAAA S0,i .t/ D exp @ .1 y0,i / @zi ci i ˆ ˆ ˆ
Published online 30 April 2015 in Wiley Online Library
Influenza vaccine efficacy trials: a simulation approach to understand failures from the past Anne Benoit,a,b * Catherine Legrand,a and Walthère Dewéb The success of a seasonal influenza vaccine efficacy trial depends not only upon the design but also upon the annual epidemic characteristics. In this context, simulation methods are an essential tool in evaluating the performances of study designs under various circumstances. However, traditional methods for simulating time-to-event data are not suitable for the simulation of influenza vaccine efficacy trials because of the seasonality and heterogeneity of influenza epidemics. Instead, we propose a mathematical model parameterized with historical surveillance data, heterogeneous frailty among the subjects, survey-based heterogeneous number of daily contact, and a mixed vaccine protection mechanism. We illustrate our methodology by generating multiple-trial data similar to a large phase III trial that failed to show additional relative vaccine efficacy of an experimental adjuvanted vaccine compared with the reference vaccine. We show that small departures from the designing assumptions, such as a smaller range of strain protection for the experimental vaccine or the chosen endpoint, could lead to smaller probabilities of success in showing significant relative vaccine efficacy. Copyright © 2015 John Wiley & Sons, Ltd. Keywords: seasonal influenza; vaccine efficacy; clinical trials; simulation
1. INTRODUCTION Seasonal flu epidemic is a worldwide phenomenon. Yearly vaccination is recommended by the World Health Organization, especially for the populations at risk of complications such as the elderly. Clinical efficacy of a new seasonal influenza vaccine is usually studied in a large phase III randomized multicentered trial. Such a study is typically conducted during one or two influenza seasons: subjects are enrolled and randomized just before the influenza season to be vaccinated either with the vaccine of interest or with a comparator [1]. The clinical cases and their time of onset are then collected until the end of the season. In the season 2008–2009, a large multi-countries trial in the elderly population (over 65 years old) failed to show improved efficacy against all influenza A and B strains (excluding the H1N1 pandemic strain) of an experimental adjuvanted vaccine versus the reference vaccine [2]. Because of the study population, no placebo group could be included in the trial, and all subjects were vaccinated, either with the experimental vaccine or with the reference one. Possible reasons for the failure of this trial include the following:
294
The endpoint was too ambitious: it had been assumed that the adjuvanted vaccine would give protection against the vaccine strains but also against other strains (excluding the pandemic strain). The two vaccines had lower efficacy against placebo (absolute efficacy) than expected. The absolute efficacy of the experimental vaccine was lower than expected, while the absolute efficacy of the reference vaccine was as expected, resulting in a lower relative efficacy defined as the efficacy of the experimental vaccine compared with the reference vaccine.
Pharmaceut. Statist. 2015, 14 294–301
Details about the computation of absolute and relative vaccine efficacy (VE) can be found in the Appendix. To learn from the past and to make further steps into flu vaccine development, it seems important to understand what happened. Indeed, when designing and analyzing VE trials, complex issues related to the flu particularities have to be dealt with. Influenza viruses are constantly evolving, which makes intensity of the seasonal epidemics and circulating strains not (fully) predictable [3]. As many complex factors must be properly accounted for, sound, detailed, evidence-based planning is required to design and conduct powerful clinical trials able to demonstrate efficacy of an experimental vaccine. Therefore, clinical trials may fail to show significant VE for various reasons other than no real efficacy, for example, because of low virus circulation or because of mismatch between the circulating and vaccine strains. In such a complex context, simulation studies are particularly useful to understand the failure of past trials to prove VE, to investigate the performances of various designs, and ultimately to help design high-quality trials. Simulating flu VE trials data comes back to simulating time-to-event data (time-to-flu episode). Indeed, the VE can be defined as 1 minus the risk reduction between an experimental and a reference group. Classical data generation techniques for time-to-event data are usually based on simple models, such as the Cox proportional hazards (PH) model, assuming a parametric baseline function [4]. However, we will show that such
a
Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université catholique de Louvain, Brussels, Belgium
b
GSK Biologicals, Rixensart, Belgium
*Correspondence to: Anne Benoit, Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université catholique de Louvain, Brussels, Belgium. E-mail: [email protected]
Copyright © 2015 John Wiley & Sons, Ltd.
A. Benoit, C. Legrand and W. Dewé models cannot appropriately take into account the facts that flu yearly epidemics are complex and depend on many factors. The particularities of influenza, such as seasonality, heterogeneity of the individuals and their level of exposure [5], virus circulation, mechanisms of vaccine protection [6,7], and regular mismatches between vaccine and circulating strains are not captured by these models. To be able to use simulations to study the impact of those particularities, more flexible models should therefore be preferred to generate data. We therefore propose a mathematical model inspired from the epidemiological literature [8]. Our objective is to generate data consistent with the predominant characteristics of seasonal influenza to assess trial designs and data analysis methodologies. Section 2.1 presents our approach. We describe the characteristics of influenza, and we discuss why we think that they are not reflected in classical data generation methods. In Section 2.2, we detail our model components and our algorithm. Based on this, a simulation study is performed in Section 3 to show how it can help in better understanding the failure of the trial mentioned earlier. We simulate data for similar trials, but adding a placebo group, and use these to investigate the impact of the choice of the trial main endpoint, the range of protection of the experimental vaccine, the presence of an immune portion in the population, and the real efficacy levels, absolute and relative.
2. DESCRIPTION OF THE METHOD 2.1. Particularities of influenza In this section, we first shortly present the traditional Cox PH model. We then describe the particularities of influenza and its spreading dynamic. We explain why they are not taken into account in this model and how we include those characteristics in our proposed model. We have selected influenza characteristics that appear to be relevant with regard to the outcome of VE trials. Our objective is to build a data generation algorithm suitable to compare designs and statistical models for VE trials. We intend our model to be relatively straightforward to use while remaining flexible enough to reflect various scenarios [9,10]. 2.1.1. Cox PH model. Simulating flu VE trials requires a model: the PH Cox regression [11] is frequently used in presence of data. Any covariate is supposed to have a multiplicative effect with respect to the baseline hazard function (Equation. (1)), and given the covariates, subjects are assumed to have the same risk of event. 0
.tjXi / D 0 .t/ eˇ Xi , i D 1, : : : , n
(1)
where .tjXi / is the hazard at time t for subjects with covariate values Xi , 0 .t/ is the baseline hazard function, and ˇ is a vector of fixed effects. In data analysis, the baseline hazard 0 .t/ shape is often left unspecified. However, for simulating data, parametrical assumptions are often made about this quantity. VE is defined as the reduction in risk: VE D 1 where D cine effect.
eˇvaccine
(2)
the year but exhibit a marked seasonal increase, typically during the winter months [12]. As a result, clinical influenza incidence curve over time is almost flat during most of the year and present one or more peaks between November and March in the Northern Hemisphere. Traditional parametric hazard distributions, such as exponential, Weibull, and lognormal densities, are characterized by one or two parameters [13]. Those distributions are characterized by a constant or unimodal hazard function. However, the choice of the right distribution to obtain the right timing and intensity for the peak is neither flexible nor straightforward. Three-parameter extension of those distributions [14,15] and polyhazards models [16] have been developed to improve the flexibility of the hazard functions, allowing multimodal hazards. The piecewise exponential distribution [17,18] is another flexible distribution for modeling time-to-event data. The difficulty in generating data when using these flexible parametric distributions lies in the selection of the parameters and in the case of the piecewise model, the determination of the hazard change points. Therefore, in the context of a seasonal infectious disease such as influenza, a parametric baseline hazard shape is not recommended to generate data. Instead, we propose to use incidence historical data, available at epidemic surveillance websites such as FluNet [19] and Sentinelles [20]. This methodology will be detailed in Section 3 of this paper. An important advantage of this proposal is that it makes the inclusion of the multiplicity of the strains causing yearly epidemics straightforward. Indeed, influenza is not a single, genetically stable virus: several strains co-circulate, and the viruses mutate continuously. The repartition of the strains and their level of circulation differ between geographical regions resulting in varying times of occurrence and magnitudes of epidemic peaks [1]. Generating data using a parametric baseline hazard function for different influenza subtypes and geographical regions would require the selection of parameters for each strain and for the assumed correlation structure between the regions. Instead, we use country-specific and strain-specific historical incidence data to take into account the geographical aspects of the circulation and the repartition of the strains.
2.1.3. Subject fragility. Not everybody is equally affected by influenza. Individual characteristics, such as age or preexisting condition, influence the probability of infection and the severity of the disease. For example, people who have weaker immune systems, such as the elderly, sick people, or young children, have lower vaccine response than healthy adults [21]. Traditional methods of simulation for time-to-event assume an equal risk for all subjects given the covariates. In the seasonal influenza context, this is not verified as unobserved sources of heterogeneity, such as subject fragility, must be taken into account. We propose to include subject weakness and contact rate through random effects. .tjzi , xi / D zi .tjxi /, i D 1, : : : , n
(3)
is the hazard ratio and ˇvaccine is the vac-
Pharmaceut. Statist. 2015, 14 294–301
where zi is an unobservable realization of a nonnegative random variable with a given probability density function fz . The random variable z is often called a frailty (as people with a higher value will have a higher hazard, and thus be more ‘frail’), and
Copyright © 2015 John Wiley & Sons, Ltd.
295
2.1.2. Seasonality and geographical regions heterogeneity. In temperate climates, influenza infections are characterized by seasonality. The disease is thought to exist at a low level throughout
A. Benoit, C. Legrand and W. Dewé model (3) is then referred to as the frailty model [22]. A classical choice for the frailty component is the continuous strictly positive one-parameter gamma distribution. 2.1.4. Contact. Influenza infection is conditional to exposure to the virus, more specifically to contact with an infected person. Subjects who have higher contact rates are more likely to encounter the virus and thus to develop the infection. The daily contact rate can be modeled from a discrete distribution. Selecting a distribution admitting mass at 0 allows for the inclusion of subjects who have no risk of exposure to the virus. It has been shown that average daily contact rate was best modeled through a negative binomial distribution [23], and this is what we will use in our work. For the purpose of our model, we consider that the number of daily contact for one individual is constant across time. 2.1.5. Vaccine protection mechanisms. The traditional model (Equation (1)) actually assumes that all the vaccinated individuals are equally partially protected against the virus via the factor exp.ˇvaccine / acting multiplicatively on the baseline hazard. However, the existence of two mechanisms of vaccine protection, leaky and all-or-none, has been argued. The leaky mechanism corresponds to the traditional model. In the all-or-none model, vaccination is assumed to provide a fraction of the vaccinated people with complete immunity. Most probably, vaccine protection mechanism is a mixture of partial protection and complete protection. The traditional model, however, only reflects the leaky model. In our model, we introduce the two protection mechanisms (Figure 1). Consequently, individuals from the experimental group have either a complete or a partial protection against the studied viruses. Furthermore, we consider that some subjects could be previously immune to the infection, for example, because of prior vaccination. We consider that a portion 0 (0 > 0) of the subjects from the reference group (placebo or active control) is completely protected. This fraction is increased by an additive efficacy term VE in the experimental group, resulting in a fraction 1 of non-susceptible subjects in this group. We therefore use two efficacy parameters [24]. First, VEs represents the vaccine protection in the experimental versus the reference group in the susceptible population. Second, VE represents the proportion of the vaccinees totally protected that is induced s /.11 / , by the vaccine. Total efficacy is computed as 1 .1VE1 0 where 1 D 0 C VE , the proportion of non-susceptible in the vaccinees. In the case of a leaky vaccine, 1 D 0, and VE resumes to VEs . If the vaccine only acts as an all-or-none protection, VES D 1 0, and VE simplifies to 1 1 10 . 2.2. Simulation model
296
Based on the previous points, we generate times of disease occurrence through a mixture cure model based on the principle that there are three conditions for any subject to acquire influenza: first, he or she must be susceptible; second, he or she must enter in contact with an infected person; and third, transmission of the virus must occur [25]. We consider a sample of n independent individuals participating in a phase III VE trial. The subjects are potentially exposed to the infectious agent of interest over a period of time Œ0, T, where T is the stopping time of the trial. So, we only consider censoring because of the termination of the trial. However, the model
Copyright © 2015 John Wiley & Sons, Ltd.
Figure 1. Illustration of mechanisms of vaccine efficacy (VE) (adapted from Halloran et al. [7]): distribution of susceptibility in the reference (top) and experimental vaccine (bottom) groups of subjects. A fraction 0 of the subjects are completely protected in the reference (placebo or active control) group. This fraction is increased by VE in the experimental group, resulting in a fraction of totally protected subjects of 1 . The level of susceptibility of the remaining 1 1 experimental group subjects is reduced by a factor VEs compared with the reference group ones.
could easily be extended to the case of non-informative random censoring. Subjects are randomized to receive either the experimental or the reference vaccine. In our model, each subject has contact with other persons at a rate of ci contact per unit of time. The probability at time t that a contact is infectious for influenza strain k is defined by the proportion of the population that is infected, that is, the prevalence of the disease caused by strain k at time t, pk .t/. If a susceptible reference group person makes a single contact with a person infected with the pathogen of interest, then he or she becomes infected with probability i , which is the transmission probability to a reference group person. If a person from the experimental makes a single contact with an infected person, then that individual becomes infected with rate .1 VEs / i . The fragility term for subject i is noted zi . It acts as a multiplier of the instantaneous hazard. So, 0,k .t/, 1,k .t/ are the instantaneous hazards for strain k at time t for the reference and experimental group subjects, respectively, and are computed as (
0,i,k .t/ D .1 y0,i /zi ci i pk .t/ 1,i,k .t/ D .1 y1,i / f.1 VEs / zi ci i pk .t/g
(4)
with i D 1, : : : , n, t D 1, : : : , T, y0,i Bernouilli .0 / for the reference group and y1,i Bernouilli .0 C VE / for the experimental group.
Pharmaceut. Statist. 2015, 14 294–301
A. Benoit, C. Legrand and W. Dewé The overall survival function for subject i, Si .t/, for any strain, can be derived as 8 0 0 111 0 t ˆ X Z ˆ ˆ ˆ @ pk .u/ duAAA S0,i .t/ D exp @ .1 y0,i / @zi ci i ˆ ˆ ˆ
