Acta Biotheoretica

, Volume 60, Issue 4, pp 333–356

Aggregation and Competitive Exclusion: Explaining the Coexistence of Human Papillomavirus Types and the Effectiveness of Limited Vaccine Conferred Cross-Immunity

Authors

    • The University of Notre Dame Australia
Regular Article

DOI: 10.1007/s10441-012-9161-5

Cite this article as:
Waters, E.K. Acta Biotheor (2012) 60: 333. doi:10.1007/s10441-012-9161-5

Abstract

Human Papillomavirus (HPV) types are sexually transmitted infections that cause a number of human cancers. According to the competitive exclusion principle in ecology, HPV types that have lower transmission probabilities and shorter durations of infection should be outcompeted by more virulent types. This, however, is not the case, as numerous HPV types co-exist, some which are less transmissible and more easily cleared than others. This paper examines whether this exception to the competitive exclusion principle can be explained by the aggregation of infection with HPV types, which results in patchy spatial distributions of infection, and what implications this has for the effect of vaccination on multiple HPV types. A deterministic transmission model is presented that models the patchy distribution of infected individuals using Lloyd’s mean crowding. It is first shown that higher aggregation can result in a reduced capacity for onward transmission and reduce the required efficacy of vaccination. It is shown that greater patchiness in the distribution of lower prevalence HPV types permits co-existence. This affirms the hypothesis that the aggregation of HPV types provides an explanation for the violation of the competitive exclusion principle. Greater aggregation of lower prevalence types has important implications where type-specific HPV vaccines also offer cross-protection against non-target types. It is demonstrated that the degree of cross-protection can be less than the degree of vaccine protection conferred against directly targeted types and still result in the elimination of non-target types when these non-target types are patchily distributed.

Keywords

Human Papillomavirus vaccinationCompetitive exclusionCompetitive coexistencePatchinessSIS model

1 Introduction

Human Papillomavirus (HPV) is a sexually transmitted infection (STI) that causes a wide range of human cancers and genital warts (Regan et al. 2010; The FUTURE I/II Study Group 2010; The FUTURE II Study Group 2007).The Papillomavirus genus is divided into multiple species, and these are further divided into multiple types (Shih-Yen et al. 1995). The competitive exclusion principle in ecology states that when similar organisms compete for the same resource in the same way, a more fecund or virulent organism should exclude a less fit relative (Hardin 1960). It has been demonstrated that this principle holds for multiple strains of viruses (Bremermann and Thieme 1989) and specifically for multiple strains of STIs (Castillo-Chavez et al. 1996). It is thus not clear why many comparatively rare HPV types manage to persist in the environment in the presence of highly prevalent types that seem to be more persistent and transmissible, such as HPV type 16 (Schiffman et al. 2009).

It has been noted that the risk of contracting a particular HPV type can differ substantially geo-spatially, even within the same country, and may not always be explained by differences in sexual behaviour (Orozco-Colin et al. 2010; Du et al. 2010). A recent Mexican study found that one HPV type only occurred in one geographic region, despite there being no difference with other geographic areas in reported sexual partners or the occurrence of STIs other than HPV (Orozco-Colin et al. 2010). More generally, it has been observed that less common HPV types show much more variation in spatial distribution than common HPV types, being highly prevalent in some areas and rare in others (for example HPV 45 in Africa and Asia versus western countries) (Clifford et al. 2003, 2005; De Vuyst et al. 2009). Spatial variability therefore is clearly important in the transmission of HPV, even within a single country. This paper explores whether a patchy broad scale spatial distribution of individuals infected with different types may offer a sufficient explanation for how HPV is able to violate the competitive exclusion principle.

Spatial distribution could also be important in the potential impact of HPV vaccination programmes. Vaccination programmes targeting specific types of HPV (vaccine types—a bivalent vaccine targets HPV 16 and 18; a quadrivalent vaccine targets HPV 6, 11, 16 and 18) are now being employed in a number of countries (Bogaards et al. 2011; Donovan et al. 2011; Lefevere et al. 2011). The highly geographically variable, less common types mentioned above are not targeted by the vaccine and are referred to as non-vaccine types. It has already been noted that the high geographic variability observed in HPV incidence could have implications for the results that can be achieved by vaccination in different geographic areas, and possibly influence the inclusion of types that are generally rare, but common in some locations, in future polyvalent vaccines (Orozco-Colin et al. 2010).

Simple forms of deterministic population processes such as compartmental disease models and the logistic growth equation do not make any allowance for different distributions of individuals in the population in geo-space, in effect implicitly assuming that individuals are discretely uniformly distributed in geo-space (Anderson and May 1991; Kuno 1988). In a logistic growth model, this means that individuals reproduce at the same rate regardless of spatial location, having the same number of offspring and the same number of partners (Kuno 1988; Yamamura 1989). By extension, in a compartmental disease model, this implies that there is the same number of infected individuals in any one place, as they are uniformly distributed in the population. This assumption is unrealistic, which has led to the use of some other probability distributions to describe spatial distribution. As spatial distribution is by necessity discrete, the Poisson and negative binomial distributions have been particularly useful (Southwood 1978). The negative binomial is an approximation of patchy distributions, because it models overdispersed copunt data (variance greater than the mean). Ecologists may prefer to call the negative binomial a contagious distribution as the term overdispersed is used with the opposite meaning in some biological disciplines (Southwood 1978). Probability distributions such as the negative binomial and Poisson are called theoretical because they will never perfectly describe the distribution of individuals in geo-space, but they have the advantages of being reasonable approximations to real data in many cases and of having known mathematical forms (Southwood 1978).

There is a substantial body of theory in ecology dealing with the spatial distribution (the extent to which clustering in geo-space can fit probability distributions) and heterogeneity of animals and plants, with the subject underpinning important chapters and texts in ecological literature (Southwood 1978; MacArthur and Wilson 1967; Pielou 1969). Lloyd’s mean crowding index is just one of several ecological measures of spatial heterogeneity that has naturally been drawn on for modelling disease in human and animal populations (Barlow 1991; Keeling 2000; Lloyd 1967; Lloyd-Smith 2007). This index is attractive for differential equation and statistical models of disease transmission because it has been linked to a number of well-defined statistical distributions that can fit count data (such as disease incidence) in differential equation and regression models (Iwao 1968; Kuno 1988).

Iwao’s mean crowding-mean density relation (also Iwao’s patchiness regression) is a linear relationship describing the relationship between mean crowding and the mean number of individuals in one location (here after called mean density) (Iwao 1968). Iwao’s linear mean crowding - mean density relation has two parameters, that have been demonstrated to take on particular values where individuals are distributed according to the discrete uniform, Poisson and negative binomial distributions (Iwao 1968; Kuno 1988). The same linear relationship between mean crowding and mean density was later used in differential equation models to incorporate heterogeneity in the number of individuals over geo-space (Kuno 1988). Despite its potential utility, as far as I am aware, Iwao’s linear relation between mean concentration and mean density has not been used previously in models of infectious diseases in humans, though the mean crowding index itself, rather than Iwao’s linear relation, has been used in models of sexually transmitted bovine tuberculosis in possums (Barlow 1991) and also appears in a modified form as an expression of overdispersion in the number of sexual partners available in some HIV models (May and Anderson 1987).

This paper presents a deterministic transmission model that incorporates the differential equation form of Iwao’s mean crowding-mean density relationship to model spatial distribution of HPV infected individuals, and to explore the effects of spatial heterogeneity on transmission and the effect of vaccination. The transmission model presented is based on the compartmental susceptible-infected-susceptible (SIS) framework. The SIS model was used because, though evidence is conflicting (Regan et al. 2010) it is possible that HPV infection may confer no long lasting immunity (Lu et al. 2010, 2011; Trottier et al. 2010; Viscidi et al. 2004). In particular, re-infection rates are similar to initial infection rates (Trottier et al. 2010). Evidence for immunity is generally linked to sero-conversion in the literature (Ho et al. 2002; Malik et al. 2009) but sero-conversion may be uncommon, particularly for some sites of HPV infection (Paaso et al. 2011). Furthermore, it is still unclear whether even a majority of those infected sero-convert at all (particularly for types other than HPV 16), (Skjeldestad et al. 2008) and whilst the duration of seropositivity is unclear, at least some people lose seropositivity in less than 2 years (Carter et al. 2000). Because studies examining immunity link it to sero-conversion and it is apparent that not everybody sero-converts, immunity may only exist in some people. Therefore, the SIS model has often been used to model HPV transmission (Canfell et al. 2004; Goldie et al. 2003, 2004; Kulasingam et al. 2007; Sanders and Taira 2003; Taira et al. 2004) and this paper takes the same approach as it is probable that the SIS model is a reasonable description of the natural history of HPV for at least a sizeable proportion of people. Forthcoming work will extend the model presented in this paper to incorporate long-term immunity.

The effect of patchy distributions of HPV infected individuals in geo-space on the transmission and vaccination is explored in this paper by linking clustering of infected individuals to three theoretical distributions; Poisson (also called random), negative binomial (also called contagious), and uniform. Extending the model to two groups of HPV types (those targeted by vaccination and those not), the hypothesis that the aggregation of infected individuals is necessary to permit the coexistence of multiple HPV types is tested. Numerical simulations are used to simulate vaccination and to assess whether the system tends towards different post-vaccination equilibria when no cross-protection or limited cross protection against non-vaccine HPV types is assumed.

The local stability conditions of the models are demonstrated and compared to those of the standard SIS model. Finally, the plausibility of the levels of clustering required for the coexistence of multiple HPV types in the model is examined by a comparison with the degree of clustering of low grade cervical lesions observed in screening data from two Australian states.

2 Methods

2.1 Application of the Mean Crowding: Mean Density Relationship to HPV

Let a real world population consist of N sexually active individuals spread over a large geographic area, where N is a large number—too large to know the exact composition of the population at any point in time. To estimate how many people are infected with an HPV type X, Qsamples are taken from N yielding Q estimates of the number of individuals infected with HPV X. Almost certainly every estimate will be different, because in a real population not every individual in every location and at every time will have the same risk of infection with HPV X. Thus the estimated incidence of infection with X can be modelled as a random variable with a mean and variance.

Where the variance in the number of individuals infected with HPV X is greater than the average number of infected individuals, the distribution of the incidence of HPV X infection will be overdispersed or patchy, indicating that infected individuals are temporally or spatially aggregated (Southwood 1978). Aggregation particularly occurs where a resource necessary for survival is rare or ephemeral (Hartley and Shorrocks 2002). It is clear that the resource that an individual infected with HPV X needs to reproduce (to propagate another individual infected with HPV X) is an individual susceptible to HPV X who is willing to have sex with them. The willingness and sexual availability of a susceptible individual will vary with both space and time as not all individuals in all locations and times are equally sexually available. Susceptible individuals therefore represent a resource for infectious people that in the real world is both comparatively rare (many people have no more than one sexual partner per year) (de Visser et al. 2003) and ephemeral as not all locations and times are equally attractive for sexual encounters.

The clustering of individuals infected with HPV X can be modelled using Lloyd’s intra-typic mean crowding index (hereafter I*), which is the average number of additional individuals in the same area as an individual of the same type (Iwao 1968; Kuno 1988; Lloyd 1967). In the context of STIs, ambit can be interpreted as being the number of other individuals infected with X that are in the sexual network of an individual who is also infected.

I* is expressed by the formula
$$ I^{*}=\frac{\sum_{q=1}^{Q} I_q(I_q - 1)}{\sum_{q=1}^Q I_q}. $$
(1)
In Eq. (1) let Iq be the number of people infected with HPV in area q of Q (in other words the crude incidence of infection with X) (Lloyd 1967). The minus 1 term is important, because it is what makes the formula express the average number of infected individuals in addition to another infected individual (the −1 is literally, minus one individual). The quantity (I* + 1) corrects for this one individual, and has been dubbed mean concentration, and can be simpler than I* to work with mathematically (Iwao 1976). Mean concentration is therefore
$$ I^{*}+1 =\frac{\sum_{q=1}^{Q} I_q^2}{\sum_{q=1}^{Q} I_q}. $$
(2)
Equations (1) and (2) can be linked to the distribution of infected individuals in space without making a geographically explicit model. Iwao devised a simple linear relationship
$$ I^{*}+1 =m\bar{I}+c+1 $$
(3)
that here describes the relationship between mean concentration and mean density (the average number of individuals in a unit of habitat, \(\bar{I}\)) under different conditions of aggregation (Iwao 1968; Kuno 1988).The slope and intercept parameters of Eq. (3) are denoted m and c to stress their linear relationship and to distinguish them from the infectious disease parameters which I later give greek letters. In Eq. (3) m is a slope parameter (the “density-contagiousness coefficient”) representing how mean crowding changes with increasing mean density and c (the “index of basic contagion”) is the average number of individuals in the same type in the ambit of an infected individual, interpreted here as the number of other infected individuals in their sexual network (Iwao 1968). It has been demonstrated that m and c take on particular values when spatial distribution is modelled using different probability distributions, in particular, the uniform, Poisson and negative binomial distributions (Iwao 1968; Kuno 1988). The values of m and c under these distributions are given in Table 1. In general c is greater than or equal to −1 and m is greater than or equal to 1, with increasing values of both indicating that infected individuals are more aggregated in the population (i.e. they are more likely to have contact with each other) and that they are less likely to partner with (and hence transmit infection to) susceptible individuals. The use of the parameters m and c in compartmental disease models to model the spatial dispersion of infected individuals is now demonstrated.
Table 1

Values of m and c with different assumed distributions of infected individuals (Iwao 1968; Kuno 1988)

Distributional assumptions

m

c

Uniformly distributed infected individuals

1

−1

Distribution of infected individuals is regular (underdispersed) but not uniform

0–1

−1

Poisson distributed infected individuals

1

0

Poisson distributed clusters of infected individuals

1

>0

Contagiously (negative binomially) distributed infected individuals

>1

0

Contagiously distributed clusters of infected individuals

>1

>0

2.2 Compartmental Disease Models: The Single Disease Model

Let Iq be the number of individuals infected with HPV, regardless of type, and Sq the number of individuals susceptible to HPV (regardless of HPV type) in an area q at time t. By using the relation Sq = Nq − Iq, where Nq is the number of individuals of either type in area q, the basic compartmental SIS model
$$ \dot{I} = (\alpha N - \gamma)I - \alpha I^2 $$
(4)
can be generalised to
$$ \dot{I} = \sum_{q=1}^{Q}(\alpha N_q - \gamma)I_q- \alpha I_q^2 $$
(5)
In Eqs. (4) and (5), α is the transmission rate and γ is the recovery rate. After some algebraic manipulation it is apparent that Eq. (5) can be rewritten in terms of mean concentration as
$$ \dot{I} = \sum_{q=1}^{Q}(\alpha N_q - \gamma)I_q- \alpha I_q (I^*+1) $$
(6)
and that by Eq. (3), Eq. (6) can be further rewritten as
$$ \dot{I} = \sum_{q=1}^{Q}(\alpha N_q - \gamma)I_q- \alpha I_q (m\bar{I}+c+1). $$
(7)
To study the asymptotic behaviour of the system it is permissible to write Kuno (1988).
$$ \dot{I} = (\alpha N - \gamma)I- \alpha I (mI+c+1). $$
(8)

By substituting m = 1 and c = −1, which are values of m and c under the assumption that there is no aggregation of infected individuals (see Table 1), into Eq. (8) and rearranging the equation it is observed that the fundamental nature of the compartmental SIS model is not changed, because this yields the original SIS model in Eq. (4).

In the SIS model if α N / γ > 1 the system tends towards an endemic equilibrium, and if α N / γ ≤ 1 the disease becomes extinct (Wang 2009). The validity of this threshold condition for other values of m and c were explored algebraically and the local stability of endemic and disease free equilibria determined.

2.3 Extending the Model to Multiple HPV Types

Infection with two circulating groups of HPV types, HPV X and Y, is modelled. It has become apparent that many or most HPV infected individuals harbour more than one HPV type (Chaturvedi et al. 2011; Plummer et al. 2011). Since this model will be used later in this paper to investigate the effects of vaccination, let IXq be the number of individuals infected with HPV , in whom at least one infecting HPV type is a type for which a vaccine becomes available (group X). Let IYq be the number of individuals infected with HPV who do not harbour at least one vaccine targeted type, and Sq the number of individuals susceptible to all types in area q of Q at time t.

Assume that the recovery rate in group i of X or Y is γi and the transmission rate of types i is αi. Letting Sq = Nq − IXq − IYq, Eq. (6) can be extended to the basic Lotka-Volterra competitive system
$$ \begin{aligned} \dot{I}_X &=\sum_{q=1}^Q (\alpha_X N_q -\gamma_X-\alpha_X {I_Y}_q ) {I_X}_q-\alpha_X (m_X \bar{I}_X+c_X+1) {I_X}_q\\ \dot{I}_Y &=\sum_{q=1}^Q (\alpha_Y N_q -\gamma_Y-\alpha_Y {I_X}_q ) {I_Y}_q -\alpha_Y (m_Y \bar{I}_Y+c_Y+1) {I_Y}_q \end{aligned} $$
(9)
Asymptotically (Kuno 1988)
$$ \begin{aligned} \dot{I}_X &= (\alpha_X N-\gamma_X-\alpha_X I_Y ) I_X-\alpha_X (m_X I_X+c_X+1) I_X\\ \dot{I}_Y &=(\alpha_Y N -\gamma_Y-\alpha_Y I_X ) I_Y-\alpha_Y (m_Y I_Y+c_Y+1) I_Y \end{aligned} $$
(10)

Note that in the absence of either HPV X or HPV Y (IX or IY equals zero)the dynamics of infection with another type is governed by Eq. (8) and the under the assumptions of uniformly distributed infected individuals (mi = 1, ci =  −1) this further reduces to a basic logistic equation identical in form to the normal SIS model in Eq. (4). The stability conditions and equilibria of Eq. (10) were determined analytically and verified numerically and are described in the Appendix. Analytical and numerical work was assisted by MapleTM 15 (MapleSoft, Waterloo, ON).

2.4 Simulations: Vaccination Against Multiple HPV Types

Numerical simulations were used to analyse the effect of varying values of mi and ci on the impact of vaccination against HPV. The values of parameters used in simulations are summarised in Table 2, and their calculation described in detail below.
Table 2

Parameter values for numeric simulations using two group model

Parameter

Description

Value

π1

Proportion women vaccine eligible each year during catch-up period

0.292

π2

Proportion women vaccine eligible each year after catch-up period

0.010

θ1

Proportion eligible women vaccinated (two-doses) annually during catch-up period

0.558

θ2

Proportion eligible girls vaccinated (two-doses) annually after catch-up period

0.730

ψX

Vaccine efficacy against infection with HPV X

0.840

ψY

Vaccine efficacy against infection with HPV Y

0 or 0.462

mX

See Sect. 2.1

3.000

cX

See Sect. 2.1

2.000

αX

Transmission rate of HPV X

0.001

γX

Recovery rate of HPV X

1.0092

mY

See Sect. 2.1

3.900

cY

See Sect. 2.1

3.000

αY

Transmission rate of HPV Y

0.0009

γY

Recovery rate of HPV Y

0.9252

Estimates of relevant parameters vary considerably in the literature. Numerous studies have examined the clearance rates of different HPV types (recovery rates γX and γY) (Arima et al. 2010; Insinga et al. 2007a, b; Winer et al. 2005, 2011; Giuliano et al. 2002, 2008, 2011; Sycuro et al. 2008; Trottier et al. 2008). Generally infections with the high risk vaccine types 16 and 18 are of longer duration than infections with other high risk types (Liaw et al. 2001; Molano et al. 2003; Richardson et al. 2003). The estimates of Trottier et al. of 75.3 (high risk infections including vaccine HPV types 16 and 18) and 78.0 (other high risk) clearances per 1,000 women- months were used for the recovery rates γX and γY (Trottier et al. 2008). The target endemic population level prevalence of infection with HPV X (\(\acute{I}_X,\) representing HPV infected individuals infected with at least one of HPV 16 or 18) was 2.3 % and the target prevalence of HPV Y (\(\acute{I}_X,\) representing individuals infected solely with non-vaccine oncogenic HPV) was 1.4 % (the mean of non-vaccine oncogenic types) (Dunne et al. 2007). Given the γi values taken from Trottier et al. Eq. (10) were solved for appropriate values for αi. For both HPV X and Y these were both approximately 0.001. Partner change rates were assumed to be subsumed within this overall transmission rate.

The characteristics of the vaccination programme were assumed to be similar to those of the Australian vaccination programme using a quadrivalent HPV vaccine. Briefly, it was assumed that for a period of 3 years both 12–13 years old girls in high schools and young women aged up to 26 could receive free vaccination (called the catch-up programme - catch up was administered both through schools and general practice). After 3 years only 12–13 years old schoolgirls received vaccination (Donovan et al. 2011; Gertig et al. 2011).Vaccination was assumed to act by reducing the proportion of N that was susceptible to infection with HPV i by a fraction ϕij, defined as
$$ \phi_{ij}=\theta_j \pi_j \psi_i $$
(11)

In Eq. (11) ψi is the proportion of potential infections with HPV i that are prevented in those vaccinated. Let j = 1, 2 where 1 refers to the catch-up programme and 2 refers to the ongoing school based vaccination programme after the catch-up programme; πj is the proportion of the eligible population which is vaccinated (coverage) in period j, and θj is the proportion of the population eligible in period j. It is assumed that the vaccine has no effect on the duration of infection with HPV i if administered to an individual already infected (Vandepapeliere et al. 2005). Currently available information provides strong evidence that the duration of immunity conferred by vaccination is long (The FUTURE I/II Study Group 2010; The FUTURE II Study Group 2007), and the Australian vaccination programme therefore administered the vaccine in up to three doses over 6 months (Gertig et al. 2011) with no plans to offer a booster. The presence of long immunity due to vaccine conferred infection, combined with the possibility of no or little immunity due to natural infection (Trottier et al. 2010) makes HPV an interesting and rare infectious agent.

The modelled population of size N was assumed to be between 12 and 60 years of age and to be comprised of equal numbers of males and females, where age in years was uniformly distributed. For the first 3 years of vaccination π1 was equal to the proportion of the population that was female and between 12 and 26 years of age to reflect catch-up, and afterwards π2 was equal to the proportion of the population that was female and 12–13 years of age.

The estimate of vaccine coverage, θj, was based on figures for two doses of a quadrivalent vaccine in Australia (Gertig et al. 2011). For the first 3 years θ1 was equal to the average estimated two dose coverage in the catch up and school based programme combined, and afterwards θ2 was equal to the estimated two dose coverage of the school based programme only. Vaccine efficacy against targeted types of ψX = 0.84 was assumed, based on the reduction in incident infection with vaccine types during follow up in the quadrivalent vaccine randomised control trial (Mao et al. 2006).

Data on the degree to which vaccination protects against HPV types other than 16 and 18 (cross-protection) vary, with both vaccine type (bivalent vs quadrivalent) and the diversity of oncogenic types being important (Brown et al. 2009; Herrero 2009). Two different scenarios regarding the effectiveness of vaccination against preventing transmission of and infection with HPV Y (representing non-vaccine oncogenic types) were examined. In the first, it was assumed that vaccination conferred no reduction in risk of HPV Y infection (no-cross-protection). In the second, it was assumed that vaccination reduced the risk of infection with HPV Y by 46.2 % (ψY = 0.462) (Herrero 2009). This value was equal to the reduction in incident infection with HPV 31 observed in those vaccinated with the quadrivalent vaccine in randomised control trials, and was chosen because the confidence intervals reported for protection by the quadrivalent vaccine against non-vaccine oncogenic types other than HPV 31 overlapped zero, suggesting they are less robust estimates (Herrero 2009).

Values of mi from 0 - 4 and values of ci from −1-4 were trialled and values that did not result in a pre-vaccination competitive co-existence equilibrium with both HPV X and Y present discarded. These ranges encompass common values of these parameters (Iwao and Kuno 1968). The final values given in Table 2 were those that produced a competitive coexistence equilibrium with effective density closest to prevalence observed in the literature for HPV 16/18 and the average prevalence of non-vaccine oncogenic HPV (Dunne et al. 2007).

2.5 Comparison of Modelled Estimates of Aggregation Permitting Coexistence to Real-World Data

To assess their plausibility, the values of mi and ci used were compared to estimates of the geo-spatial clustering of cytologically detected low grade cervical lesions (LSIL) derived from Australian data. The Australian states of South Australia (SA) and Victoria report the numbers of LSIL per statistical local area (SLA), and this information is published in the Social Health Atlas of Australia 2011, available online (http://www.publichealth.gov.au/interactive-mapping/). In both SA and Victoria the variance in the number of LSIL occurring per SLA was greater than the mean, indicating over-dispersion and suggesting that a negative binomial distribution could offer a reasonable description of the clustering of LSIL in different geographic locations. The dispersion parameter k of the negative binomial distribution was estimated for counts of LSIL at the state level and for the two state capital cities (Adelaide and Melbourne) using the formula
$$ k=\frac{\bar{x}^2}{(s^2-\bar{x})} $$
(12)
where s2 is the sample variance in the number of LSIL and \(\bar{x}\) is the mean (Southwood 1978; Bliss and Owen 1958; Fisher 1941).

Using the calculated values of k, the value of mi that would be required to fit Eq. (2) to the LSIL data was estimated using the formula mi = 1 + 1/k (Iwao and Kuno 1968). The values of m estimated from the LSIL data were compared to the model estimates mi. Though the model is a model of infection, not disease, the LSIL data was used for comparison as LSIL are mostly attributable a low level HPV infection (Solomon et al. 2002), and are the least severe manifestation of HPV infection for which data of the appropriate level of spatial detail are available.

3 Results

3.1 Single Disease Model: Implications for Vaccination and HPV Ecology

When HPV is modelled as a single infectious agent using the extended SIS model given in Eq. (8), the endemic equilibrium density of infected individuals is dramatically reduced with increasing aggregation of infected individuals. This drives a requirement for increased reproductive efficiency with increasing values of the parameters m and c. Increasing values of the density contagiousness coefficient m reduces the total number of infected individuals present at endemic equilibrium, representing the increasingly patchy distribution of pockets of infection with increasing prevalence (mean crowding increases with mean density). Increasing values of c, representing more infected individuals in the effective ambit (sexual network) of an infected individual increases the required transmission rate α, indicating that an individual in a network with more infected individuals needs to have either a greater number of partners or a more virulent infection to infect more than one other individual (reproductive threshold >1). This is summarised in Table 3 and demonstrated in Fig. 1) and has direct implications for the required efficacy of a future non-type-specific or broad spectrum vaccine.
Table 3

Stability for the endemic equilibrium (\(\acute{I}\)) for the single disease model

Parameters

Equilibrium density

Stability conditions

m = 1, c =  −1

\(\acute{I}=N-\frac{\gamma}{\alpha}\)

\(\frac{\alpha N}{\gamma} >1\)

m = 1, c = 0

\(\acute{I}=N-\frac{\gamma}{\alpha}-1\)

\(\frac{\alpha N}{\gamma} >1+\alpha\)

m = 1, c > 0

\(\acute{I}=N-\frac{\gamma}{\alpha}- c-1\)

\(\frac{\alpha N}{\gamma} >1+\alpha c +\alpha\)

m > 1, c = 0

\(\acute{I}=\frac{N-\frac{\gamma}{\alpha}-1}{m}\)

\(\frac{\alpha N}{\gamma} >1+\alpha\)

m > 1, c > 0

\(\acute{I}=\frac{N-\frac{\gamma}{\alpha}- c-1}{m}\)

\(\frac{\alpha N}{\gamma} >1+\alpha c +\alpha\)

https://static-content.springer.com/image/art%3A10.1007%2Fs10441-012-9161-5/MediaObjects/10441_2012_9161_Fig1_HTML.gif
Fig. 1

Endemic equilibrium densities of infected individuals decrease with increasingly patchy distributions; a uniformly distributed infected individuals (m = 1, c = −1); b poisson distributed infected individuals (m = 1, c = 0); c contagiously distributed infected individuals (m = 2, c = 0). For all trajectories the initial conditions were 120 infected individuals

In a patchily distributed population of infected individuals (m > 1) where a general HPV vaccine was available, a potentially lower proportion \(1-\frac{m}{\frac{\gamma N}{\alpha}}\) of people would need to be vaccinated to eliminate the disease. Note that if m = 1 this is equivalent to the vaccination threshold in the standard SIS model, which assumes uniformaly distributed infected individuals. It follows that a vaccine might be a factor of m less efficacious at preventing transmission in a population with highly clustered infected individuals than when infection is uniformly distributed through the population.

Setting \(\dot{I}=0\) in Eq. (8) and solving for I shows that for all values of m and c, the model possesses two solutions. These occur when the equilibrium density of infected individuals, \(\acute{I},\) is equal to zero (disease free equilibrium) or to \(\frac{\alpha N-\gamma-\alpha c-\alpha)}{\alpha m}\) (endemic equilibrium). Note that the endemic equilibrium density of infected individuals will be reduced by increasing values of m and c required to fit non-uniform distributions of infected individuals: results are summarised for different distributions in Table 3. Assuming a uniform distribution of infected individuals (m = 1, c = −1) \(\acute{I} = N-\frac{\gamma}{\alpha}\) which is the same endemic equilibrium solution as in the basic SIS model (Keeling and Rohani 2008; Wang 2009).

The threshold value to prevent HPV from becoming extinct increases with aggregation of infecteds. If \(\frac{\alpha N}{\gamma} \leq 1+\alpha c + \alpha,\) the system tends towards the disease free equilibrium at the limit. If \(\frac{\alpha N}{\gamma} > 1+\alpha c + \alpha\) the system tends towards the endemic equilibrium density of infected individuals. Assuming a uniform distribution of infected individuals (m = 1, c = −1) the threshold condition reduces to the same as in the normal SIS model in Eq. (4). It is apparent that increasing values of c, representing more infected people in an average infected individual’s sexual network (ambit), make it more difficult for HPV to avoid extinction (see Table 3).

The local stability of the equilibrium solutions of Eq. (8) are the same as in the basic SIS model. In all cases the second derivative evaluated at the endemic equilibrium is negative, indicating a stable endemic equilibrium has been reached. Similarly, in all cases when the second derivative is evaluated at I = 0 its value is positive indicating that the disease free equilibrium is unstable.

3.2 Lotka-Volterra System with Two Groups of HPV

There are four equilibria for the two group model given in Eq. (10): the disease free equilibrium where both groups of HPV are extinct; two competitive exclusion equilibria (Y is endemic and X is extinct or the reverse), and competitive co-existence (both X and Y at below endemic equilibrium). These correspond to the four equilibria in a standard Lotka-Volterra competitive system and their stability conditions are summarised in Table 4. A more detailed analysis is given in the Appendix. If \(\frac{\alpha_X N}{\gamma_X} \leq 1+\alpha_X c_X + \alpha_X\) and \(\frac{\alpha_Y N}{\gamma_Y} \leq 1+\alpha_Y c_Y + \alpha_Y\) the system tends towards the dual extinction equilibria; if \(\frac{\alpha_i N}{\gamma_i} > 1+\alpha_i c_i + \alpha_i\) for one group of HPV i, the system tends towards the equilibrium with group i HPV being endemic. These are the same conditions as required in the one group model in Eq. (8). For both types to be present at competitive co-existence (both X and Y present at the endemic equilibrium) the conditions are not so simple. In particular, the threshold conditions are not the only conditions to maintain this state; the ratio of the two threshold conditions and the magnitude of mi and ci are also important.
Table 4

Equilibria and stability conditions for HPV X (\(\acute{I}_X\)) and HPV Y (\(\acute{I}_Y\)) under different assumptions about clustering

Equilibrium

Fixed point

Stability

Dual extinction

\(\acute{I}_X=0,\acute{I}_Y=0\)

αiN ≤ γi + αici + αi

Exclusion of Y by X

\(\acute{I}_X=\frac{N-\frac{\gamma_X}{\alpha_X}-c_X-1}{m_X}, \acute{I}_Y=0\)

\(\frac{\alpha_Y N}{\gamma_Y} < 1+\alpha_Y c_Y +\alpha_Y\)

Exclusion of X by Y

\(\acute{I}_X0,\acute{I}_Y=\frac{N-\frac{\gamma_Y}{\alpha_Y}- c_Y-1}{m_Y}\)

\(\frac{\alpha_X N}{\gamma_X} < 1+\alpha_X c_X +\alpha_X\)

Competitive coexistence

\(0 > \acute{I}_X < \frac{N-\frac{\gamma_X}{\alpha_X}-c_X - 1}{m_X},\)

\({\frac{\alpha_{X} N}{\gamma_X}}>1+\alpha_X c_X +\alpha_X, \)

 

\(0 > \acute{I}_Y < \frac{N-\frac{\gamma_Y}{\alpha_Y} -{c}_Y - 1}{m_Y} \)

\(\frac{\alpha_Y N}{\gamma_Y} > 1 +\alpha_Y c_Y +\alpha_Y, \)

  

mX or mY > 1, cXcY ≥ 0

If both groups of HPV infected individuals become equally aggregated with changing density (mX = mY) the larger of the two thresholds \(\frac{\alpha_i N}{\gamma_i} > 1+\alpha_i c_i + \alpha_i\) dominates and group i excludes the other group, as in the normal Lotka-Volterra system. Under normal circumstances, where independence of transmission is assumed as in Eq. (10), competitive co-existence should not be possible, but here competitive coexistence can occur provided the less prevalent HPV types can infect individuals who are less uniformly dispersed throughout the susceptible population. By doing this however, they make themselves more vulnerable to potential vaccine effects as noted above and described below. As in standard Lotka-Volterra, in the competitive coexistence equilibrium the total number of infections due to both HPV X and Y should be greater than the number infected at an equilibrium with only HPV X or HPV Y present (see Fig. 2 and the Appendix for more information).
https://static-content.springer.com/image/art%3A10.1007%2Fs10441-012-9161-5/MediaObjects/10441_2012_9161_Fig2_HTML.gif
Fig. 2

The point of competitive coexistence equilibrium (the intersection of the two nullclines) lies above the line connecting the single carrying carrying capacities (the K′ connector), showing that the sum of IX and IY is greater than the number of infecteds present in a system with only IX or IY present

3.3 Simulations: HPV Vaccination

It was possible to maintain competitive coexistence (both HPV X and Y present) given the values of αi and γi in Table 2 whilst using a range of values of the clustering parameters mi and ci, provided that mY > mX. The parameter values mX = 3.0, cX = 2, mY = 3.9 and cY = 3.0 resulted in prevalence below the one-type endemic equilibrium (as is expected at the competitive coexistence equilibrium) but closest to the target prevalences taken from the literature.

HPV X halved in prevalence by 2020 under the assumption of no cross protection and vaccination starting in 2007, eventually being eliminated (see Fig. 3); this is an indication that vaccination lowered transmission so that \(\frac{\alpha_X N}{\gamma_X} \leq 1+\alpha_X c_X + \alpha_X\) (see Table 4 line 3), meaning that the system tended towards the extinction equilibrium at the limit. Under the same assumption of zero cross-protection, HPV Y increased in prevalence as HPV X decreased, excluding HPV X by 2050. This is because at the pre-vaccination competitive coexistence equilibrium, both the number of individuals infected with HPV X and the number of individuals infected with HPV Y are less than they would be if they were able to exclude the other HPV group (see Table 4 line 4). The requirement that \(\frac{\alpha_X N}{\gamma_X} > 1+\alpha_X c_X + \alpha_X\) and \(\frac{\alpha_Y N}{\gamma_Y} > 1+\alpha_Y c_Y + \alpha_Y\) to maintain competitive coexistence with both HPV X and Y at below their maximum level is violated when vaccination causes \(\frac{\alpha_X N}{\gamma_X} \leq 1+\alpha_X c_X + \alpha_X\) shifting the system to a competitive exclusion equilibrium.
https://static-content.springer.com/image/art%3A10.1007%2Fs10441-012-9161-5/MediaObjects/10441_2012_9161_Fig3_HTML.gif
Fig. 3

Numerical results from Eq. (10) under different assumptions about cross-protection using parameters from Table 2. When cross-protection against non-vaccine types (HPV Y) exists, both types can eventually be eliminated even though the vaccine cross-protection against HPV Y is low (ψY in 2 of 46.2 %). This is because a less effective vaccine can produce elimination in the presence of aggregation of infected individuals (see analysis of the single disease model). In the absence of cross protection non-vaccine types are able to achieve their maximum equilibrium density as HPV X tends to extinction

Where vaccination protects to some degree (cross-protection) against infection with HPV Y, both HPV X and Y tend towards extinction, indicating that for i vaccination led to the condition \(\frac{\alpha_i N}{\gamma_i} \leq 1+\alpha_i c_i + \alpha_i\). HPV Y declined in prevalence more rapidly than HPV X, even though vaccine efficacy against HPV Y was lower (46 % versus 84 %). This is explained by the patchier distribution of Y (mY = 3.9 > mX = 3.0), and confirms the analysis of the one type model (see Sect. 3.1) suggesting that a lower efficacy vaccine would be required to cause elimination in a population where infected individuals were very clustered. Thus in the presence of a vaccine that affects both HPV X and Y, the less prevalent and more clustered HPV Y declines in prevalence more rapidly.

An important implication of these simulations is that competitive coexistence is unlikely to be maintained in the presence of vaccination. In the case of vaccination affecting only one group of HPV, the model converges to the equilibrium with only the other HPV group present. In the presence of cross-protection, the reduction in the basic reproduction number of both HPV X and Y can causes both HPV groups to become extinct if the non-target HPV is more patchily distributed.

3.4 Comparison of Modelled Clustering to Real World Clustering

The values of the dispersion parameter k inferred from the values of mX and mY for the modelled HPV X and Y were both close to zero, suggesting that individuals infected with either type were very clustered in the modelled population. Estimated values of k describing the clustering of LSIL cases in SA and Victoria were lowest when data from SA and Victoria were combined, but on the other hand were very high (k > 2) when metropolitan areas were examined in isolation (see Table 5). The dispersion parameters, k, inferred from model parameters were most similar to the values estimated from the data derived from larger geographical areas. This indicates that the parameter values in Table 2 can be most plausibly used to model the patchiness in the distribution of HPV infection for a general population covering a large area where geographic variability in the incidence of HPV related conditions is high.
Table 5

Comparison of model and data based estimates of the density contagiousness coefficient (m) and the dispersion parameter (k) of the negative binomial distribution

Source of estimate

m

k

Model—HPV X

3.00

0.50

Model—HPV Y

3.90

0.33

Data—All areas

2.48

0.75

Data—All Victoria

2.00

0.99

Data— metropolitan Melbourne

1.23

4.36

Data—SA alone

1.67

1.48

Data—metropolitan Adelaide

1.25

3.67

As real world data (LSIL—counts of cervical low-grade intraepithelial lesions) encompass more geographically heterogeneous areas, the values of m increase and k decrease, indicating a patchier distribution of LSIL and approaching the values required by the model to allow for coexistence of multiple HPV types

4 Discussion

The single disease model does not differ materially from the standard compartmental SIS model, but rather extends it. The parameter c approximates the number of infected individuals in the ambit (broadly interpreted as social space) of another infected individual, and the parameter m models the changes in the average number of infected individuals per infected individual with increasing prevalence. The average number of infected individuals per infected individual per area corresponds to Lloyd’s mean crowding index, an ecological measure of aggregation. The use of these parameters models a scenario where some HPV infected individuals are more likely than others to have contact with other infected individuals, because HPV prevalence varies spatially. The implications of this model for HPV are that if a vaccine became available that did not differentiate between HPV types it could have a lower efficacy to reduce the prevalence of HPV if it was demonstrated that individuals infected with HPV regardless of type were highly spatially or temporally clustered. An analysis of variability in cytologically detected LSIL counts from two mainland Australian states (SA and Victoria) was conducted, without taking into account what HPV types might have caused the lesions. This suggested that LSIL incidence was highly patchy (fitted k of the negative binomial distribution less than 2), meaning that analytical results from the one - type model are potentially important for understanding the possible effectiveness of a future broad spectrum vaccine. Because the parameters in our model can be linked to known probability distributions, such as the negative binomial, the single disease model in this paper links traditional deterministic compartmental models to probability distributions used in the spatial analysis of disease clusters.

As well as linking compartmental modelling with spatial modelling, the model of aggregation of infected individuals established in this paper will be particularly useful for deterministic models used for exploring the theory and ecology of STIs. Models used for these purposes have provided important insights whilst being based on quite broad assumptions and no or little data (e.g. Castillo-Chavez et al. 1999). More recent papers that have provided important insights about STIs have sometimes made the broad assumption of homogeneous mixing (Elbasha and Galvani 2005; Swinton et al. 1992) but homogeneous mixing should be a last resort because it is an unrealistic assumption for STIs (Anderson and May 1991; Keeling and Rohani 2008). The model used in this paper presents a simple but more realistic alternative to homogeneous mixing for theoretical modelling by introducing heterogeneity via the parameters m and c.

Whilst the single disease model closely resembles the logistic SIS model, the two-group HPV model closely resembles the Lotka-Volterra model developed by Kuno (1988). It differs from this model in the assumption of completely independent transmission of HPV X and Y, whereas Kuno allowed for direct competition between two animal species in his system, and by the incorporation of the assumption that infected individuals are part of a larger population of N individuals. Its equilibria correspond to the equilibria in the standard Lotka-Volterra competitive system, with similar stability requirements. Because independent transmission is assumed (all HPV types use the same resource in the same way), according to ecological theory, competitive exclusion of the less prevalent HPV types should be assured, so the existence of the competitive coexistence equilibrium is significant. In Kuno’s model, a less effective competitor can coexist with a more efficient opponent if it has a patchier distribution (Kuno 1988). In the model presented here, the existence of the competitive coexistence equilibrium is permitted by greater aggregation of individuals infected with the less prevalent HPV type. Because this paper models infectious disease transmission rather than reproduction in animal populations, however, the implications of the existence of a competitive coexistence equilibrium are different to Kuno’s model, particularly when vaccination is considered.

The model predicts that in the presence of a vaccine that confers cross-protection against HPV Y (non-vaccine HPV) in addition to primary protection against HPV X (vaccine HPV) both groups may become extinct. On the other hand, if a vaccine offers no protection against HPV Y, non-vaccine HPV Y will reach endemic equilibrium as HPV X approaches extinction due to vaccination. These results will hold provided that the degree of vaccine efficacy (and cross-protection, when present) are sufficient to reduce the ratio of transmission and recovery rates below the threshold values required for endemicity.

The major limitations of both versions of the model are their simplicity; they are very much first steps at applying the concepts of clustering experimented with by Kuno in a deterministic framework to infectious disease modelling. Despite their simplicity, however, parameters from and equilibrium solutions to the models (because the two-group model simplifies to the single disease model for each type) produced appropriate prevalence of modelled groups of HPV types. The stratification of HPV types into vaccine and non-vaccine types is a simplification, but a necessary one to demonstrate the characteristics of the novel model presented in this paper algebraically. The basic principles demonstrated can be extrapolated to more complicated scenarios.

The values of the density contagiousness coefficient (m) that produced appropriate prevalence in simulations were >1, suggesting that patchy distributions of infectives might be necessary to permit coexistence in nature (see Table 1). Estimates of the negative binomial parameter k arising from the values of m used were much less than 1, indicating a high degree of patchiness. Analysis of LSIL data from SA and Victoria showed that a negative binomial distribution with a value of k < 1 would also describe the counts of LSIL across these two states. Therefore, the simple models presented could provide plausible underlying mechanisms for some of the complex variation in low-level HPV infection observed in nature.

The suggestion that the negative binomial distribution with a very low value of k (indicating a high degree of aggregation) could describe the distribution of HPV infected individuals has implications for HPV surveillance. Where a phenomenon is overdispersed it is much more difficult to accurately estimate its prevalence (Iwao and Kuno 1968; Southwood 1978). It has been suggested that using the mean crowding - mean density relationship parameters m and c can account for patchy distributions and produce better estimates of required sample size in simple random and sequential sampling (Iwao and Kuno 1968). Future research could examine ways in which these parameters, as used in the mechanistic model in this paper, could be used to inform requirements for HPV surveillance to account for unusually high prevalence of HPV amongst some spatially distinct areas (e.g. Orozco-Colin et al. 2010).

Another major assumption of the model is the use of the SIS framework. Biological and epidemiological justifications for using this model are given in the introduction to this paper, and there is some uncertainty over whether an SIS, SIRS or SIR model is most appropriate for HPV. Aside from these reasonable justifications, the use of the SIS model has some advantages that deserve mentioning. Mathematically, the SIS model can be reduced to a simple logistic equation, a well studied function that will be familiar to readers of this paper from either ecology or infectious disease modelling. Similarly, the expanded Lotka Volterra system used to model HPV X and Y simply consists of two extended logistic equations. These attributes allow the results and equations presented in this paper to be more easily compared and related to the logistic and Lotka-Volterra equations of Kuno, which in part inspired this work, than would be the case if an SIR framework was used. Use of the SIS model allows for easy generalisation of the model established in this paper to other STIs (such as gonorrhoea and Chlamydia) which often have short or non-existent immune periods (Castillo-Chavez et al. 1999; Hyman and Li 1997) and to diseases such as HIV where infection is lifelong by setting recovery rates to zero (Hilker et al. 2007). The models in this work can certainly be extended to an SIR and SIRS framework, but this is beyond the scope of the introductory model presented for the first time in this paper and will be addressd in a future publication.

5 Conclusion

This paper establishes a novel model for the transmission of single or multiple agents that links standard compartmental models with ecological theory that dscribes the complex spatial patterns of disease transmission observed in nature. The model is used to study the transmission of vaccine and non-vaccine HPV. Importantly, it was found that different degrees of aggregation of infected individuals, as represented by the mean crowding—mean density relationship, could explain the coexistence of multiple HPV types, with patchier distributions of less virulent types allowing them to coexist with more common and more virulent types. Numerical and algebraic analysis of the models showed that a relatively low efficacy vaccine could reduce HPV prevalence where clustering of infected individuals is high. By extension, because lower prevalence types need to be distributed more patchily to coexist with more prevalent types, this might result in the eradication of non-vaccine types in the presence of a modest degree of cross-protection conferred by a vaccine targeting other types. Numerical simulations also suggested that in the absence of such cross protection, the system tends towards a competitive exclusion equilibrium with non-vaccine types present at higher levels than in the competitive coexistence equilibrium.

The dispersion parameters in the models presented here are easily related to the parameters of probability distributions that describe the heterogeneous distribution of infected individuals in the real world. Comparison to the degree of clustering observed in HPV related disease incidence over two Australian states shows that the models presented may be realistic for larger populations covering geographically heterogeneous areas. The models presented are therefore a potentially useful advance in HPV modelling, suggesting that the spatial distribution of HPV may on its own facilitate the coexistence of multiple HPV types, and may affect the transmission of HPV and the requirements for an HPV vaccination programme.

Acknowledgments

Mr Edward K Waters would like to thank Drs David G Regan, David J Philp and Andrew J Hamilton for their helpful advice and support in bringing this work to publication. He would also like to thank Dr Lia Hemerik, associate editor for Acta Biotheoretica for her helpful comments. This paper was funded from the following sources: the Australian Government Department of Health and Ageing; Australian Research Council (ARC) Linkage Project (LP0883831) which included contributions from CSL Ltd and Victorian Cytology Service Inc. The views expressed in this publication do not necessarily represent the position of the Australian Government.

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© Springer Science+Business Media B.V. 2012