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

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.

Appendix

Here the local stability conditions for Eq. (10) and given in Table 4 are derived. The Jacobian is

$$ \left( \begin{array}{cc} \alpha_{X} N - \gamma_{X} -\alpha_{X} I_{Y} - \alpha_{X}(2{m}_{X} I_{X} + {c}_{X} +1) &-\alpha_{X} I_{X}\\ -\alpha_Y I_Y & \alpha_Y N - \gamma_Y -\alpha_Y I_X - \alpha_Y(2{m}_Y I_Y + {c}_Y +1) \end{array} \right) $$

The eigenvalues of the Jacobian at (0,0) are α _X N − γ _X  − α _X c _X  − α _X and α _Y N − γ _Y  − α _Y c _Y  − α _Y . Only if both $\frac{\alpha_{i}{N}}{\gamma_i} \geq 1+alpha_{i}{c_{i}} -\alpha_i$ the equilibrium is locally stable; otherwise it is unstable.

Using the notation $\acute{I}_i$ to represent the exclusion equilibrium density, at (0,$\acute{I}_Y$) HPV X is extinct and the number of individuals infected with HPV Y is at $N-\frac{\alpha_Y}{\gamma_Y}$. According to the competitive exclusion principle this implies that $\acute{I}_X < \acute{I}_Y, \frac{\alpha_X N}{\gamma_X} < \frac{\alpha_Y N}{\gamma_Y}$ and therefore also rα _X N − γ _X  < α _Y N − γ _Y . The equilibrium may be stable if $\frac{\alpha_X N}{\gamma_X} < 1+\alpha_X c_X +\alpha_X,$ that is, it converges to the single disease model. The exclusion equilibrium where HPV Y is extinct and the number of individuals infected with HPV X is at endemic equilibrium will be stable under similar conditions.

When both HPV X and HPV Y are present at equilibrium (competitive coexistence) their equilibrium densities must be below their exclusion equilibrium levels, which reflects the extent to which their level of aggregation limits their reach into the susceptible population. The determinant of the Jacobian at this point is $\alpha_{X}\alpha_{Y}\acute{I}_{X} \acute{I}_{Y}\left(m_{X}m_{Y}-1\right)$. The determinant is >0 when m _X m _Y  > 1, so if m _X and m _Y  > 1 the equilibrium is stable. This suppoerts Kuno’s (1988) conclusion that aggregation, as expressed by values the slope of Iwao’s linear relation >1, can facilitate coexistence.

The partial derivatives and the eigenvalues of the Jacobian also become extremely complicated. It is also instructive to substitute $\acute{I}_i=\frac{N-\frac{\gamma_i}{\alpha_i}-c_i -1}{m_i}$ for all $\acute{I}_i$. These substitutions make it clear that for competitive coexistence to occur, whichever group of HPV is less prevalent must be able to infect individuals who are less uniformly dispersed throughout the susceptible population. For example, if $\acute{I}_X > \acute{I}_Y,m_Y > \frac{\acute{I}_X}{\acute{I}_Y}$. A few other conditions are obvious; α _i N ≤ γ _i  + α _i c _i  + α _i for all i, or else one of the equilibria involving extinction would have been achieved. Under these conditions, provided that all c _i  >  − 1 and all m ≥1, the coexistence equilibrium may be stable. When an equilibrium is stable the point of intersection of the nullclines of the competitive equations lies above the K′ connector - a line connecting the two competitive exclusion equilibria ($I_i=\frac{N-\frac{\gamma_i}{\alpha_i}- c_i-1}{m_i},I_j=0$) (see Table 4, lines 2–3). The K′ connector is so called because K′ is traditional notation for carrying capacity. A consequence of this is that the sum of I _X and I _Y at stable competitive coexistence is greater than the number of infecteds of a single type present at a competitive exclusion equilibria. These results are shown graphically in Fig. 2.