Dans le modèle mathématique SIR, qui décrit la dissémination d'une maladie par contact direct dans une population, il est possible d'obtenir une dynamique cyclique lorsque l'on renouvelle continûment les individus susceptibles. Cette dynamique est cependant amortie vers un point d'équilibre stable. L'inclusion de distributions réalistes pour les durées de latence ou infectieuse, comme celle de composantes stochastiques, fragilise cette propension à l'oscillation.

En revanche, dès lors que l'on étend la modélisation pour incorporer plusieurs populations en interaction, des dynamiques source–puits peuvent voir le jour et expliquer la survenue d'épidémies. La prise en compte des pathogènes en compétition ou en interaction, par exemple plusieurs souches ou une diversité virale, permet également de voir apparaître des cycles plus ou moins réguliers, voire des trajectoires chaotiques.

Dans des réseaux complexes, l'existence de cycles est encore peu abordée, bien qu'il ait été décrit que, dans certaines conditions de croissance du réseau, des cycles dans l'incidence pourraient être présents.

L'évolution démographique propre peut être motrice dans l'existence de cycles, par exemple en imposant un remplacement de la population susceptible lent par rapport à la dynamique de la maladie.

La régularité du forçage saisonnier suggère également que la relation hôte–pathogène ait pu en être affectée. Effectivement, nombre de maladies transmissibles ont un cycle saisonnier, que celui-ci soit induit par les vacances scolaires ou par d'autres modifications, même ténues, des comportements. La résistance maximale face à l'invasion par un autre pathogène mènerait à cependant à exacerber les variations saisonnières, voire à perdre la synchronisation pour adopter un cycle pluriannuel. L'influence du climat est également démontrée, qu'il crée les conditions d'existence par un vecteur ou par des modalités encore indéterminées.

Il n'existe pas encore de théorie unique pour expliquer la totalité des manifestations cycliques et récurrentes des épidémies. La prise en compte des multiples acteurs dans des systèmes réalistes sera source de travaux dans les prochaines années.

In the famous SIR model, the human population is split in three subgroups: (S)usceptibles to infection, (I)nfectious and (R)emoved by cure or death, with the following differential equations describing contamination, cure and removal [20]:

The occurrence of oscillations is a direct consequence of the form of the interaction between susceptible and infected individuals (i.e. the law of mass action) in a finite population, leading to a critical threshold in the number of susceptible individuals below which the incidence of new cases fails to compensate those removed; the dynamics of births, which replenishes the susceptible compartment with time and allow for renewed increase in prevalence. This intrinsic periodic behaviour was first thought to account fully for recurrent epidemics [22], all the more because almost yearly periods may be obtained with reasonable parameter values. However, the oscillations damp with time and fail to explain sustained seasonal cycles [23]. Inclusion of realistic distributions for the duration of the infectious period (in the SIR model, an exponential distribution is assumed) shows that, while damping may take considerable time, it is nevertheless present [24]. Introducing stochastic components changes the behaviour of models with respect to perpetuation in an essential way, since there is a possibility of disease extinction [25]. A stochastic treatment to the SIR model leads to undamped oscillations in incidence, but the disease goes extinct with probability 1 [26]. Furthermore, taking into account realistic distributions for the epidemic period even hasten this process, as the troughs are more pronounced [24]. Therefore, although the simple SIR model provides for both perpetuation and recurrence, it fails short to provide a satisfying explanation to observed infectious disease dynamics.

Having multiple coupled populations somewhat alleviates the problem of extinction. Measles in the UK has provided ample evidence of source dynamics explaining recurrences in isolated populations. In London, for example, measles never goes extinct, since the size of the children population is large enough to perpetuate the disease, with a simple seasonal pattern (see also extrinsic factors below). On the contrary, small places show scattered epidemics, interspersed by irregular time intervals [27]. These ‘meta-population’ dynamics may have vastly diverging effects depending on the disease. For example, measles dynamics tends to be synchronized through these interactions, while whooping cough, with an infectious period approximately thrice longer, is desynchronized [28]: this is explained by a subtle interplay between the duration of the infectious period, the coupling between populations and seasonal forcing. Perpetuation may be more likely with desynchronized epidemics in a meta-population approach.

A very different meta-population dynamics is seen in vector-borne diseases. In dengue and malaria, the pathogen is transmitted by mosquitoes. The disease dynamics is therefore subject to the population dynamics of both species. Interestingly, this translated into an almost three-year interepidemic period for both diseases, in very good agreement with the prediction of a SEIR model [29]. More generally, multi-host infectious diseases may be associated with original perpetuation and recurrence characteristics. This is so because multi-host pathogens have generally reservoir hosts in which they persist with little clinical manifestation, and cause outbreaks outside this reservoir [30]. Even if malaria was one of the first diseases submitted to a mathematical treatment, there is still considerable work to gain further insight into the emerging zoonotic threat [31].

In the preceding, a limited view of the interaction between pathogens and humans was adopted, taking no notice of diversity in the pathogen population. This may be true for diseases like measles, where little viral diversity exist. However, the influenza virus effectively escapes immunity by constant changes at key antigenic sites [32], causing an effective influx of susceptible individuals. The joint study of pathogen diversity and epidemic transmission is the subject of much current research [33].

When several strains of a pathogen co-circulate, we may expect independent behaviour and coexistence at levels determined by the relative fitness of each strain. However, this does not account for potential interference between strains: competing for susceptible hosts [34], reducing susceptibility to infection by genetically ‘close’ strains [35] or transmissibility after an initial infection [36]. With these refinements, unconstrained coexistence may be the rule at low levels of cross immunity, but only non-interfering strains may coexist at very high levels of cross immunity. Furthermore, in a range of intermediate levels of cross immunity, periodic or chaotic cycling between strains is possible [35–38]. In this case, the secular variation seen in the prevalence of the strains of some pathogens may result from complex interplay due to cross immunity. Examples conforming to these predictions may include lineages of N. meningitidis and serotypes of group-A streptococcus [39]. If the co-circulating strains have different pathogenicity, periodic or recurrent outbreaks will occur [40].

It is noteworthy that such rich dynamics may also be observed in a very particular case of interference known as ‘antibody-dependent enhancement’ [41,42]. In this case, previous infection does not act to reduce susceptibility or transmission to a new infection, but on the contrary may increase transmissibility during a second infection and infectiousness. This has been proposed in the case of dengue (although it is debated [40]), and for other viruses as well (among others coronaviruses of the SARS type [43]). Modelling shows that antibody enhancement may lead to undamped cyclical epidemics with different strains, much like more classical interactions.

Diversity in pathogens is, like transmission, a dynamic process, and should be considered in a ‘phylodynamics’ approach [33]. Models were proposed in this respect where strain diversity changes with time given a constant mutation rate, accompanied by a decrease in cross immunity with increasing genetic distance [44,45]. In a detailed analysis of the behaviour of this model, it appears that combining short infectious duration with long-lived immunity induced by infection (total immunity against the same strain, and partial immunity to other strains) lead to the occurrence of renewed outbreaks with time, where at each time the dominant strain is different from that in earlier epidemics [45]. Calibrating the model to reflect influenza, it was found that the simulated phylodynamics were reminiscent of that observed in real influenza [44]: several closely related genetic strains co-circulating at any given time, with a continuous drift [46].

An assumption of the basic SIR model is that of ‘homogeneous mixing’, whereby all infectious may contact all the susceptible individuals. This also means that little variation is expected in the number of secondary cases borne from two different infectious cases, since the effective contacts take the form of a completely random graph. Studying the impact of heterogeneity in transmission was first addressed by splitting compartments according to individual characteristics. The impact of such structuring was manifest in the study of sexually transmitted infections (STI), and led to the influential concept of ‘core groups’ having more numerous partners than the rest of the population [47]. These groups were essential to explain the maintenance of STIs in a population where it would have otherwise disappeared.

The recent epidemic of SARS has put forward that in other diseases as well, the number of secondary cases could be highly variable. In fact, most cases were without descent, while few cases had a disproportionately high number of descendants [48]. The so-called ‘superspreaders’ have since then been recognized as fairly common in many communicable diseases, appearing as the norm rather than the exception [49]. While superspreaders may have a definite role in introducing a disease in a population, and be targeted first to prevent its spread, it is not clear as of today how their role would extend in perpetuating a disease.

Superspreaders echo findings from complex network theoretical epidemiology. In those, the topology of contact networks has been changed from completely random to include small world or scale-free properties. A major finding of these studies has been the strong dependence of epidemic spread on network structure [50]. For example, contrary to random networks, infinite power-law networks do not display threshold behaviour for the persistence of a disease, even if a threshold-like phenomenon may be recovered in finite-size networks [51]. Provided that susceptible individuals are introduced or recovered ones lose their immunity, the perpetuation of any disease is theoretically granted in such networks. Indeed, any initial number of infected individuals may lead to large-scale epidemics. Oscillatory dynamics in prevalence may appear in growing networks [52], while in other cases perpetuation exists without periodicity.

The mere size of the population has a profound effect on the perpetuation of diseases. Large populations enable perpetuation, whereas the disease generally disappears in small populations. The critical community size corresponds to the threshold below which perpetuation is not possible [53]. Changing birth rates may profoundly affect the pattern of recurrence of diseases. This is best exemplified for measles epidemics, where sustained annual epidemics require high birth rates. In the UK, decreasing birth rates have been associated with the occurrence of large epidemics every other year [54]. Importantly, external stimulation of the model through the changing birth rates was sufficient for the trajectory to hop between attractors having cycles of various lengths [54]. The same profound effect of birth rates was seen for smallpox. Indeed, the intrinsic period of this disease, in a population of constant size, is 3 to 4 years; however, inspection of past time series show preferentially 5-year cycles [13], or even longer. The explanation of this period lengthening has been linked to extrinsic processes, involving for example the occurrence of famine and high wheat price [13].

Many human activities exhibit a seasonal component, and this may have implications in disease transmission. For example, childhood diseases exhibit marked seasonality, although patterns may change between places. Incidence increases during the whole school year, and drops in the summer. A very good candidate to explain this seasonality pattern is the forcing imposed by school holidays, and it has been repeatedly found that this leads to improved fit to observed data [55]. For diseases with long infectious periods, school-term forcing leads to annual cycles. When the duration of the infectious period decreases, as well as the effective number of secondary cases, longer cycles may occur [56]. Large differences in contact rates also lead to longer cycles. On the contrary, most respiratory diseases exhibit a seasonal pattern with incidence increase during the cold season. Importantly, this is not a characteristic of the disease only, as it is well documented that the seasonal component increases with latitude: tropical countries experience similar attack rates as temperate countries, only the epidemic period is diffuse, instead of concentrated during a few weeks [5].

Up to now, no single theory has been sufficient to explain all seasonal patterns. First, there may be changes in contact rates as a consequence of crowding in colder seasons. It was recently proposed that even subtle changes in contact rate could lead to large amplitude changes due to resonance between the intrinsic cycle of the disease and that in contact rates [57]. In any case, seasonal forcing in the contact rate in SIR-type models introduces extremely rich dynamics [54]. It has also be suggested, and found in mice, that seasonal changes in immunity of the host were at stake [5].

Given that there is so much evidence for seasonal changes, one may wonder on the advantage of seasonal variation in incidence for perpetuation of a disease. We must recall first that the dynamics of the SIR model intrinsically leads to oscillations in the number of susceptible individuals. If there is a good correlation in the intrinsic cycle of the disease and seasonality in contact rates (i.e. if the pathogen has been selected so), the pathogen becomes more firmly established as the magnitude of seasonal changes increases. However, the same argument leads to favour an increase in duration between recurrences of the disease, and ultimately an escape from the seasonal behaviour [58]. Further theoretical results are required to resolve these apparent paradoxes.

It is well known that the environment plays a key role in the distribution of human diseases [59], through climatic factors. Could global processes be associated with the occurrence of disease in particular places? While not causal determinants, global processes could favour environmental conditions leading to recurrence or perpetuation. El Niño occurs irregularly (cycles between two and seven years). The El Niño cycle is associated with increased risks of some of the diseases transmitted by mosquitoes, such as malaria, dengue, and Rift Valley fever [60], but also of cholera [61]. Candidate mechanisms linked to the effect of environment on the vector, or on water movement have been proposed. More unexpectedly, the cold event of the El Niño oscillation (or La Niña) is also associated with higher influenza mortality [62].

Other environmental factors, like vegetation indices, have been linked to the occurrence of vector-borne diseases like the Rift Valley fever disease [63]. These indicators would be extremely helpful for predicting and organizing containment measures: this is a way forward for the coming years [64].