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Multi-pathogen Systems
Introduction
The spread of pathogens is a complex and multifaceted phenomenon, thus epidemiol-ogy has established as an interdisciplinary research field, drawing attention from many scientific areas, including mathematics and computer science. The use of mathematical models has proven valuable in the fight against infectious diseases. The reasons behind this success are multiple. First, epidemic models provide a way to formulate and investi-gate mechanistic hypotheses about how diseases spread at the population scale. Second, mathematical models can make quantitative predictions about the temporal evolution of a disease. Models can thus be used both for epidemic assessment and for guiding public health interventions.
Most mathematical models focus on a single pathogen, ignoring interactions between pathogens. Although this is appropriate in some cases, certain problems require a multi-pathogen/strain perspective. For example, the occurrence of pathogen-pathogen inter-actions can hamper the efficacy of treatment [71, 72, 73] and public health measures such as vaccination [17, 74, 19]. Therefore, it is important to understand pathogen-pathogen interactions and their impact on epidemiological and macro-ecological patterns. This jus-tifies accounting for multiple strains/pathogens and their mutual interactions into math-ematical models.
In this chapter we review some of the most common and well-studied interactions between pathogens. We then discuss the importance of accounting for pathogen in-teractions in epidemiology and public health, highlighting possible biases arising from their neglect. Finally, we introduce compartmental models as well as the fundamental mathematical tools required to describe disease spread. We devote particular attention to the description of additional challenges that arise when accounting for multiple pa-thogen/strains and their mutual interactions.
Interaction mechanisms
As a first approximation we may distinguish between competitive and synergistic inter-actions. Competitive interactions play a fundamental role in shaping pathogen commu-nities [75, 76]. Competition can arise due to limited shared resources or space. Nasal colonization by S. aureus has been shown for example to prevent invasion from other S. aureus strains because of the limited amount of attachment sites [77]. Some pathogens can also harm competitors by producing chemical compounds. S. pneumoniae for example produces oxygen peroxide [78], which is harmful to other commensal bacteria. Finally, some infections confer some degree of cross-immunity against other pathogens/strains, effectively protecting the host from secondary infections [79]. Owing to their antigenic similarity, S. pneumoniae serotypes can elicit serotype-specific cross-immunity [80, 81]. In-fluenza strains also compete for hosts through cross-immunity; indeed, this mechanism is a fundamental driver of recurrent patterns of seasonal influenza [82].
Pathogens may also interact in a synergistic way, facilitating their mutual spread. For example, primary influenza infections can temporarily increase susceptibility to bacte-rial infections by, e.g., S. pneumoniae and Neisseria meningitidis [83, 84, 85]. Within-host mechanisms responsible for such facilitation are multiple, ranging from increased bac-terial adherence to host cells [86] to impairment of host immune defenses [87, 88]. An-other paradigmatic synergistic interaction between pathogens is the one between HIV and Mycobacterium tuberculosis, which appear to fuel each other’s spread in Sub-Saharan Africa [89, 72]. On one hand, HIV exacerbates every aspect of tuberculosis infection, increasing also susceptibility [90]; on the other hand tuberculosis impairs highly active antiretroviral therapy, making it difficult to treat patients co-infected with HIV and Tu-berculosis [91].
In other cases, however, the nature of a given interaction cannot be identified as purely competitive or purely cooperative. Let us consider for example the case of immune-mediated cross-reactions between Dengue serotypes. Primary Dengue infections grant full immunity against the same serotype and short-term protection against other serotypes [92]. However, secondary Dengue infections are much more severe than primary ones, suggesting a complex immunological scenario. Late disease enhancement may be ex-plained by a non-linear association between cross-protection and antibody concentration, according to a mechanism known as antibody-dependent enhancement [93]. As a con-sequence, the nature of cross-reactive interactions between Dengue serotypes shifts from protective to enhancing as antibody count decreases over time.
Besides interactions taking place within hosts, pathogens may also interfere at the population level [94]. Sickness arising from some primary infection is likely to be fol-lowed by convalescence. Owing to the reduced number of contacts, convalescent indi-viduals are less exposed to secondary infections and they can thus be considered as being effectively removed from the pool of susceptibles. This can hamper the spread of other pathogens, as potential hosts are temporarily unavailable. This kind of interference has been suggested to affect apparently unrelated diseases such as measles and pertussis, which happen to affect the same pool of hosts (children in this case) [95]. Fatal diseases may interfere in a similar manner, owing to the increased mortality and quarantine mea-sures, which ultimately result in the depletion of the susceptible pool [96].
Implications for epidemiology and public health
In the previous section we discussed several mechanisms through which two or more pathogens can interact. We show here that interactions between pathogens can affect their mutual spread as well as epidemic assessment and public health projections.
Interactions can affect spatial and temporal spreading patterns. Cross-protection due to previous immunity can prevent multiple diseases to co-circulate within the same pop-ulation; for example, past immunity to chikungunya [97] and Dengue [98] was found to displace or reduce the size of Zika epidemics. Also, cross-protection due to seasonal influenza has been suggested to tighten constraints on the timing of pandemics [99]. In some cases, the depletion of the shared susceptible pool can also induce out-of-phase relationships between diseases and alter disease periodicity [95, 100].
S. pneumoniae, S. aureus and Neisseria gonorrhoeae bacterial populations are character-ized by a multiplicity of strains with varying profiles of antibiotic resistance. Under-standing the mechanisms behind the emergence and maintenance of resistance in bac-terial populations is fundamental to quantify the impact of resistance on public health. This necessarily requires a proper accounting of the ecological factors that shape bacterial ecosystems, including interactions among different strains [13, 14, 15, 16].
A correct interpretation of eco-epidemiological patterns is also fundamental to assess vaccine effectiveness. Vaccines usually target only a subset of circulating strains. The heptavalent pneumococcal vaccine (PCV7), for example, targets only 7 out of the over 90 known pneumococcus serotypes. In such cases, ignoring eventual competitive inter-actions between target and non-target types may lead to an overestimation of vaccine effectiveness. PCV7 resulted for example in replacement of target types by non-target types [17, 18]. Sporadic increases in the prevalence of non-target HPV types have been reported as well [101, 102, 103, 104], although replacement is still matter of debate [105, 106].
The burden of emergent diseases such as Zika may also depend on past exposure to other pathogens. Zika recently emerged in the Pacific area and in South America [8], sharing considerable geographical overlap with Dengue. Cross-reactions between Zika and Dengue suggest that exposure to the latter may modulate Zika outbreaks by either protecting against it or by triggering antibody-dependent enhancement [107, 108, 109, 98].
Modeling multi-pathogen interactions
Compartmental models
Mathematical models describing infectious disease spread date back to 1766 [110]. In his seminal paper, Bernoulli analyzed mortality due to smallpox, discussing the beneficial impact of prevention strategies against the disease. The foundations of modern epidemi-ology were laid by Ross in 1911 [111], who developed the first mathematical model of malaria transmission.
Epidemiological models are often formulated in terms of compartments, i.e. groups of hosts sharing similar properties such as the health status [20]. Upon certain events like, e.g., infection and recovery, individuals transit from one compartment to the other. Within this framework, it is possible to study the temporal evolution of a disease by tracking the number of individuals within each compartment.
The Susceptible-Infected-Susceptible (SIS) model includes just two categories of indi-viduals, namely susceptibles (S) and infected (I) (Fig. 1.1 A). The SIS model is character-ized by two elementary reactions: S + !Ib I + I , the first reaction refers to the infection of a susceptible upon contact with an infected, whereas the second refers to the recovery of an infected into a susceptible. Infection and recovery occur at rates b and m respectively. If we assume that every individual contacts, on average, ¯ other individuals chosen fully at random, the evolution equation for the k density of infected individuals rI (t) is given by: r˙I = ¯ (1.2) mrI + bk(1 rI )rI , the two terms on the right side represent recovery and infection respectively. Notice that, in the case of a closed population, a single variable, namely the density of infected r I , completely characterizes the state of the system. Dynamical trajectories arising from Eq. (1.2) settle around different values as the control parameter R0 = b/m (the basic re-productive number) is varied. By setting r˙I = 0 in Eq. (1.2), we obtain two possible fixed points, namely rI = 0 and rI = 1 R0 1. The first solution corresponds to a disease-free state, whereas the second one corresponds to an endemic state with stationary preva-lence. If R0 < 1 the disease dies out immediately, whereas if R0 > 1 it becomes endemic. At R0 = 1 the model thus undergoes a dynamical transition which changes its qualitative behavior. R0 has an important epidemiological interpretation: it is the average number of secondary cases originating from a single infected seed in a fully susceptible population.
There is a connection between the SIS model and the theory of critical phenomena in statistical physics. More precisely, the disease-free and endemic states can be regarded as two distinct phases of a physical system such as a ferromagnet. According to this analogy, the critical point at R0 = 1 can be regarded as a phase transition. Furthermore, because total prevalence varies continuously at the transition, a physicist would denote this phase transition as continuous or “second order ».
For some diseases, recovery grants lifelong immunity, preventing re-infection by the same pathogen. This situation is described by the Susceptible-Infected-Recovered (SIR) model (Fig. 1.1 B), where infected individuals transit to the recovered (R) compartment. The SIRS model describes yet another scenario, where immunity wanes over time (Fig. 1.1 C).
FIGURE 1.1: Elementary compartmental models. Transition schemes for (A) the Susceptible-Infected-Susceptible (SIS) model, (B) the Susceptible-Infected-Recovered (SIR) model and (C) the Susceptible-Infected-Recovered-Susceptible (SIRS) model.
Compartmental models that are formulated in terms of deterministic ordinary differ-ential equations (ODE) such as Eq. (1.2) rely on a few critical assumptions. First, they completely ignore stochasticity arising from transmission, recovery and demographic processes. Nonetheless, stochastic fluctuations are usually negligible in the limit of large population size (their typical magnitude is inversely proportional to population size). Thus, the deterministic ODE approach is expected to perform well for large populations. Second, they assume a homogeneously-mixed population where every individual has the same probability to meet everyone else. Within the homogeneous mixing framework, the rate of new infected cases is given by the mass-action law, as if individuals were inter-acting chemical molecules in a well-stirred vessel. As we will see in the next chapter, the homogeneous mixing hypothesis is a crude approximation of real contact patterns.
We can use compartmental models to describe multi-pathogen/strain systems as well. This usually requires additional compartments. Let us consider the case of NS mutually-excluding strains, each following SIS dynamics. Here, the number of compartments scales linearly with NS since we have just one infected compartment per strain (plus a compartment for susceptible individuals). In principle, this model may be characterized by 2NS parameters, namely the NS infection rates bi and the NS recovery rates mi. Such model may be used as a basis for more complex models. In Fig. 1.2 we show a variant of the previous model, based on SIR dynamics instead of SIS, where infected compartments are coupled to each other. Transitions of the kind Ii ! Ij can model a whole range of pro-cesses, including mutations and superinfection (new infections can clear previous ones). In the original formulation of this model [112], transitions form a nested pattern where Ii can mutate into Ij only if j > i. Here, increasing strain number i may be associated, for example, to increasing levels of drug resistance.
If recovery grants some degree of cross-immunity, one should in principle track each host’s immune status as well as the set of past exposures [113]. Although exact, this approach is not amenable to analytical techniques as it requires monitoring the state of each individual. An alternative is to adopt either a status-based or a history-based ap-proach [114, 115, 116]. In the former case, one defines compartments which track the set of strains a host is immune to, whereas in the second case one subdivides individuals according to their set of past infections. While history-based models usually require a large number of variables when NS is large – typically 2NS –, certain status-based need much less variables in order to be fully specified. For example, authors in [117] were able to derive a status-based model that requires just 2NS state variables. This model is based on the assumptions that only a fraction of individuals successfully mount an immune response upon infection and that immunity results in reduced transmissibility.
Ecological insights from multi-strain models
Models of multiple, interacting strains or pathogens are key to address a wealth of eco-logical and epidemiological questions. Under which conditions do two or more interact-ing species co-exist within a population? How common is a species relative to others? The problem of co-existence is a central one in disease ecology and a highly non-trivial one. Indeed, simple models of competing strains usually fail to predict co-existence [118, 13]. Rather, they predict competitive exclusion, i.e. total domination by the fittest pa-thogen species or by the one with the largest R0. However, observed co-circulation pat-terns of, e.g., drug-sensible and drug-resistant strains in S. pneumoniae [119, 120, 121, 122] and S. aureus [123, 124] populations, suggest the existence of mechanisms promot-ing population-level co-existence. Mathematical models have shed light on a few pos-sible such mechanisms. Some mechanisms favor the partitioning of strains into non-overlapping niches, thus minimizing competition. These include, for example, reduced mixing between host sub-groups [22, 23, 24, 25] and heterogeneity in duration of car-riage [26]. In some cases, cross-immunity can also promote co-existence by neutralizing fitness differences, curtailing the competitive advantage of certain strains. For example, non-specific cross-immunity, built through multiple, consecutive carriage events, and in-terference by Haemophilus influenzae have been suggested to promote diversity in S. pneu-moniae serotypes by reducing differences in carriage duration [27, 28].
Multi-strain models shed light on the emergence of invading pathogens. Indeed, new pathogens may appear in a population as a result of external importations, spillover from an animal reservoir or mutations. In all these cases, emergence may be affected not only by environmental and host-related factors, but also by interference with resi-dent pathogens. Multi-strain mathematical models have been used to understand for instance the emergence of antibiotic resistance, providing insights about factors leading to co-existence of sensitive and resistant strains as well as their frequencies under differ-ent treatment regimes [23]. Models also allowed studying the probability of reassortant influenza strains developing into a global pandemic, the frequency of replacement of circulating seasonal strains and the timing of pandemic emergence [125, 115, 126, 99].
Insights from multi-strain models can also help the design of public health interven-tions. Multi-strain models can be employed for example to investigate the long-term impact of vaccination on a pathogen population. In particular, mathematical models can be used to predict post-vaccination size and composition of a pathogen population in order to assess the risk of strain replacement [127, 128, 19].
Open challenges in modelling multi-strain dynamics
The diversity of multi-strain interactions opens up for several theoretical challenges. One problem of paramount importance concerns the link between within-host dynamics and population-level models. In fact, while within-host processes drive the outcome of strains’ interactions, accounting for the full complexity entailing such processes typically requires high-dimensional models that hardly scale to more than a few pathogenic vari-ants. Thus, in order to allow for tractable models, researchers make simplifying assump-tions about disease interactions. The challenge is then to define models that represent a compromise between tractability and realism [129].
For example, models dealing with strains that confer only partial cross-immunity should specify how past infections affect the outcome of future exposure and/or trans-mission events and how immunity is built over time [114, 117, 115, 116]. As already discussed, a full description of such a system would amount to monitoring both the im-mune status and history of each host [113]. Here, model reduction is possible only under specific assumptions about strain interactions.
Not all interaction mechanisms have drawn equal attention in the literature. In fact, because interactions among strains are typically based on assumptions, most works have focused so far on just a few outcomes of multiple infections, notably co-infection and super-infection. Authors in [130, 131] have shown however that, even in the case of just two co-circulating pathogens, within-host dynamics can give rise to a large number of outcomes after simultaneous or sequential inoculations with different pathogens. The population-level impact of such infection patterns remains largely unexplored and thus calls for further research work.
Authors in [129] have also pointed out the importance of incorporating heterogeneities in host population structure within multi-strain models. In this thesis we elaborate on this aspect and make extensive use of individual-based and spatially-structured mod-els in order to assess the impact of the heterogeneous host behaviour on the ecology of interacting strains.
Epidemics on Networks
Introduction
Compartmental models are pervasive in mathematical epidemiology [21, 20]. As seen in Chapter 1, this mathematical framework allows fundamental understandings about factors driving epidemic spread and enables the investigation of outbreaks and the mod-eling of incidence data for a wide range of diseases [132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145]. However, compartmental models make several simplifying assumptions, e.g. homogeneous mixing (Fig. 2.1 A), which may represent a limitation in some cases.
Network epidemiology provides the right framework to account for the complex-ity entailed by contact structure and epidemic dynamics. Within this framework, hosts are represented as nodes in a network while their mutual connections are represented by edges (Fig. 2.1 B). During recent years we have witnessed significant advances in network epidemiology, supported by an increasing amount of data and theoretical ad-vances [29]. Advanced technology has improved our ability to track individual activity, yielding a high throughput of data about different kinds of social interactions, e.g. face-to-face proximity [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42] and sexual encounters [43, 44, 45], and human movements [46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. Theoretical development has allowed dealing with several types of heterogeneities emerging from these data, as well as novel frameworks to model complex network architectures, e.g. multi-layer net-works [146, 147, 148, 149, 150, 151]. More recently, the availability of time-stamped con-tact data sparked substantial interest in temporal networks, where nodes and edges vary in time (Fig. 2.1 C) [152, 153, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 154, 155, 156, 157, 158].
At the same time, the theory of epidemic processes on networks has progressed as well [29]. Several mathematical frameworks have been proposed [159, 160, 161, 162, 163, 164], allowing for a comprehensive understanding of the role of network structure and dynamics on epidemic spread. Some of the most important analytical results concern the epidemic threshold [165, 166, 160, 167], prevalence [161, 168, 169, 170, 171, 172, 173] and immunization protocols [174, 175, 176, 177, 178, 179]. Several works have addressed the dynamics of interacting pathogens on networked populations. Particular attention has been devoted to a small set of interactions, namely cross-immunity [57, 58, 59], mutual exclusion [60, 63, 64] and increased susceptibility [65, 66, 67, 68], and different network substrates have been considered, including multi-layer [69, 70] and metapopulation net-works [61, 62]. Research work in this direction is however at the beginning. Indeed, the majority of works deal with just two pathogens that either compete or cooperate.
This chapter is organized as follows: in Section 2.2 we give an overview of main sources of networked data that are relevant for epidemic spread. In Section 2.3 we in-troduce basic network concepts and tools. In Section 2.4 we turn our attention to the mathematical description of epidemic dynamics on networks. Finally, in Section 2.5 we review the literature about multi-strain/pathogen dynamics on networks, describing the outcome of competitive and cooperative interactions.
Network data in epidemiology
Early studies of human contacts originated in the context of social science in order to better understand human behavior and social relationships [180]. Contact data from dif-ferent studies may display different levels of detail. For example, a few studies yield only summary statistics about respondents, e.g. their number of individual contacts [181]. In other studies, respondents are asked to report also about characteristics of their acquain-tances. Age information, for instance, is often reported [182, 183, 184], and may be used to inform mixing rates in age-structured models [185]. Finally, studies carried at the indi-vidual level sometimes report on the identity of respondents’ acquaintances, potentially enabling the reconstruction of the underlying contact network [44, 186, 30].
Early attempts to measure individual contacts were based on contact-tracing [187, 188, 189, 190, 191, 192, 182, 44, 193]. The latter aims to identify potential transmission routes by asking subject individuals to report on their past relationships. Contact-tracing is also a standard control tool used, for example, to identify asymptomatic infected in-dividuals in the case of sexually transmitted diseases [194, 195, 196, 192]. Another tra-ditional approach relies on contact diaries [186, 183, 184, 197]: respondents receive a personal diary in order to progressively record their contacts.
Recently, advances in communication technologies allowed measuring contacts with unprecedented temporal and spatial resolution. For example, a recent experimental setup makes use of dedicated sociometric sensors based on Radio Frequency Identification (RFID) technology [31]. RFID devices act as beacons which periodically broadcast a sig-nal that is eventually “listened » by neighboring RFIDs. Whenever two RFID sensors successfully reach each other, a Close Proximity Interaction (CPI) is automatically regis-tered. Experimenters can manually tune the strength of the signal, thus selecting only social interactions occurring within a given radius from the sensor. RFID devices have been used in different settings, such as schools [32, 33], conferences [34], museums [35], workplaces [36], households [37, 38] and hospitals [39, 40, 41, 42]. Other technologies include Bluetooth and/or WiFi signals in mobile phones [154, 155, 156, 157, 158].
The importance of contacts is becoming increasingly recognized in hospital settings, as evidenced by the increasing amount of studies accounting for contact networks [198]. These works have focused on the properties of network structure, the epidemiological relevance of measured contacts and on the impact of contact structure on epidemic spread and prevention and control strategies [199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 41, 42, 209, 210]. On a different scale, increasing attention is devoted to networks of hospitals linked by patient transfer [211, 212, 213, 214, 215].
Further network data is becoming available about human mobility at multiple scales. Air traffic data, for instance, encodes information about long-range movements and is of direct importance for understanding the rapid spread of pandemic diseases [216, 217]. At a shorter scale, census data provides information about commuting trips between neighboring locations. Regional patterns of seasonal influenza [218] and measles [219] have been studied in relation to commuting patterns. More recently, massive data about mobile phone calls provided unique opportunities in the study of human mobility [54, 55]. Indeed, call data records enabled remarkable insights not only about properties of human behavior [53, 52], but also about the impact of human mobility on the spread of several diseases, including malaria [220, 221], cholera [222, 223] and HIV [224].
Table of contents :
Introduction
1 Multi-pathogen Systems
1.1 Introduction
1.2 Interaction mechanisms
1.3 Implications for epidemiology and public health
1.4 Modeling multi-pathogen interactions
1.4.1 Compartmental models
1.4.2 Ecological insights from multi-strain models
1.5 Open challenges in modelling multi-strain dynamics
2 Epidemics on Networks
2.1 Introduction
2.2 Network data in epidemiology
2.3 Network analysis
2.3.1 Basic definitions
2.3.2 Topological properties
2.3.3 Temporal networks
2.3.4 Null models
2.3.5 Generative models
2.4 Epidemics on networks
2.4.1 Static networks
2.4.2 Temporal networks
2.5 Multi-strain dynamics on networks
2.5.1 Competitive interactions in SIS dynamics
2.5.2 Competitive interactions in SIR dynamics
2.5.3 Cooperative interactions in SIS dynamics
2.5.4 Cooperative interactions in SIR dynamics
2.5.5 Multiple pathogens on networks
3 Host Contact Dynamics and Diversity in Pathogen Strains
3.1 Introduction
3.2 Ecological modeling of multi-strain dynamics
3.2.1 Ecological characterization of multi-strain populations
3.3 Case study: S. aureus spread in hospitals
3.3.1 Overview of S. aureus ecology
3.3.2 The I-Bird experiment
3.4 First article: Host contact dynamics shapes richness and dominance of pathogen strains
3.5 Additional results: neutral dynamics
3.5.1 Domination time and attack rate
3.5.2 Impact of bursty contacts on strain diversity
3.6 Additional results: non-neutral dynamics
3.6.1 Simulation results
3.6.2 Analytical derivation of the condensation threshold
3.6.3 Numerical characterization of the s < sc regime
3.7 Conclusions
4 The Interplay of Cooperative and Competitive Interactions
4.1 Introduction
4.2 Pathogen-pathogen interactions and their impact on multi-strain ecosystems
4.3 Factors affecting co-existence between multiple strains
4.4 Second article: Interplay between competitive and cooperative interactions in a three-player pathogen system
4.5 Conclusions
5 Conclusions and Perspectives
A Multi-strain model
A.1 Model description
A.2 Simulating transmission
B Condensation threshold on networks
B.1 General framework
B.2 Analytical results
