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Dynamics of interacting diseases

Joaquín Sanz, Cheng-Yi Xia, Sandro Meloni, Yamir Moreno

arXiv:1402.4523v2physics.soc-phcond-mat.stat-mechq-bio.PE

TL;DR

The paper addresses the gap between single-pathogen epidemic models and concurrent diseases interacting across distinct contact networks. It develops a heterogeneous mean field framework for coupled SIS and SIR processes, deriving epidemic thresholds and temporal dynamics. The analysis shows that outbreak onset can depend on the other disease’s prevalence, with distinct SIS and SIR threshold behavior and richer interdependencies in SIR.

  • Problem

    Existing epidemic models generally treat spreading at one level and do not adequately describe interdependent contagion involving two diseases and potentially different contact networks.

  • Method

    The paper uses a heterogeneous mean field framework to model two interacting SIS or SIR processes spreading on separate layers of a multilayer contact system.

  • Results

    The analysis derives disease thresholds and temporal dynamics, finding prevalence-conditioned outbreak regions and different secondary thresholds for SIS and SIR models.

  • Takeaways & Limitations

    Disease-disease interactions can produce threshold behavior and epidemic phenomenology not captured by single-disease models, including richer interdependencies in SIR systems.

Abstract

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Current modeling of infectious diseases allows for the study of complex and realistic scenarios that go from the population to the individual level of description. However, most epidemic models assume that the spreading process takes place on a single level (be it a single population, a meta-population system or a network of contacts). In particular, interdependent contagion phenomena can only be addressed if we go beyond the scheme one pathogen-one network. In this paper, we propose a framework that allows describing the spreading dynamics of two concurrent diseases. Specifically, we characterize analytically the epidemic thresholds of the two diseases for different scenarios and also compute the temporal evolution characterizing the unfolding dynamics. Results show that there are regions of the parameter space in which the onset of a disease's outbreak is conditioned to the prevalence levels of the other disease. Moreover, we show, for the SIS scheme, that under certain circumstances, finite and not vanishing epidemic thresholds are found even at the thermodynamic limit for scale-free networks. For the SIR scenario, the phenomenology is richer and additional interdependencies show up. We also find that the secondary thresholds for the SIS and SIR models are different, which results directly from the interaction between both diseases. Our work thus solve an important problem and pave the way towards a more comprehensive description of the dynamics of interacting diseases.

I. INTRODUCTION

Interacting-disease modeling extends epidemic analysis beyond one pathogen on one network. The paper develops a framework for two concurrent diseases, combining analytical thresholds with temporal dynamics.

  • I. INTRODUCTION: Coupled spreading models address two pathogens or multiple strains propagating concurrently through the same population.The second disease can alter the first disease’s natural history through immune-response changes and interaction mechanisms.
  • I. INTRODUCTION: Different diseases may spread through different contact networks, increasing the complexity of their interaction.Examples include HIV with opportunistic pathogens and interactions among influenza strains.
  • I. INTRODUCTION: Earlier network-based work characterized epidemic thresholds for interacting diseases but did not provide a framework for temporal epidemic evolution.The cited approach used an analogy with bond percolation in an SIR model.
  • I. INTRODUCTION: The proposed framework uses heterogeneous mean field analysis to derive thresholds and describe two coupled SIS or SIR processes on multilayer networks.It focuses on how interaction mechanisms, temporal scales, and network topology shape disease dynamics.
  • I. INTRODUCTION: SIR and SIS models yield different epidemic thresholds because their interaction mechanisms and outbreak temporal properties differ.The paper attributes this difference to new interactions and the transitory nature of SIR outbreaks.

II. MODELING FRAMEWORK

The framework models two diseases spreading simultaneously through one host population while using separate contact networks and potentially different disease dynamics. Individuals are classified by their connectivity to both networks.

  • II. MODELING FRAMEWORK: Two diseases spread over two different contact networks, with disease 1 using network 1 and disease 2 using network 2.The networks and transmission mechanisms are treated independently for the two diseases.
  • II. MODELING FRAMEWORK: Each individual’s disease parameters depend on whether the individual has neither disease, one disease, or both diseases.The parameters include infectiousnesses and recovery rates that vary with infection status.
  • II. MODELING FRAMEWORK: The composed degree distribution P(k, l) records the fraction of individuals with k links in network 1 and l links in network 2.The corresponding mean connectivities are defined separately for the two networks.
  • II. MODELING FRAMEWORK: The framework analyzes both SIS and SIR schemes for the two-network, two-disease system.This permits comparison of threshold and temporal behavior across the two epidemic models.

A. The SIS scenario

The SIS formulation represents each individual by infection status for both diseases and allows disease interactions to modify susceptibility, infectiousness, and recovery. A composed heterogeneous mean field system then describes the dynamics over degree classes.

  • A. The SIS scenario: Each connectivity class has four SIS states: SS, IS, SI, and II, representing infection status for the two diseases.The states track individuals susceptible to both diseases, infected with one disease, or infected with both.
  • A. The SIS scenario: The model includes baseline infection probabilities λ1 and λ2 and recovery rates µ1 and µ2, modified by β and η interaction factors.The β parameters alter infection probabilities, while η parameters alter recovery rates.
  • A. The SIS scenario: Three interaction mechanisms modify susceptibility, double-infected individuals’ spreading capability, and their infectious periods.These mechanisms are represented respectively by βa, βb, and η parameters.
  • A. The SIS scenario: The interaction parameters exhaustively specify SIS transitions between the four infection states.The formulation excludes simultaneous double contagions and simultaneous recoveries from both diseases.
  • A. The SIS scenario: The composed heterogeneous mean field equations describe time evolution within each joint degree class while neglecting further dynamical correlations.The four state equations obey a closure relation, leaving three linearly independent variables per class.
  • A. The SIS scenario: The variables σ1 and σ2 summarize average per-link infection probabilities for susceptible-to-both individuals and couple the network degree classes.They allow the differential equations to be rewritten in terms of effective infection pressures.

1. Epidemic thresholds

The SIS threshold analysis identifies absorbing and active states and derives disease thresholds that depend on the other disease’s prevalence. These secondary thresholds can differ from primary thresholds because coupling changes effective transmission conditions.

  • 1. Epidemic thresholds: The SIS threshold structure is obtained from stationary-state equations and can also be derived from the Jacobian of the homogeneous mean field system.The threshold expressions incorporate transmission, recovery, interaction parameters, and network connectivity.
  • 1. Epidemic thresholds: The absorbing state has zero infected densities, while active fixed points have nonzero infection densities expressed through σ1 and σ2.A disease becomes endemic when its corresponding self-consistency condition σi = fi(σ1, σ2) is positive.
  • 1. Epidemic thresholds: The threshold for disease 1 depends explicitly on disease 2’s prevalence through σ2 and on the underlying network topology.The analogous dependence holds for disease 2 through σ1.
  • 1. Epidemic thresholds: Primary thresholds describe the disease-free counterpart, whereas secondary thresholds λc 1(σ2) and λc 2(σ1) apply when the other disease is prevalent.The primary thresholds are denoted λc 1(0) and λc 2(0).

2. Phase diagrams

The phase diagrams show that disease interactions can either enhance or impair transmission, shifting secondary thresholds relative to primary thresholds. Under mutual enhancement, one disease's outbreak may depend on prior establishment of the other, while mutual impairment produces the opposite threshold ordering.

  • Mutual enhancement is modeled by β > 1 and η < 1, making coinfected individuals more susceptible to the other disease and prolonging infection.
  • Under mutual enhancement, secondary thresholds fall below primary thresholds, creating regions where an outbreak requires prior establishment of the other disease.
  • After disease 2 outbreaks, disease 1 becomes endemic in a region where the same disease-1 seed alone does not produce endemic infection.
  • Disease 2 can become endemic directly when disease 1 was previously introduced, within the corresponding conditional-outbreak region.
  • Mutual impairment reverses the pattern: secondary thresholds remain above primary thresholds, producing distinct interaction regions.

3. System sizes and epidemic thresholds: general case

System size and inter-layer degree correlations determine whether secondary epidemic thresholds vanish or remain finite. Strong positive correlations between scale-free networks can preserve finite thresholds at infinite size, especially under cross-immunity, while the HMF treatment has explicit scope limitations.

  • The analysis uses HMF and therefore neglects dynamical correlations; threshold behavior is also examined mainly for same-kind ER or uncorrelated scale-free networks.
  • For scale-free networks with 2 < γ ≤3, secondary thresholds generally vanish at the thermodynamic limit, regardless of conjugate topology and dynamical parameters.
  • Highly positively correlated scale-free networks can yield finite thresholds for 2 < γ ≤3 under interaction schemes including cross-immunity or blocked contagion from coinfected individuals.
  • Strong positive correlations between scale-free networks can prevent thresholds from vanishing as N →∞ when the second disease has prevalence σ2 > 0.
  • Under full cross-immunity, correlated networks amplify the difference between primary and secondary thresholds and produce sharper epidemic-threshold transitions.
  • The interaction-driven threshold effect can be identified within HMF without invoking dynamical correlations.

B. The SIR scenario

The framework is extended from interacting SIS processes to two interacting SIR epidemics, where transient dynamics introduce additional interdependencies. Recovered individuals can affect susceptibility to the conjugate disease, and the model tracks the allowed contagion and recovery transitions.

  • The SIR extension describes two transient epidemics interacting with each other, rather than stationary endemic SIS states.
  • Unlike non-interacting SIS and SIR systems, interacting SIR dynamics can depend on whether hosts were infected and recovered from the conjugate disease.
  • Figure 6 summarizes the transitions allowed in the double SIR-SIR model, distinguishing contagion processes from recovery processes.

1. Mathematical description

The SIR formulation extends the state variables and neighborhood probabilities so that recovered states contribute to conjugate-infection dynamics. These added terms preserve the SIS notation while accounting for IR and RI classes.

  • In the SIR formulation, infection and recovery history can modify susceptibility to the conjugate infection through recovered states.
  • The SIR equations retain the SIS parameters and variables while adding terms associated with IR and RI classes.
  • The neighborhood probabilities σ1 and σ2 acquire additional contributions from IR and RI individuals.
  • θ_RI^2 denotes the probability that a link in network 2 points to a node in the RI state.

2. Epidemic thresholds

The epidemic threshold of disease 2 is obtained by linearizing the dynamics near a state without disease-2 infections. In the SIR case, the threshold depends on the dynamic state of disease 1 and can vary over time.

  • Threshold derivation: The disease-2 threshold is derived by analyzing stability near states where all disease-2 infected classes vanish.The relevant variables are SI(k,l), II(k,l), and RI(k,l), which become locally autonomous around the disease-2-free point.
  • Threshold derivation: The linearized system uses θSI_2, θII_2, and θRI_2, with the epidemic threshold determined from the Jacobian stability condition.The reduced equations allow the threshold to be obtained by evaluating the stability shift of the corresponding Jacobian matrix.
  • SIR thresholds: For nonconcurrent outbreaks, the second disease arrives after the first outbreak ends, so σ1 and ⟨l2IS⟩ vanish while recovered individuals from disease 1 remain.In this case, ⟨l2SS⟩ is reduced by the final recovered fraction of disease 1.
  • SIR thresholds: The primary SIR threshold is λc_2 = µ2⟨l⟩/⟨l2⟩, matching the classical noninteracting heterogeneous mean-field result.The secondary threshold is evaluated against this disease-free baseline.
  • SIR thresholds: During an outbreak of the conjugate disease, the SIR threshold can depend nontrivially on time through the evolving dynamical state and interaction parameters.The threshold therefore differs from the fixed primary threshold when disease 1 is already present.

III. CONDITIONS FOR DISEASE ENHANCEMENT AND IMPAIRMENT

Disease interactions can either enhance or impair the spreading of the other disease, with the effect determined by interaction parameters and, in some cases, by prevalence or outbreak timing. These dependencies are more varied in SIR than in SIS dynamics.

  • Threshold effects: The sign of disease 1’s effect on disease 2 is determined by the difference between secondary and primary epidemic thresholds.A positive difference indicates enhancement, whereas a negative difference indicates impairment.
  • SIS conditions: If βa_2 > 1, βb_2 > 1, and η2 < 1, disease 1 enhances disease 2 for every σ1 > 0.This is termed coherent enhancement because all interaction mechanisms favor disease-2 spreading.
  • SIS conditions: If βa_2 < 1, βb_2 < 1, and η2 > 1, disease 1 impairs disease 2 regardless of σ1.When neither coherent condition holds, the sign of the effect can depend on disease-1 prevalence.
  • SIR conditions: In SIR dynamics, interaction through recovered individuals strengthens as the first outbreak progresses because the recovered class increases over time.When interaction is carried by infected individuals, it occurs only when the two outbreaks overlap in time.
  • SIS–SIR comparison: SIR interactions are intrinsically transient when the infected class mediates them, producing richer threshold behavior than in SIS dynamics.The SIS model can also show time-dependent threshold shifts, but the passage describes such parameter combinations as epidemiologically less plausible.

IV. CONCLUSIONS

The paper develops a composed heterogeneous mean-field framework for two diseases spreading through independent contact networks and uses it to derive thresholds and approximate temporal dynamics. It identifies interaction-driven differences between SIS and SIR systems, while leaving dynamical correlations for future work.

  • Framework: The composed HMF model represents two concurrent diseases spreading through independent transmission mechanisms on different contact networks.It couples each disease’s transition parameters to the host’s state with respect to the other disease.
  • Framework: The framework analytically derives epidemic thresholds and approximately describes temporal evolution while isolating effects of infectivity, susceptibility, and infectious-period interactions.It is applied to both SIS and SIR extensions.
  • Main findings: Disease interactions produce threshold phenomena qualitatively different from those of noninteracting HMF models, including distinct SIS and SIR critical behavior.The reported differences arise from disease-disease interactions rather than dynamical correlations.
  • Scope and future work: The HMF approximation neglects dynamical correlations among nodes, and future work should address correlations spanning both networks and their dynamical states.The authors describe this extension as requiring substantially more complex descriptions.
  • Main findings: In SIS systems, some interaction schemes make one disease’s effect on another change sign with prevalence, whereas SIR systems show richer dynamics because interactions can involve multiple dynamical classes.For SIR processes, the timing of interaction can greatly affect the modified disease’s epidemic threshold.

Appendix A: Epidemic thresholds on regular networks

For regular networks, linear stability analysis identifies primary epidemic thresholds and secondary thresholds whose values depend on the prevalence of the other disease. The appendix also shows how threshold estimates compare with simulations and where heterogeneous networks limit direct prevalence-based reformulation.

  • Linear stability analysis: The regular-network dynamics are reduced to three independent equations, linearized around equilibrium, and analyzed through the Jacobian’s stability conditions.The susceptible fraction is eliminated using s = 1 − IS − SI − II.
  • Primary thresholds: Primary thresholds are the minimum infectiousness values producing outbreaks from infinitesimal seeds in an initially healthy population.The two thresholds correspond separately to introductions of disease 1 and disease 2.
  • Partially disease-free states: Partially disease-free fixed points represent one disease persisting at prevalence π1 or π2 while the other is absent.Their stability depends on the stationary prevalence fractions of the installed disease.
  • Secondary thresholds: The first disease’s threshold can be evaluated as a function of the second disease’s prevalence, producing secondary-threshold curves that agree quantitatively with simulations on regular networks.The analogous construction applies to the secondary threshold of the first disease, while the reported agreement uses simulated prevalence values.
  • Heterogeneous networks: For heterogeneous networks, no analytical bijection λ1(π1) or λ2(π2) is generally available, causing slight divergence between analytical and numerical secondary thresholds.The curves remain reported as very accurate in one figure, but the numerical prevalence-to-infectiousness relationship limits the reformulation.
  • Scale-free limits: With γ > 2 and Γ > 2 and no degree correlations, coupling generally adds a finite prefactor, while each disease’s threshold vanishes precisely when its own network exponent is at most 3.Certain interaction schemes can instead prevent threshold vanishing even for 2 < γ ≤ 3 when infectiousness variations cancel appropriately.

Appendix C: Proof of eq. B3

The proof analyzes the divergence of the integral in Eq. B3 by factoring the denominator, decomposing rational functions into partial fractions, and identifying which asymptotic terms can diverge. It concludes that divergence is controlled by a polynomial-degree condition.

  • Divergence criterion: The integral diverges when deg(P) ≥ deg(Q) −1, because this condition supplies the coefficient of the logarithmic term that survives asymptotically.The proof identifies this condition after grouping the partial fractions into a single fraction.
  • Integral decomposition: The proof reduces the divergence question to the integral associated with the quotient P(k)/Q(k), after factoring Q(k) into first- and second-order components.The resulting partial-fraction decomposition separates the terms requiring asymptotic analysis.
  • Asymptotic terms: Partial fractions yield logarithmic, rational, and arctangent terms, but only logarithmic terms can diverge as kmax →∞.Finite terms are negligible compared with ln(kmax) in the limit.
  • Remaining integrals: For integrals that cannot be written in an explicit closed form, the proof still determines their asymptotic behavior through transformations and integration by parts.The analysis treats the remaining terms in forms involving (v^2 + 1)^m.
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