MBRD-CI: Co-Infection in Multiplex Networks

• Co-Infection Model (MBRD-CI) is a framework that simulates two interacting pathogens on a multiplex metapopulation network using reaction-diffusion dynamics.
• It integrates layer-specific diffusion and cross-diffusion to capture infection clustering, hotspot formation, and Turing (or Turing-Hopf) instabilities.
• The model distinguishes co-infection from superinfection while quantifying thresholds, amplitude effects, and the impact of network structure on epidemic spread.

Searching arXiv for the core MBRD-CI papers and a few closely related co-infection antecedents. arxiv_search(query="MBRD-CI co-infection model multiplex bi-virus reaction-diffusion", max_results=10) arxiv_search(query="Co-Infection Model MBRD-CI", max_results=10) The Co-Infection Model (MBRD-CI) is the co-infection member of the Multiplex Bi-Virus Reaction-Diffusion (MBRD) framework for two interacting pathogens on a spatial network. In this formulation, a host may be infected by both viruses at once, and the dynamics are posed on a multiplex metapopulation network in which each node is a population center and different layers represent susceptible, mono-infected, and co-infected densities. MBRD-CI combines nonlinear epidemiological reaction terms with layer-specific diffusion and cross-diffusion, and it is used to study infection clustering, large-scale spatial distributions, hotspot formation, and Turing or Turing-Hopf instability in two-pathogen systems (Yu et al., 21 Aug 2025, Yu et al., 3 Sep 2025).

1. Formal definition and compartmental organization

MBRD-CI is formulated on a four-layer multiplex network with state variables $S_i$, $I_i$, $J_i$, and $C_i$, representing susceptible density, pathogen 1 infected density, pathogen 2 infected density, and co-infected density at region $i$. The framework introduces a fourth layer used only for modeling MBRD-CI and describing the density of co-infections at each region. That layer has no edges, because $C_i$ can be derived from the other layers. In the reaction-diffusion formulation, $I$ and $J$ denote the total pathogen-associated densities, including both mono- and co-infections, while $C$ explicitly tracks the co-infected population (Yu et al., 21 Aug 2025).

The defining epidemiological assumption is that hosts can carry both pathogens simultaneously and that, in MBRD-CI, no pathogen steals hosts from the other. This distinguishes co-infection from the corresponding super-infection model, in which pathogens cannot coexist in the same host and a more virulent strain can displace the other. The model is therefore intended for settings where simultaneous carriage, rather than displacement, is the central interaction mechanism (Yu et al., 21 Aug 2025).

Feature	MBRD-CI	MBRD-SI
State variables	$(S_i,I_i,J_i,C_i)$	$I_i$0
Core interaction	co-infection dynamics only	superinfection dynamics but no co-infection
Extra coupling terms	$I_i$1	$I_i$2
Additional layer	fourth layer for $I_i$3, with no edges	none

The susceptible population is modeled with a logistic/Allee-like growth term,

$I_i$4

where $I_i$5 is the natural growth rate, $I_i$6 is the carrying capacity, and $I_i$7 is the critical spatial carrying capacity density. Recovery parameters $I_i$8, infection-related removal parameters $I_i$9, and natural mortality $J_i$0 appear explicitly in the reaction terms (Yu et al., 21 Aug 2025).

2. Governing equations and transmission structure

At node $J_i$1, the MBRD-CI equations are

$J_i$2

These equations appear in the network introduction of MBRD-CI and in the later study of hotspot growth and long-term prevalence (Yu et al., 21 Aug 2025, Yu et al., 3 Sep 2025).

The parameters $J_i$3 and $J_i$4 govern mono-infection transmission, while $J_i$5 describe transmission involving co-infected hosts. The paper classifies pathogen interaction regimes as non-interaction, mutual enhancement, enhancement/inhibition asymmetry, and mutual inhibition, based on relations among $J_i$6, $J_i$7, $J_i$8, and $J_i$9 (Yu et al., 21 Aug 2025).

A central structural feature is that the co-infected layer does not diffuse. Only the $C_i$0, $C_i$1, and $C_i$2 layers experience diffusion, and the $C_i$3 layer has no edges because the relative density for each node on the $C_i$4 layer can directly be calculated from the densities of the corresponding nodes on the other three layers. This modeling choice makes the co-infected population dynamically explicit while keeping inter-node transport tied to the susceptible and mono-infected layers (Yu et al., 21 Aug 2025).

3. Multiplex diffusion, cross-diffusion, and instability analysis

For each diffusing layer, the network transport operator is the negative combinatorial Laplacian,

$C_i$5

with $C_i$6. Ordinary diffusion appears through $C_i$7, while cross-diffusion enters the susceptible equation through $C_i$8 and $C_i$9. The model interprets these coefficients behaviorally: if $i$0, susceptibles move toward low pathogen-1 density; if $i$1, susceptibles move toward high pathogen-1 density; similarly for $i$2 and pathogen 2. In the later numerical paper, negative cross-diffusion rates $i$3 are explicitly interpreted as meaning that susceptible individuals gravitate toward areas with many infections (Yu et al., 21 Aug 2025, Yu et al., 3 Sep 2025).

Linearization around a steady state $i$4 yields a quartic characteristic polynomial,

$i$5

The paper gives the corresponding coefficient relations

$i$6

It then states necessary conditions for instability, in the paper’s notation,

$i$7

These are obtained from Vieta relations and the Routh-Hurwitz criterion (Yu et al., 21 Aug 2025).

The same analysis introduces the quartic discriminant $i$8. If $i$9, then $C_i$0 must have three real roots alternating in sign for Turing instability to occur, and $C_i$1 is a necessary condition for Turing-Hopf instability. In the terminology used in the MBRD papers, Turing instability produces stationary spatial pattern formation, while Turing-Hopf instability produces spatial patterns with temporal oscillation (Yu et al., 21 Aug 2025).

4. Hotspot formation, co-transmission thresholds, and network effects

The later numerical study investigates how MBRD-CI behaves under different epidemiological and network conditions, including virulence, diffusion rates, cross-diffusion, and network topology. It explicitly studies lattice (LA), Watts-Strogatz (WS), and Barabási-Albert (BA) networks, and reports that different average degrees in each layer alter epidemic spread, hotspot formation, and saturation times (Yu et al., 3 Sep 2025).

For MBRD-CI, Example 2 uses

$C_i$2

On an LA12-LA12-LA4 multiplex network, patterns appear by $C_i$3, persist through $C_i$4, and show collapse by $C_i$5. The paper states that it observes fine-grained Turing patterns with a maze-like structure, and that the hotspots are still at the same locations between times 250 and 500 (Yu et al., 3 Sep 2025).

A second set of numerical results concerns the co-transmission parameter $C_i$6. The paper defines the $C_i$7 threshold of a co-infection parameter configuration to be the largest value of $C_i$8, with all other parameters fixed, such that the maximum $C_i$9-spread index over time is less than $I$0 and converges to $I$1. It also states that larger $I$2 should require a larger $I$3 threshold. In amplitude studies, small $I$4 is insufficient to sustain strong co-infection clusters, intermediate $I$5 maximizes oscillations or amplitude, and very large $I$6 homogenizes the system and suppresses pattern formation. The text states that the average amplitude peaks approximately when $I$7 (Yu et al., 3 Sep 2025).

The network-structural findings are equally specific. The paper reports that, in some settings, larger overall average degrees can lead to larger clusters but also slower amplitude growth. Low average degree of the layer corresponding to the more virulent pathogen tends to increase the time before saturation. For Example 5 and Example 8, the combinations LA12-LA4-LA4 and LA4-LA4-LA4 are highlighted as beneficial, and the paper concludes that, to minimize the spread of both infections, it is most important to reduce the movement of individuals infected by either, and limiting the movement of susceptible individuals is less important (Yu et al., 3 Sep 2025).

5. Relation to earlier co-infection theory

MBRD-CI belongs to a broader theoretical lineage of multi-pathogen epidemic models, but its distinctive contribution is the joint use of multiplex network structure, explicit co-infected densities, and reaction-diffusion transport. Earlier network models established exact threshold and phase-diagram results for co-infection without this spatial multiplex transport layer. In "Cooperative epidemics on multiplex networks" (Azimi-Tafreshi, 2015), two diseases spread on a two-layer multiplex network with edge overlap, and infection with one disease enhances susceptibility or transmissibility to the other. That model is solved exactly with generating functions and overlap renormalization into supernodes, yielding exact self-consistency equations for $I$8 and the giant co-infected cluster size $I$9, with the main conclusions that cooperation lowers epidemic thresholds, the outbreak can change from continuous to hybrid, and a tricritical point exists at $J$0 in the symmetric Erdős–Rényi case (Azimi-Tafreshi, 2015).

A different antecedent is the sequential dependence model of "Interacting epidemics and coinfection on contact networks" (Newman et al., 2013), in which disease 2 can spread only through hosts who have already been infected by disease 1. On configuration-model networks, that paper derives exact large-$J$1 outbreak sizes and thresholds, including

$J$2

for the first epidemic and a conditional threshold $J$3 for the second. Its principal conceptual result is that disease 2 can be controlled either directly, by lowering $J$4, or indirectly, by lowering $J$5 enough that disease 1 infects too few nodes to sustain disease 2 (Newman et al., 2013).

The distinction between discontinuous and genuinely first-order transitions was clarified in "First-order phase transitions in outbreaks of co-infectious diseases and the extended general epidemic process" (Janssen et al., 2016). That work shows that the homogeneous mean-field co-infection model of Chen, Ghanbarnejad, Cai, and Grassberger coincides with the homogeneous limit of the extended general epidemic process, and argues that the discontinuous outbreak seen in homogeneous mean-field theory is a spinodal transition rather than a first-order transition with phase coexistence. A true first-order transition with phase coexistence appears only when spatial inhomogeneities are retained (Janssen et al., 2016). Relative to these earlier models, MBRD-CI shifts the emphasis from exact percolation thresholds or homogeneous mean-field discontinuities to multiplex reaction-diffusion pattern formation and hotspot growth.

6. Interpretation, scope, and neighboring co-infection paradigms

The MBRD papers explicitly distinguish co-infection from superinfection, but the wider literature shows that co-infection is not synonymous with mutual enhancement. "Antagonistic coinfection in rock-paper-scissors models during concurrent epidemics" (Menezes et al., 9 May 2025) studies a spatial stochastic model in which simultaneous occurrence of coinfection reduces the probability of host mortality. In that system, antagonism enhances species population growth, reduces the average probability of healthy organisms becoming infected, decreases the characteristic length scale of the spatial patterns by about $J$6 from $J$7 to $J$8, and, when combined with mobility restriction, can increase organisms’ life expectancy by up to $J$9 (Menezes et al., 9 May 2025). This establishes that multi-pathogen interaction can be inhibitory as well as facilitative.

A second neighboring paradigm concerns scale reversal between individual-level and population-level effects. In "Interactions between chronic diseases: asymmetric outcomes of co-infection at individual and population scales" (Gorsich et al., 2017), an age-structured BTB–brucellosis model finds that BTB increases brucellosis transmission risk by a factor of $C$0 on average, while brucellosis does not measurably change the risk of acquiring BTB in the empirical fit. Yet, at the population level, the presence of brucellosis lowers the endemic prevalence and basic reproduction number of BTB, with modeled values $C$1 without brucellosis and $C$2 with brucellosis present, and endemic prevalence of BTB falling from $C$3 to $C$4 (Gorsich et al., 2017). This broader evidence is compatible with the MBRD-CI emphasis on interaction structure: the sign and scale of co-infection effects depend on the specific coupling terms that are retained.

Within its own stated scope, MBRD-CI is deterministic, posed on undirected multiplex networks, and does not include vaccination, age structure, cross-immunity, weighted or directed movement, time-varying networks, or vector-borne transmission structure. The co-infected layer is non-diffusive by construction. The papers nevertheless present the framework as applicable beyond epidemiology, including information propagation, malware propagation, urban transportation networks, and election forecasting (Yu et al., 21 Aug 2025, Yu et al., 3 Sep 2025).