Multiplex Bi-Virus Reaction-Diffusion (MBRD)

• MBRD is a multiplex network framework that models the spatio-temporal evolution of two interacting pathogens using reaction-diffusion equations coupled with cross-diffusion forces.
• It distinguishes between super-infection (MBRD-SI) and co-infection (MBRD-CI) formulations, capturing epidemic features such as Turing instabilities, hotspot formation, and strain dominance.
• Numerical experiments show that parameter sensitivity, network topology, and negative cross-diffusion critically shape epidemic patterns and enable targeted control strategies.

The Multiplex Bi-Virus Reaction-Diffusion framework (MBRD) is a class of multiplex metapopulation network models for the spatio-temporal evolution of two interacting pathogens, with diffusion and cross-diffusion acting on layer-specific graphs and nonlinear reaction terms governing transmission, recovery, death, super-infection, or co-infection. In the 2025 formulations, MBRD appears in two concrete closures: the super-infection model (MBRD-SI) and the co-infection model (MBRD-CI), both of which were introduced to capture epidemic pattern formation, Turing and Turing-Hopf instabilities, hotspot growth, coexistence, and strain dominance on multiplex networks (Yu et al., 21 Aug 2025).

1. Conceptual definition and modeling scope

MBRD treats a population as a collection of size-normalized patches or regions indexed by $i=1,\dots,N$. Each patch carries susceptible hosts and two pathogen-associated compartments. In the super-infection closure, the state is $(S_i,I_i,J_i)$, where $I_i$ and $J_i$ denote mono-infection by virus 1 and virus 2. In the co-infection closure, a fourth compartment $C_i$ tracks hosts carrying both pathogens simultaneously. The framework combines compartmental epidemic kinetics with multiplex transport: susceptibles and infected subpopulations move on distinct network layers, and the motion of one compartment can depend on gradients of another through cross-diffusion.

This construction is intended for settings in which pathogen interaction is not reducible to a single effective strain. In MBRD-SI, virus 2 can displace virus 1 through a super-infection coefficient $\sigma$. In MBRD-CI, mono-infected and co-infected classes interact through the parameters $\beta_{10}$, $\beta_{02}$, $\beta_{12}$, and $\alpha_{12}$, allowing direct co-transmission and co-infection mortality. The reported applications extend beyond epidemiology to information propagation, malware diffusion, and urban transportation networks, where the same reaction-diffusion structure is reinterpreted in non-biological terms (Yu et al., 21 Aug 2025).

A distinct but related line of work in molecular communication developed a two-species nonlinear reaction-diffusion PDE system and explicitly outlined how such a system could be generalized to an $(S_i,I_i,J_i)$0-species multiplexed reaction-diffusion setting. This suggests a methodological antecedent for multi-species reaction-diffusion computation, although the epidemic MBRD framework itself is formulated in the 2025 multiplex bi-virus network papers rather than in the molecular communication setting (Jamali et al., 2019).

2. General multiplex reaction-diffusion structure

In its network-diffusion form, the general MBRD framework is written for each compartment $(S_i,I_i,J_i)$1 as

$(S_i,I_i,J_i)$2

Here $(S_i,I_i,J_i)$3 is the graph Laplacian for the layer governing movement of compartment $(S_i,I_i,J_i)$4, and $(S_i,I_i,J_i)$5 is the cross-diffusion coefficient by which compartment $(S_i,I_i,J_i)$6 moves up or down the gradient of $(S_i,I_i,J_i)$7. The same framework is expressed in continuum form by replacing graph Laplacians with diffusion and cross-diffusion operators such as $(S_i,I_i,J_i)$8 and $(S_i,I_i,J_i)$9 (Yu et al., 3 Sep 2025).

The susceptible dynamics in both closures include an Allee-logistic growth term

$I_i$0

with per-capita growth rate $I_i$1, carrying capacity $I_i$2, and Allee threshold $I_i$3. Infection terms are normalized by local total population expressions such as $I_i$4 or $I_i$5, while recovery, disease-induced death, and background removal enter through $I_i$6-, $I_i$7-, and $I_i$8-terms. Boundary conditions are implemented by graph Laplacians with zero-row sums, which enforce the no-flux condition in the discrete network formulation. Initial data are typically chosen as the disease-endemic steady state plus small noise.

The multiplex structure is not merely a notational refinement. The framework assigns separate layers to $I_i$9, $J_i$0, and $J_i$1, so spatial transport is intrinsically heterogeneous across epidemiological classes. Cross-diffusion coefficients $J_i$2 and $J_i$3 in the super-infection notation, or $J_i$4 and $J_i$5 in the co-infection notation, allow susceptible motion to be biased by infected gradients. Negative cross-diffusion is especially important in the reported instability mechanisms because it can draw susceptibles toward infected regions and thereby amplify spatial inhomogeneities.

3. MBRD-SI: super-infection closure

The MBRD-SI model tracks susceptible density $J_i$6, hosts infected by strain 1 $J_i$7, and hosts infected by strain 2 $J_i$8. Its network ODE system is

$J_i$9

The biological interpretation is asymmetric. Virus 2 gains an advantage through the super-infection term proportional to $C_i$0, while virus 1 is correspondingly penalized. In the parameterization adopted in the papers, $C_i$1 represents how much more easily virus 2 infects hosts already infected by virus 1. This asymmetry produces a coexistence-versus-exclusion structure that is central to the model’s long-time behavior. Specifically, large $C_i$2 leads to strain-2 dominance: $C_i$3 and $C_i$4 endemic in each hotspot (Yu et al., 3 Sep 2025).

The continuum version replaces the graph Laplacian terms by $C_i$5, $C_i$6, $C_i$7, and cross-diffusion operators $C_i$8, $C_i$9. This allows the same model class to be studied either as a PDE system or as a multiplex network dynamical system. In the published analysis, both views are used: the continuum description supports dispersion-relation calculations, while the network version supports numerical experiments on lattice, Watts-Strogatz, and Barabási-Albert layers.

4. MBRD-CI: co-infection closure

The MBRD-CI model augments the system with a co-infected class $\sigma$0. Its reaction terms are organized around the transmission channels $\sigma$1, $\sigma$2, $\sigma$3, $\sigma$4, and $\sigma$5, where $\sigma$6 and $\sigma$7 are baseline co-transmission modifiers, $\sigma$8 is the direct co-transmission rate, and $\sigma$9 is the disease-induced death rate of the co-infected class. The network system is

$\beta_{10}$0

The co-infection closure changes both the state space and the interpretation of persistence. Moderate $\beta_{10}$1 can stabilize coexistence of $\beta_{10}$2, $\beta_{10}$3, and $\beta_{10}$4, whereas very large $\beta_{10}$5 or $\beta_{10}$6 drive collapse of mono-infection clusters in favor of homogenized co-infection or extinction. In the linearized analysis, MBRD-CI yields a four-morphogen problem and a quartic characteristic polynomial, in contrast to the cubic characteristic equation of MBRD-SI (Yu et al., 3 Sep 2025).

A potentially misleading intuition is that the addition of a co-infected class necessarily implies an independently diffusing fourth layer. In the derivation presented for the 2025 model, the network-multiplex form adds layer-specific Laplacians for $\beta_{10}$7, $\beta_{10}$8, and $\beta_{10}$9 diffusion, while the co-infection layer $\beta_{02}$0 carries no independent diffusion because $\beta_{02}$1 is algebraically determined once $\beta_{02}$2, $\beta_{02}$3, and $\beta_{02}$4 are known (Yu et al., 21 Aug 2025). The general MBRD notation can accommodate $\beta_{02}$5, but the concrete MBRD-CI closure introduced in the model papers omits a separate diffusion operator for $\beta_{02}$6.

5. Instability analysis, hotspot formation, and multiplex effects

The analytical core of MBRD is a Turing and Turing-Hopf instability analysis about a spatially homogeneous endemic steady state. For MBRD-SI, small perturbations $\beta_{02}$7 satisfy the eigenvalue problem

$\beta_{02}$8

which yields a cubic

$\beta_{02}$9

For MBRD-CI, the corresponding characteristic equation is quartic,

$\beta_{12}$0

The reported zero-diffusion stability conditions are $\beta_{12}$1, $\beta_{12}$2 sum of minors $\beta_{12}$3, $\beta_{12}$4, and $\beta_{12}$5. The Routh-Hurwitz and discriminant tests are then applied as functions of $\beta_{12}$6 to separate diffusion-driven static patterning from oscillatory Turing-Hopf onset (Yu et al., 21 Aug 2025).

In qualitative terms, hotspots emerge when the usual inhibitor-fast/activator-slow separation is realized in the multiplex setting. Large differences between $\beta_{12}$7 and $\beta_{12}$8, together with negative $\beta_{12}$9 and $\alpha_{12}$0, destabilize the endemic steady state and produce spatial localization. For MBRD-SI, the instability threshold is described roughly by

$\alpha_{12}$1

with $\alpha_{12}$2 increasing as $\alpha_{12}$3 grows. Analogous threshold expressions hold in the co-infection model in terms of $\alpha_{12}$4 and $\alpha_{12}$5 (Yu et al., 3 Sep 2025).

The multiplex architecture materially alters these thresholds. If the average degrees of the $\alpha_{12}$6, $\alpha_{12}$7, and $\alpha_{12}$8 layers differ substantially, pattern-forming modes are damped. Balanced degree-regular layers produce the strongest hotspots, whereas mismatched or heterogeneous layers inhibit them. This point addresses a common simplification in multistrain network epidemics: the 2025 MBRD results do not treat layer choice as incidental. Instead, layer regularity, mean degree, and topology enter directly into the onset and persistence of spatial structure.

6. Numerical regimes, spread metrics, and extensions

The published numerical experiments make the model classes concrete. For MBRD-SI on lattice, Watts-Strogatz, and Barabási-Albert networks, one example uses

$\alpha_{12}$9

with

$(S_i,I_i,J_i)$00

Starting near the endemic steady state plus white noise, spatial “dots & stripes” in $(S_i,I_i,J_i)$01 and $(S_i,I_i,J_i)$02 emerge and amplify from $(S_i,I_i,J_i)$03, and the amplitude $(S_i,I_i,J_i)$04 grows unbounded until local collapse $(S_i,I_i,J_i)$05, $(S_i,I_i,J_i)$06 in hotspots. In the companion instability paper, a closely related super-infection example on a $(S_i,I_i,J_i)$07 LA12 lattice yields stable spot-like clusters in the $(S_i,I_i,J_i)$08-layer at $(S_i,I_i,J_i)$09 (Yu et al., 3 Sep 2025).

For MBRD-CI, a representative experiment uses

$(S_i,I_i,J_i)$10

with

$(S_i,I_i,J_i)$11

The reported outcome is maze-like Turing spots in $(S_i,I_i,J_i)$12, together with persistent mono-infection structure in $(S_i,I_i,J_i)$13 and $(S_i,I_i,J_i)$14, amplifying until collapse by $(S_i,I_i,J_i)$15. In the earlier model paper, a mixed LA12/LA4 lattice under comparable parameters produces stripe and labyrinth patterns in the $(S_i,I_i,J_i)$16-layer at $(S_i,I_i,J_i)$17 (Yu et al., 21 Aug 2025).

Parameter sensitivity is explicitly nontrivial. In MBRD-SI, $(S_i,I_i,J_i)$18, so larger $(S_i,I_i,J_i)$19 suppresses pattern reversion. In MBRD-CI, $(S_i,I_i,J_i)$20 versus $(S_i,I_i,J_i)$21 is non-monotonic and peaks at $(S_i,I_i,J_i)$22, indicating that intermediate co-transmission is the most destabilizing regime. Layer-degree effects are similarly selective: patterns form only when

$(S_i,I_i,J_i)$23

and higher overall degree gives larger but slower hotspots. Barabási-Albert layers yield faster saturation and weaker dependence on layer degree, suggesting hub-driven super-spread during “holiday” mobility (Yu et al., 3 Sep 2025).

The point-source experiments introduce a spread index

$(S_i,I_i,J_i)$24

Within MBRD-SI, the $(S_i,I_i,J_i)$25-spread peak is weakly sensitive to source distance, whereas $(S_i,I_i,J_i)$26-saturation slows for closer sources. With varying time lag $(S_i,I_i,J_i)$27, the reported fits are $(S_i,I_i,J_i)$28 and $(S_i,I_i,J_i)$29-saturation time $(S_i,I_i,J_i)$30. In MBRD-CI, all three indices saturate later as $(S_i,I_i,J_i)$31 grows, and increasing $(S_i,I_i,J_i)$32 lowers $(S_i,I_i,J_i)$33 while advancing its peak time; increasing $(S_i,I_i,J_i)$34 speeds $(S_i,I_i,J_i)$35-saturation, with a mutual-enhancement regime beyond a co-transmission threshold (Yu et al., 3 Sep 2025).

The broader significance claimed for MBRD is not limited to two-pathogen epidemiology. The same formalism is proposed for rumor-versus-rumor dynamics, malware-versus-malware propagation, and multiplex transportation congestion. Negative cross-diffusion is interpreted there as the tendency of naïve nodes to seek influential clusters or, in other application domains, as movement shaped by competing signals or congested routes. Proposed extensions include optimal-control formulations with vaccination or quarantine rates, environmental forcing through time-periodic reaction terms, higher-order networks such as hyperedges and simplicial complexes, and data-driven parameter estimation via physics-informed neural nets from spatial incidence maps. The simulations also suggest that limiting mobility of infected subpopulations through high $(S_i,I_i,J_i)$36 and $(S_i,I_i,J_i)$37 is more effective in slowing multi-pathogen spread than blanket reductions in $(S_i,I_i,J_i)$38, although this remains a numerical implication of the reported experiments rather than a general theorem (Yu et al., 3 Sep 2025).