--- title: Multiplex Bi-Virus Reaction-Diffusion Models url: https://www.emergentmind.com/topics/multiplex-bi-virus-reaction-diffusion-models-mbrd type: topic --- # Multiplex Bi-Virus Reaction-Diffusion Models Multiplex Bi-Virus Reaction-Diffusion Models (MBRD) are a family of models for the spatio-temporal spread of two interacting pathogens or contagions on node-aligned multiplex networks, combining within-node reaction terms—such as transmission, recovery, virulence, competition, super-infection, or co-infection—with layer-specific transport operators. In the literature, MBRD spans several mathematically distinct but compatible formalisms: competitive continuous-time bi-virus SIS systems on directed multiplex infection graphs, exact fluid-limit ODEs on multipartite metapopulations, graph and PDE reaction-diffusion systems with supra-Laplacians and cross-diffusion, threshold-based co-diffusion models on multiplex social networks, and shared-resource epidemic systems with exclusive infection constraints [1901.00765; 1306.6198; 1406.6401; 1806.11526; 2011.07569; 2508.15740; 2509.03374]. ## 1. Network representations and state spaces A recurrent structural assumption is that all layers share the same node set while allowing layer-specific connectivity. In the continuous-time competitive SIS backbone, virus \(1\) and virus \(2\) spread on their own directed, strongly connected infection graphs, with irreducible nonnegative infection matrices \(B^{(1)},B^{(2)}\in\mathbb{R}^{n\times n}\), diagonal healing matrices \(D^{(1)},D^{(2)}\), and heterogeneous node-wise rates. Infection fractions are \(x_i(t),y_i(t)\in[0,1]\), susceptibility is \(s_i(t)=1-x_i(t)-y_i(t)\), and no co-infection per individual is allowed, so \(x_i(t)+y_i(t)\le 1\). The positively invariant domain is \[ D=\{(x,y)\mid x\ge 0,\; y\ge 0,\; x+y\le 1\}. \] The neighbor graph is the union of the two infection graphs [1901.00765]. In the exact multipartite fluid-limit formulation, the node set is partitioned into islands \(V_1,\dots,V_M\), with complete bipartite connectivity between connected islands and no intra-island edges. For \(K\) strains, the limiting fractions \(y_{ik}(t)\) satisfy \(\sum_{k=1}^K y_{ik}(t)\le 1\), again encoding exclusive infection through a shared susceptible capacity [1306.6198]. The explicit MBRD-SI and MBRD-CI frameworks use multiplex metapopulation networks with layers \(S\), \(I\), \(J\), and, in the co-infection case, \(C\). Here \(S_i\) denotes susceptible density, \(I_i\) pathogen-1 infected density, \(J_i\) pathogen-2 infected density, and \(C_i\) coinfection density. Diffusion acts on \(S\), \(I\), and \(J\) layers through graph Laplacians \(L^{(S)},L^{(I)},L^{(J)}\), while the \(C\) layer has no edges and no diffusion. Cross-diffusion appears as off-diagonal diffusion blocks in the \(S\) equation, driven by \(I\) and \(J\) layer Laplacians [2508.15740; 2509.03374]. A distinct multiplex interpretation arises in shared-resource models. There, each virus has a person-person layer \(B^{(k)}\) and a person-resource layer defined by resource-to-node infection rates \(b^{(k)}\), node-to-resource contamination weights \(c^{(k)}\), and resource cleaning rate \(\delta_w^{(k)}\). The augmented matrices \[ B_w^{(k)}= \begin{bmatrix} B^{(k)} & b^{(k)}\\ \delta_w^{(k)}c^{(k)} & 0 \end{bmatrix}, \qquad D_w^{(k)}= \begin{bmatrix} D^{(k)} & 0\\ 0 & \delta_w^{(k)} \end{bmatrix} \] embed the resource as an additional layer-like state while preserving exclusive infection at population nodes [2011.07569]. ## 2. Competitive SIS backbone on multiplex graphs The canonical continuous-time bi-virus competitive SIS dynamics on a network of \(n\) groups are \[ \dot{x}_i = -\delta^{(1)}_i x_i + (1-x_i-y_i)\sum_{j=1}^n \beta^{(1)}_{ij}x_j, \qquad \dot{y}_i = -\delta^{(2)}_i y_i + (1-x_i-y_i)\sum_{j=1}^n \beta^{(2)}_{ij}y_j, \] or, in vector form, \[ \dot{x}=-D^{(1)}x+\big(I-\operatorname{diag}(x)-\operatorname{diag}(y)\big)B^{(1)}x, \qquad \dot{y}=-D^{(2)}y+\big(I-\operatorname{diag}(x)-\operatorname{diag}(y)\big)B^{(2)}y. \] Writing \(X=\operatorname{diag}(x)\) and \(Y=\operatorname{diag}(y)\), this is equivalently \[ \dot{x}=\big(-D^{(1)}+B^{(1)}-XB^{(1)}-YB^{(1)}\big)x, \qquad \dot{y}=\big(-D^{(2)}+B^{(2)}-YB^{(2)}-XB^{(2)}\big)y. \] Under \(x(0),y(0)\in[0,1]^n\) with \(x(0)+y(0)\le 1\), the set \(D\) is positively invariant [1901.00765]. This system inherits the single-virus SIS threshold structure. For \[ \dot{z}=\big(-D+B-\operatorname{diag}(z)B\big)z, \] if \(s(-D+B)\le 0\), then \(z=0\) is the only equilibrium and is globally asymptotically stable on \([0,1]^n\); if \(s(-D+B)>0\), there are exactly two equilibria, \(z=0\) and a unique endemic equilibrium \(z^*\gg 0\), with \(z^*\) asymptotically stable on \([0,1]^n\setminus\{0\}\) [1901.00765]. The multipartite fluid-limit bi-virus system has the closely related form \[ \frac{d}{dt}y_{ik}(t)=\Big(\sum_{j\sim i}\overline{\gamma}^{k}_{ji}y_{jk}(t)\Big)\Big(1-\sum_{m=1}^{K}y_{im}(t)\Big)-\mu^k_i y_{ik}(t), \] which is an exact macroscopic limit of a normalized Markov jump process for large islands. In this representation, the adjacency-driven inflow is modulated by the same occupancy factor \(1-\sum_m y_{im}\), so the exclusivity constraint is structurally identical even though the microscopic derivation differs [1306.6198]. A common misconception is that all MBRD models are Laplacian diffusion systems. In the 2019 bi-virus model, “diffusion” is represented implicitly by adjacency-weighted infection terms \(B^{(k)}x\) rather than by explicit spatial Laplacians; the paper states that there are no explicit spatial Laplacian diffusion terms. The multipartite fluid-limit model likewise yields adjacency-gated infection inflows rather than gradient-driven fluxes [1901.00765; 1306.6198]. ## 3. Explicit reaction-diffusion and cross-diffusion formulations The explicit MBRD frameworks introduced in 2025 separate reaction kinetics from graph diffusion and cross-diffusion. For the super-infection model MBRD-SI, the supra-Laplacian block structure is \[ \mathcal{L}_{\mathrm{SI}}= \begin{pmatrix} d_{11}L^{(S)} & d_{12}L^{(I)} & d_{13}L^{(J)}\\ 0 & d_{22}L^{(I)} & 0\\ 0 & 0 & d_{33}L^{(J)} \end{pmatrix}, \] where \(d_{11},d_{22},d_{33}\) are self-diffusion rates and \(d_{12},d_{13}\) are cross-diffusion coefficients acting only on the susceptible equation [2509.03374]. The node-wise MBRD-SI equations are \[ \begin{aligned} \frac{dS_i}{dt} &= rS_i\left(1-\frac{S_i}{K}\right)\left(\frac{S_i}{A}-1\right) -\frac{(\beta_1 I_i+\beta_2 J_i)S_i}{S_i+I_i+J_i} +\gamma_1 I_i+\gamma_2 J_i-\mu S_i\\ &\quad + d_{11}\sum_{j=1}^N L_{ij}^{(S)}S_j + d_{12}\sum_{j=1}^N L_{ij}^{(I)}I_j + d_{13}\sum_{j=1}^N L_{ij}^{(J)}J_j,\\ \frac{dI_i}{dt} &= I_i\left(\frac{\beta_1 S_i}{S_i+I_i+J_i}-\mu-\alpha_1-\gamma_1-\frac{\sigma\beta_2 J_i}{S_i+I_i+J_i}\right) +d_{22}\sum_{j=1}^N L_{ij}^{(I)}I_j,\\ \frac{dJ_i}{dt} &= J_i\left(\frac{\beta_2 S_i}{S_i+I_i+J_i}-\mu-\alpha_2-\gamma_2+\frac{\sigma\beta_2 I_i}{S_i+I_i+J_i}\right) +d_{33}\sum_{j=1}^N L_{ij}^{(J)}J_j. \end{aligned} \] Superinfection enters through the replacement terms \(\pm (\sigma\beta_2 I_iJ_i)/(S_i+I_i+J_i)\): pathogen \(2\) steals hosts from pathogen \(1\) [2508.15740]. MBRD-CI adds a dedicated coinfection compartment \(C_i\), transmission modifiers \(\beta_{10},\beta_{02},\beta_{12}\), coinfection virulence \(\alpha_{12}\), and recovery \(\gamma_1+\gamma_2\). Diffusion again acts on \(S\), \(I\), and \(J\), while \(C\) remains reaction-only. The effective incidence denominator is \(S_i+I_i+J_i-C_i\), because \(I\) and \(J\) include \(C\) [2508.15740; 2509.03374]. These graph systems admit continuous-space analogs with homogeneous Neumann boundary conditions by replacing graph Laplacians with \(-\Delta\). The susceptible PDE takes the form \[ \partial_t S = R_S(S,I,J,C)+\nabla\cdot\big(d_{11}\nabla S+d_{12}\nabla I+d_{13}\nabla J\big), \] while \(I\) and \(J\) satisfy standard diffusion equations and \(C\) remains reaction-only [2509.03374]. A broader multiplex reaction-diffusion theory predates MBRD-SI and MBRD-CI. For two species \(u,v\) on a two-layer multiplex, \[ \dot{u}^K_i=f(u_i^K,v_i^K)+D_u^K\sum_{j=1}^{\Omega}L_{ij}^K u_j^K+D_u^{12}(u_i^{K+1}-u_i^K), \] \[ \dot{v}^K_i=g(u_i^K,v_i^K)+D_v^K\sum_{j=1}^{\Omega}L_{ij}^K v_j^K+D_v^{12}(v_i^{K+1}-v_i^K), \qquad K=1,2. \] This formalism provides the supra-Laplacian and perturbative spectral machinery later reused in bi-virus settings [1406.6401]. ## 4. Thresholds, equilibria, and instability mechanisms In the multiplex competitive SIS model, the healthy state \((x,y)=(0,0)\) always exists and is the unique equilibrium, globally asymptotically stable on \(D\), if and only if \[ s(-D^{(1)}+B^{(1)})\le 0 \quad\text{and}\quad s(-D^{(2)}+B^{(2)})\le 0. \] If \(s(-D^{(1)}+B^{(1)})>0\) and \(s(-D^{(2)}+B^{(2)})\le 0\), there are exactly two equilibria: the unstable healthy state and a dominant virus-1 state \((\tilde{x}^{(1)},0)\), which is asymptotically stable with domain of attraction \[ \mathcal{D}_1=D\setminus\{(0,y)\mid y\in[0,1]^n\}. \] The symmetric statement holds for virus \(2\). If both spectral abscissae are positive, there exist at least three equilibria: the healthy state and the two single-virus endemic states [1901.00765]. Coexistence is strongly constrained by layer alignment. When both viruses share the same directed infection graph and homogeneous rates \(B^{(k)}=\beta^{(k)}A\), \(D^{(k)}=\delta^{(k)}I\), coexisting equilibria can exist only if \[ \frac{\delta^{(1)}}{\beta^{(1)}}=\frac{\delta^{(2)}}{\beta^{(2)}}. \] At coexistence, irreducibility forces proportionality \(x=\alpha y\), \(\alpha>0\). If \(s(A)>\delta^{(1)}/\beta^{(1)}=\delta^{(2)}/\beta^{(2)}\), there are infinitely many coexisting equilibria parameterized by \(\alpha\), and the Jacobian has a zero eigenvalue along the direction \([x;-x]\), so classical linear analysis is inconclusive [1901.00765]. The 2025 MBRD papers add diffusion-driven pattern formation to this equilibrium picture. At the disease-free equilibrium, \(S^*\) solves \[ rS\left(1-\frac{S}{K}\right)\left(\frac{S}{A}-1\right)-\mu S=0, \] and the uniform-mode thresholds, ignoring diffusion, are \[ R_0^{(1)}=\frac{\beta_1}{\mu+\alpha_1+\gamma_1}, \qquad R_0^{(2)}=\frac{\beta_2}{\mu+\alpha_2+\gamma_2}, \] with \[ R_0^{(C)}=\frac{\beta_{12}}{\mu+\alpha_{12}+\gamma_1+\gamma_2} \] for the coinfection compartment in MBRD-CI [2509.03374]. Linearization around homogeneous equilibria yields cubic and quartic characteristic polynomials. For the three-morphogen MBRD-SI case, reaction-only stability requires \[ p_1<0,\qquad p_2>0,\qquad p_3<0,\qquad p_1p_20\) and \(D\succ 0\), linearization of the equilibrium equation yields \[ \big(-D+B-X^*B-\operatorname{diag}(Bz^*)\big)\Delta z^* \approx X^*\Delta\delta +(X^*-I)(\Delta B)z^*, \] and the matrix on the left has a strictly negative inverse. Consequently, \[ \frac{\partial z^*}{\partial \delta_i}<0 \quad\text{and}\quad \frac{\partial z^*}{\partial \beta_{ij}}>0 \] elementwise. In the dominant-state bi-virus regime, \(\tilde{x}^{(1)}\) inherits the same monotonicity with respect to \(\delta_i^{(1)}\) and \(\beta_{ij}^{(1)}\) [1901.00765]. The limitations of decentralized proportional control are sharp. For a single virus, \(\delta_i(t)=k_i x_i(t)\) yields a structurally equivalent SIS system with \(\rho(K^{-1}(K+B))>1\), so the origin is unstable and a unique nontrivial endemic equilibrium exists. For the bi-virus system with \(\delta_i^{(1)}(t)=k_i^{(1)}x_i(t)\) and \(\delta_i^{(2)}(t)=k_i^{(2)}y_i(t)\), linearization at \((0,0)\) remains unstable; proportional local feedback cannot stabilize the healthy state [1901.00765]. The shared-resource framework provides constructive mitigation strategies. If one chooses \[ \delta_i^{(k)}=\beta_{iw}^{(k)}+\sum_{j=1}^n \beta_{ij}^{(k)}+\epsilon_i^{(k)}, \qquad \epsilon_i^{(k)}\ge 0, \] then \(\epsilon_i^{(k)}=0\) for all \(i\) yields asymptotic eradication, while \(\epsilon_i^{(k)}>0\) for at least one \(i\) yields exponential eradication. Applying this to both viruses guarantees convergence to the healthy state. In the bi-virus case, under \(c^{(1)}=c^{(2)}\) and \(E^{(2)}\subseteq E^{(1)}\), healing-rate design can enforce \[ (D_w^{(1)})^{-1}B_w^{(1)}>(D_w^{(2)})^{-1}B_w^{(2)} \] entrywise, making \((\tilde{y}^{(1)},0)\) the only locally asymptotically stable equilibrium; the paper interprets this as using one virus to eradicate the other [2011.07569]. Beyond epidemiology, the same formal machinery has been mapped to information diffusion, malware spread, and urban transportation. In threshold-based co-diffusion, synergy is encoded through a multivariate Hill kernel and dormancy acts as a one-directional immunity against spreading only. In MBRD-SI and MBRD-CI, superinfection corresponds to competitive replacement, while coinfection parameters \(\beta_{10},\beta_{02},\beta_{12}\) encode non-interaction, mutual enhancement, one-side enhancement or inhibition, and mutual inhibition [1806.11526; 2508.15740]. Several limitations are explicit in the literature. The 2019 bi-virus model has no explicit spatial Laplacian. The 2025 co-infection model assumes no diffusion for the \(C\) layer. The multipartite fluid-limit theory fixes the number of islands while taking a large-island limit. The threshold social-contagion model uses algorithmic adoption rules rather than a deterministic closed-form threshold equation. This suggests that “MBRD” is best understood not as a single canonical equation, but as a technically coherent class of multiplex two-contagion systems linked by exclusive or partial occupancy constraints, layer-specific transport, and spectral or mode-based criteria for eradication, dominance, coexistence, and pattern formation [1901.00765; 1306.6198; 1806.11526; 2508.15740; 2509.03374]. Source: https://www.emergentmind.com/topics/multiplex-bi-virus-reaction-diffusion-models-mbrd