Editorial: Integrative mathematical models for disease, volume II

Building on the success of the first Research Topic,'Mathematical Modeling of Diseases at Population-Level and Cellular-Level,' this second collection advances integrative mathematical modeling across biological scales. Mathematical models have become indispensable tools in infectious disease research, providing rigorous quantitative frameworks for characterizing transmission dynamics, evaluating intervention strategies, and illuminating within-host processes that govern disease severity. At the population scale, these models inform vaccination programs, quarantine design, and outbreak forecasting.At the cellular scale, they expose the dynamics of immune regulation, viral replication, and pathogen clearance. The COVID-19 pandemic made abundantly clear that progress in one domain depends critically on the other, and that no single scale of analysis is sufficient to guide an effective public health response 1. This Research Topic was conceived with a dual purpose: to bring together modeling work that operates at the interface of population and cellular biology, and to demonstrate that such integrative approaches yield insights inaccessible to single-scale analyses. The overarching goals are to deepen our mechanistic understanding of disease transmission and progression, identify the parameters that most strongly govern disease outcomes, and provide a rigorous basis for the design of targeted interventions. The four articles assembled here each address a distinct aspect of this topic, collectively spanning empirical network-based estimation of the effective reproduction number, individual-based multiscale modeling of an animal disease, feedback-driven within-host immune dynamics, and compartmental modeling of a respiratory virus.The collection opens at the macroscopic level, where the accurate, real-time measurement of transmission is paramount. Kim et al. propose a network-based method for empirically estimating the effective reproduction number R t using individual-level COVID-19 transmission records from South Korea. The authors reconstruct infector-infectee pairs from Korea Disease Control and Prevention Agency (KDCA) data spanning 2020-2021, covering 670,484 confirmed reported cases, and construct directed infection networks stratified by four age groups (0-19, 20-29, 30-59, and 60 + ) and seven cities. The empirical R t is computed as the ratio of total secondary transmission events to the number of active infectors within a rolling seven-day window, with a correction for incomplete contact tracing applied through an introduced parameter α to avoid overestimation when networks are sparse.Three characteristic patterns emerge from comparison with Cori's method 2. First, when case counts were low in the early phase of the outbreak, Cori's method overestimated R t above unity, whereas the empirical estimate remained below one, more faithfully reflecting the limited transmission actually occurring. Second, during superspreading events, the empirical R t produced sharper, more immediate peaks, reflecting its greater sensitivity to abrupt surges in transmission intensity. Third, as the Delta variant became dominant and contact tracing capacity in dense metropolitan areas was strained, the two methods converged, highlighting the value of network-based approaches in detecting subtle changes during periods of high variability. The results emphasize the value of real-time, high-resolution R t tracking, especially in regions with high variability. Moreover, the study demonstrates that regions with stringent social distancing measures saw R t fall below one within weeks, while regions with looser restrictions sustained values above one, and that-contrary to expectation-mobility trends alone were weakly correlated with R t , with superspreading at fixed locations accounting for many of the sharpest fluctuations.While Kim et al. demonstrate the power of empirical network analysis for human populations, Mufoya et al. extend network principles accross biological scales. They address a longstanding challenge in infectious disease modeling by developing an individual-based network multiscale model for foot-and-mouth disease (FMD) in cattle, grounded in the replication-transmission relativity theory 3. By explicitly coupling within-host dynamics (viral replication, epithelial cell infection, and immune responses mediated by interferon and antibodies) to a between-host spatial network in which transmission decays exponentially with distance, the model captures biological heterogeneity that aggregate models necessarily obscure. This integration is crystallized in the basic reproduction numberR 0 = max i j̸ =i β ij + ϵ i N i ϕ A i ω iwhich depicts how processes unfolding inside a single animal (cell infection rate ϵ i , viral clearance ω i , and antibody production ϕ A i ) jointly determine population-level outbreak potential alongside between-host transmission rates β ij . A notable theoretical finding is that the model admits backward bifurcation under Editorial: Integrative Mathematical Models for Disease: Volume II certain parameter regimes, implying that R 0 < 1 is necessary but not sufficient to guarantee eradication, a result with direct consequences for the design of control programmes. Sensitivity analysis confirms that viral clearance and antibody production exert the greatest influence on R 0 , formally identifying vaccination as the most potent control lever, while quarantine remains critical for suppressing between-host transmission. Network analysis of a simulated herd reveals a Poisson-like degree distribution, ruling out super-spreader dynamics and indicating that population-wide vaccination strategies are more appropriate than targeted isolation for FMD in cattle.Wang et al. address a fundamental question in infection biology: why do individuals exposed to the same pathogen recover so differently? Using a six-variable ODE model that integrates viral load, innate immunity, cellular immunity, humoral immunity, immune suppression, and IL-6, the authors demonstrate that disease outcome depends critically on the efficiency of cellular immune clearance. Weak clearance locks the system into a chronic high-inflammation state; sufficiently strong clearance permits recovery.Under continuous viral exposure, the system admits up to five coexisting steady states (Table 1). Within the bistable regime, two individuals with identical exposure may follow irreversibly divergent disease trajectories depending solely on their initial immune conditions. Sensitivity analysis reveals a notable asymmetry in the drivers of transmission. The effective contact rate β overwhelmingly dominates: a 23% reduction in β cuts R 0 by approximately 25%, whereas a 44% increase in the screening and isolation rate yields only a 7% reduction. Importantly, when β is held constant in a secondary analysis, isolation strategies appear considerably more potent, revealing that contact rate can confound the apparent effectiveness of such interventions by masking their contribution when transmission pressure is high. This interaction warrants caution when interpreting isolation efficacy in settings where contact reduction is not simultaneously controlled.Simulations yield three actionable insights. First, early contact suppression delays and flattens epidemic peaks, easing hospital strain. Second, screening and isolation of asymptomatic cases have modest systemic impact, affecting primarily the isolated class with minimal influence on population-level transmission.Third, waning immunity, modeled with a 6-month protective period, generates recurrent epidemic waves, whereas durable immune protection suppresses resurgence. These dynamics are, of course, sensitive to the assumed waning timescale, and empirical calibration of this parameter will be important for translating the model's projections to specific RSV-endemic settings.Together, these findings argue that public health strategies should prioritise contact reduction and long-lasting immunity through vaccination over labour-intensive isolation programmes. The framework offers a tractable foundation for evaluating combined interventions, and future extensions incorporating age-structured mixing or seasonally forced transmission could further strengthen its epidemiological relevance.The four contributions assembled in this Research Topic collectively advance mathematical epidemiology along two complementary axes. At the methodological level, they bring together empirical network reconstruction, individual-based multiscale modeling, nonlinear dynamical systems analysis, and compartmental epidemiology, demonstrating that each tradition offers tools the others lack. At the substantive level, they yield concrete, actionable insights: the superiority of network-based R t estimation in detecting superspreading events; the primacy of viral clearance and antibody production rates in governing FMD dynamics; the role of the innate-cellular-viral axis in sustaining chronic inflammation; and the dominance of contact suppression over isolation as an RSV mitigation strategy. Table 1 summarises the key results, policy implications, and our editorial perspectives for each study.A unifying theme across all four articles is the importance of heterogeneity (whether in population structure, spatial connectivity, host immune status, or disease state) and the inadequacy of homogeneousmixing assumptions when that heterogeneity is consequential. Equally prominent is the value of rigorous mathematical analysis: stability proofs, bifurcation theory, and sensitivity analysis transform simulation results into theorems about system behavior, enabling conclusions that generalize beyond any single parameter configuration. We hope this collection encourages further collaboration between mathematicians, epidemiologists, immunologists, and public health practitioners, and that the frameworks developed here find application in the design and evaluation of disease control programs.