We formulated a multiscale epidemic model linking within-host viral dynamics to population transmission under resistance-mediated loss of treatment efficacy. The within-host subsystem tracked target cells, infected cells, viral load, and immune effectors, while an SEIR model represents population spread, vaccination, and recovery. We established positivity and invariance, derived a quasi-steady population model through singular perturbation, and provided a finite-horizon reduction error estimate that included the effects of initial conditions and changes in control. For constant controls and frozen efficacy, a next-generation matrix yielded a control reproduction number and local and global disease-free stability thresholds. A finite-horizon control problem compared vaccination and treatment schedules under progressive efficacy decay, using numerically optimized open-loop policies and explicitly defined baseline controls. In 180-day full-model simulations at baseline resistance, the computed policy had a composite cost of 8623.88, compared with 9638.97 for constant maximal control, while cumulative incidence was 0.240 versus 0.221. A cost-component analysis and a comparison without the explicit viral-load penalty revealed how the objective weights affect these rankings. Under active treatment, resistance influences cumulative incidence, and a synthetic negative-binomial fit reproduces weekly exposure means.
Citation: Muattar Abdimuradova, Sherbek Razzaqov, Yerimbet Aitzhanov, Shirali Kadyrov. Thresholds and control in a multiscale epidemic model[J]. Mathematical Biosciences and Engineering, 2026, 23(10): 3184-3210. doi: 10.3934/mbe.2026110
We formulated a multiscale epidemic model linking within-host viral dynamics to population transmission under resistance-mediated loss of treatment efficacy. The within-host subsystem tracked target cells, infected cells, viral load, and immune effectors, while an SEIR model represents population spread, vaccination, and recovery. We established positivity and invariance, derived a quasi-steady population model through singular perturbation, and provided a finite-horizon reduction error estimate that included the effects of initial conditions and changes in control. For constant controls and frozen efficacy, a next-generation matrix yielded a control reproduction number and local and global disease-free stability thresholds. A finite-horizon control problem compared vaccination and treatment schedules under progressive efficacy decay, using numerically optimized open-loop policies and explicitly defined baseline controls. In 180-day full-model simulations at baseline resistance, the computed policy had a composite cost of 8623.88, compared with 9638.97 for constant maximal control, while cumulative incidence was 0.240 versus 0.221. A cost-component analysis and a comparison without the explicit viral-load penalty revealed how the objective weights affect these rankings. Under active treatment, resistance influences cumulative incidence, and a synthetic negative-binomial fit reproduces weekly exposure means.
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Figures(7) / Tables(7)
Muattar Abdimuradova, Sherbek Razzaqov, Yerimbet Aitzhanov, Shirali Kadyrov. Thresholds and control in a multiscale epidemic model[J]. Mathematical Biosciences and Engineering, 2026, 23(10): 3184-3210. doi: 10.3934/mbe.2026110
Multiscale flow diagram. The within-host subsystem (gray) couples to the population SEIR system (blue) through the viral-load-dependent transmission coefficient $ \beta_p(V_h) $. Treatment efficacy $ \epsilon(t) $ modulates viral production, while vaccination $ u(t) $ moves susceptible individuals to the recovered class
Full-model policy comparison at $ \alpha = 0.0252 $ day$ ^{-1} $: (A) infectious prevalence for four policies; (B) representative viral load on a symlog scale, including rebound; (C) heuristic and numerically optimized controls, with vaccination normalized by $ u_{\max} $; and (D) weekly new-exposure fractions, with the final interval truncated at day 180
Resistance sweep for all four policies: (A) cumulative new exposures; and (B) composite objective cost. The heuristic has no treatment and is independent of $ \alpha $, while the treatment-active schedules permit rebound as efficacy erodes
Paired mean cost savings (%) for the fixed numerically optimized schedules under resistance and parameter perturbations: (A) relative to the weak DFE heuristic; and (B) relative to constant maximal control. Each cell uses the same six draws; $ q $ denotes parameter perturbations
Synthetic negative-binomial calibration under $ (u, a) = (0.0015, 0.5) $: observed weekly fractions, generating mean, lowest returned likelihood candidate, and pointwise interval from successful parametric-bootstrap refits. Curve agreement does not establish parameter identifiability; the band excludes future count noise and failed refits
PRCC for cumulative new exposures using the same 250-point Latin hypercube under the threshold heuristic and fixed treatment-active policy $ (u, a) = (0.0015, 0.8) $; the heuristic's efficacy parameters have no structural effect because its treatment intensity is zero
Paired mean cost savings of fixed numerically optimized schedules against the weak DFE heuristic and constant maximal control under 12 common parameter draws at each resistance value; error bars are percentile bootstrap intervals for the simulated mean
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