Year: 2026
Type: article
Abstract: Shigellosis is a leading bacterial cause of moderate-to-severe childhood diarrhoea in sub-Saharan Africa, is increasingly resistant to first-line antibiotics, and has no widely licensed vaccine. Its control is complicated by two transmission routes - direct faecal-oral person-to-person spread, favoured by the very low infectious dose, and an indirect waterborne route through a contaminated environmental reservoir - that respond differently to intervention and to season. We formulate a periodic Lagrangian metapopulation model with susceptible-symptomatic-asymptomatic-recovered (S-I-A-R) human compartments and an environmental compartment b in each patch. The direct route is coupled across patches by a residence-time (Lagrangian) matrix, while the reservoir is transported hydrologically by an explicit graph Laplacian (Chari-Logone → N'Djamena → Lac Tchad); rains amplify chiefly the environmental route, which makes the seasonal forcing non-separable. We prove well-posedness, compute the next-generation reproduction number, and establish that it decomposes exactly into additive route kernels, K = Kdir + Kwat, with R0 = r(K) (Lemma 1); we characterise the basic reproduction number of the non-autonomous system in the sense of Wang-Zhao through the monodromy matrix, recovering the time-averaged number as the special case ε = 0. At the calibrated baseline the direct and waterborne routes contribute comparably (R0dir ≈ 1.015, R0wat ≈ 0.922; elasticities +0.52 and +0.48, summing to unity) and the corresponding reproduction numbers combine mildly sub-additively (R0avg ≈ 1.836 < R0dir + R0wat = 1.937), since the spectral radius of a sum of non-negative matrices need not equal the sum of the radii even though the kernels themselves add exactly. The direct-route floor result (Proposition 2) holds at this calibration for both R0avg and the exact Floquet threshold provided R0dir > 1: the baseline calibration supports this (R0dir ≈ 1.015), a targeted robustness sweep along the manifold consistent with the calibrated transmission intensity finds it holds for about half of the admissible parameter combinations, and the CFR calibration of Section VIII places R0dir above the threshold - but independent, patch-level ±30% uncertainty alone would leave the sign of R0dir − 1 genuinely uncertain, so we present the floor as supported by, not established by, the calibration. The exact Floquet threshold (R0Floquet ≈ 1.895) exceeds the time-averaged proxy (R0avg ≈ 1.836) because rains amplify chiefly the environmental route; a quadratic-in-σ calculation - combining the route kernels before, not after, taking spectral radii, as Lemma 1 requires - gives the heuristic bracket 1.836 < 1.895 < 1.946, and an explicit reservoir-residence correlation calculation shows the seasonal correlation between shedding and uptake is very nearly perfect (≈ 0.997) on the relevant timescale. A normalised sensitivity analysis, a fully specified optimal-control problem with three levers (WASH, hygiene, antibiotic treatment) solved by Pontryagin's principle with control-cost weights comparable to the disease-burden cost, and simulations calibrated to an explicit, reproducible parameter set illustrate the theory. Optimal effort is front-loaded and tracks the rainy-season reservoir surge, reducing symptomatic burden by ≈ 97%. A fifty-fold sweep over the relative cost of the treatment lever shows the achieved burden reduction is nearly invariant (95.6-98.0%) while the mean treatment intensity required falls about ten-fold (0.62 → 0.06): comparable epidemiological performance is achievable by leaning far more on WASH and hygiene and far less on antibiotics. Stated even more directly, banning antibiotic treatment outright and re-optimising WASH and hygiene alone costs only 1.87 percentage points of burden reduction relative to the balanced optimum, using less total control effort than the treatment-heavy programme, not more - but this substitution is a property of the epidemiology only given that WASH and hygiene remain the cheaper levers, and reverses if that cost ordering is reversed. Because a one-season objective does not price in the resistance externality of antibiotics, this near-invariance, not the treatment-heavy naive optimum, is the quantitative statement that should guide policy: the mathematically attractive treatment lever should in practice be subordinated to the transmission-blocking levers.
Source: BIOMATH
Institution: University of N'Djamena
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7225384605