Multi-route transmission of epidemics on temporal and spatial networks with adaptive behaviors

Epidemic dynamics with multi‑route transmission are strongly influenced by adaptive temporal and spatial contacts of human beings. We proposed a SEIRQAD‑UPL model that captures both human‑to‑human and human‑environment‑human transmission on a temporal and spatial network, where individuals adapt their social contacts and location visits in response to infection risk. Theoretical analysis shows that the system undergoes a forward transcritical bifurcation at the basic reproduction number, without backward bifurcation. Basic reproduction number is a linear combination of contributions from the two transmission routes. The control reproduction number and phase diagrams reveal that optimal epidemic control strategy requires a combination of individual quarantine, spatial location lockdown, and individual behavioral changes. Using Pontryagin Maximum Principle, we derived time‑varying optimal control policies. Numerical experiments show that multi‑route transmission accelerates epidemic outbreaks. Quarantine is the dominant control policy, and time‑varying cooperative policies achieve the suppression of epidemics at a lower cost than sustained full‑scale controls.