Air quality index induced nonsmooth system for respiratory infection

It has been observed that air pollution greatly affects respiratory infection and generates public health problem, but there are many challenges to quantifying the dynamics of air pollution and evaluating its impact on respiratory infections. A periodic Filippov system describing the state-dependent control strategy for air pollution, described by air quality index (AQI), is proposed. We theoretically analyze the non-autonomous Filippov subsystem for variation of AQI by converting into an autonomous Filippov system via increasing dimension of the system. We obtain that there is a unique periodic solution which is globally asymptotically stable. In particular, it shows that AQI stabilizes at either one of the periodic solutions of the free and control systems or a new periodic solution induced by the on-off control strategies, depending on the threshold levels. Then, a periodic system for respiratory infection with AQI-embedded transmission probability is formulated to examine the influence of air pollution on respiratory infection. We further obtain the control reproduction number (basic reproduction number with control strategies) of this periodic respiratory infection model. It is shown that the respiratory infection will go extinct if the control reproduction number is less than unity, while uniformly persists for larger than unity. The case study showed that the estimated threshold level coincides with the actual threshold which launches the traffic limitation measure in Xi'an, indicating the untimely density-dependent traffic limitation measure was ineffective in improving air quality. Numerically studies indicate that increasing the threshold level leads to an increase in the maximum value of the unique periodic solution for AQI and the control reproduction number for the AQI embedded SEIS model. These findings emphasize the importance of suitable threshold level to trigger interventions and suggest that untimely implementing control strategy may not effectively control air population.