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A classic SIR model with nonlinear state-dependent feedback control is proposed and investigated in which integrated control measures, including vaccination, treatment and isolation, are applied once the number of the susceptible population reaches a threshold level. The interventions are density dependent due to limitations on the availability of resources. The existence and global stability of the disease free periodic solution (DFPS) are addressed, and the threshold condition is provided which can be used to define the control reproduction number Rc for the model with state-dependent feedback control. The DFPS may also be globally stable even if the basic reproduction number R0 of the SIR model is larger than one. To show that the threshold dynamics are determined by the Rc, we employ bifurcation theories of the discrete one-parameter family of maps, which are determined by the Poincare map of the proposed model, and the main results indicate that under certain conditions a stable or unstable interior periodic solution could be generated through transcritical, pitchfork and backward bifurcations. A biphasic vaccination rate (or threshold level) could result in an inverted U-shape (or U-shape) curve which reveals some important issues related to disease control and vaccine design in bioengineering including vaccine coverage, efficiency and vaccine production. Moreover, the nonlinear state-dependent feedback control could result in novel dynamics including various bifurcations.

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Text S1. Data and methods. Table S1.Total and imported cases from 2012 to 2017. Table S2. Definitions of the parameters used in the model. Table S3. Results of the MCMC algorithm. Table S4. Correlation coefficients (P-values) between the precipitation data and the data generated from the Beta distributions. (DOCX 101 kb)

Based on the report of epidemic data sets for Hubei province and mainland China,the big data from Baidu population migration trends and distributions,we formulated the COVID-19 transmission model on complex networks for Wuhan city and the 15 surrounding cities with severe epidemics, and analyzed the possible times for resumption of work in Wuhan and its surrounding areas and the impact of resumption of work on the risk of a secondary outbreak. Firstly, we estimated the actual cumulative number of cases in Wuhan on January 23rd on the basis of the cumulative number of reported cases in other 15 cities, and obtained the control reproduction numbers for the 16 major cities in Hubei province in different periods. Our research results revealed that the early transmission risk in these areas is high and the current transmission risk is low (due to the control reproduction numbers being less than 1). By simulating the whole network model on the flow network structure and flow volume in the same period of last year, we investigated the impact of the resumption of work on February 17, February 24 and March 2 on disease infection for each city. Main conclusion showed that with strong prevention measures and self-protection the second outbreak will not be caused during a period by the resumption of work on March 2, 2020.

Surgery is the traditional method for treating cancers, but it often fails to cure patients for complex reasons so new therapeutic approaches that include both surgery and immunotherapy have recently been proposed. These have been shown to be effective, clinically, in inhibiting cancer cells while allowing retention of immunologic memory. This comprehensive strategy is guided by whether a population of tumour cells has or has not exceeded a threshold density. Conditions for successful control of tumours in an immune tumour system were modeled and the related dynamics were addressed. A mathematical model with state-dependent impulsive interventions is formulated to describe combinations of surgery with immunotherapy. By analysing the properties of the Poincar´e map, we examine the global dynamics of the immune tumour system with state-dependent feedback control, including the existence and stability of the semi-trivial order-1 periodic solution and the positive order-k periodic solution. The main results showed that surgery alone can only control the tumour size below a certain level while there is no immunologic memory. If comprehensive therapy involving combining surgery with immunotherapy is considered, then not only can the cancers be controlled below a certain level, but the immune system can also retain its activity. The existence of positive order-k periodic solutions implies that periodical therapy is needed to control the cancers. However, choosing the treatment frequency and the strength of the therapy remains challenging, and hence a strategy of individual-based therapy is suggested.