The epidemic of novel coronavirus pneumonia has spread throughout the country. The early epidemic cases in many provinces, including Shaanxi, are mainly imported cases. The latest epidemic situation has been decreasing under restrict prevention and control strategies. Accessing the efficacy of control measures, analyzing the impact of population flow on the epidemic situation are of great significance for the study of the epidemic situation in Shaanxi (or other areas with imported cases as the main cases) and the future response to emergent infectious diseases. According to the detailed data published by Shaanxi, we can obtain the transmission chains (infection tree), and the median durations from the illness onset to the first medical visit, to the admission, and then to the final confirmation. We can obtain the daily number of latent, infectious and hospitalized individuals and the spatial distribution of their state evolution. The control reproduction number of COVID-19 epidemic was determined (1.48-1.69). We develop the statistical inference method to calculate the effective regeneration number under the strict control measures in Shaanxi province. Furthermore, a novel stochastic discrete transmission model for COVID-19 was proposed, which integrates possible interventions and import cases. The parameterization of the formulated model was realized through multiple source data. Our main conclusion shows that intermittent population flow, close attention and effective isolation of the floating population can effectively reduce the risk of secondary outbreak, which consequently provides decision support for the orderly organization of returning to work/school.
Before the lock-down of Wuhan/Hubei/China, on January 23 rd 2020, a large number of individuals infected by COVID-19 moved from the epicenter Wuhan and the Hubei province due to the Spring Festival, resulting in an epidemic in the other provinces including the Shaanxi province. The epidemic scale in Shaanxi was comparatively small and with half of cases being imported from the epicenter. Based on the complete epidemic data including the symptom onset time and transmission chains, we calculate the control reproduction number (1.48-1.69) in Xi'an. We could also compute the time transition, for each imported or local case, from the latent, to infected, to hospitalized compartment, as well as the effective reproduction number. This calculation enables us to revise our early deterministic transmission model to a stochastic discrete epidemic model with case importation and parameterize it. Our model-based analyses reveal that the newly generated infections decay to zero quickly; the cumulative number of case-driven quarantined individuals via contact tracing stabilize at a manageable level, indicating that the intervention strategies implemented in the Shaanxi province have been effective. Risk analyses, important for the consideration of "resumption of work", show that a large second outbreak is expected if the level of case importation remains at the same level as between January 10 th and February 4 th 2020. However, if the case importation decreases by 30%, 60% and 90%, the second outbreak if happening will be of small-scale assuming contact tracing and quarantine/isolation remain as effective as before. Finally, we consider the effects of intermittent inflow with a Poisson distribution on the likelihood of multiple outbreaks. We believe the developed methodology and stochastic model provide an important model framework for the evaluation of revising travel restriction rules in the consideration of resuming social-economic activities while managing the disease control with potential case importation.
During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically and numerically investigate the dynamics of the proposed model. Given great difficulties caused by high nonlinearity in theoretical analysis, we separately considered subsystems with only nonlinear recovery or with only saturated media impact. For the model with only nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. Numerical simulations reveal the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, saddle-node bifurcation, homoclinic bifurcation and unstable limit cycle. So the limitation of medical resources induces rich dynamics and causes much difficulties in eliminating the infectious diseases. We then investigated the dynamics of the system with only saturated media impact and concluded that saturated media impact hardly induces the complicated dynamics. Further, we parameterized the proposed model on the basis of the COVID-19 case data in mainland China and data related to news items, and estimated the basic reproduction number to be 2.86. Sensitivity analyses were carried out to quantify the relative importance of parameters in determining the cumulative number of infected individuals at the end of the first month of the outbreak. Combining with numerical analyses, we suggested that providing adequate medical resources and improving media response to infection or individuals’ response to mass media may reduce the cumulative number of the infected individuals, which mitigates the transmission dynamics during the early stage of the COVID-19 pandemic.
During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically obtained that saturated recovery significantly contributes the occurrence of backward bifurcation and rich dynamics. Then it is reasonable to only considering nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. And numerical simulations reveals the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, Saddle-Node bifurcation, Homoclinic bifurcation and unstable limit cycle. Comparing the system with linear recovery, where the threshold dynamic are almost completely characterized by a threshold condition called the basic reproduction number, we concluded that only saturated media impact hardly induces the complicated dynamics, while the nonlinear recovery function, associated with limitation of medical resources, may induce the coexistence of the disease-free equilibrium (DFE) and a endemic state or multiple endemic states, which means that the limitation of medical resources causes much difficulties in eliminating the infectious diseases.
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