In this task, you're given a paragraph from the research paper and your task is to generate a suitable title for the research paper based on the given paper. Under 100 words is a good title length.

Q: In this paper, we analyze a delayed SEIR epidemic model in which the latent and infected states are infective. The model has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number R 0 , is less than or equal to unity. We investigate the effect of the time delay on the stability of endemic equilibrium when R 0 > 1. We give criteria that ensure that endemic equilibrium is asymptotically stable for all time delays and a Hopf bifurcation occurs as time delay exceeds the critical value. We give formulae for the direction of Hopf bifurcations and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold reduction for functional differential equations. Numerical simulations are presented to illustrate the analytical results.

A: Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period
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Q: A new porcine coronavirus SADS-CoV was recently identified from suckling piglets 20 with severe diarrhea in southern China and its genome sequence is most identical (~95% 21 identity) to that of bat α-coronavirus HKU2. It again indicates bats are the natural 22 reservoir of many coronaviruses that have great potential for cross-species transmission 23 to animals and humans by recombination and/or mutation. Here we report the cryo-EM 24 structures of HKU2 and SADS-CoV spike glycoprotein trimers at 2.38 Å and 2.83 Å 25 resolution, respectively. HKU2 and SADS-CoV spikes exhibit very high structural 26 similarity, with subtle differences mainly distributed in the NTD and CTD of the S1 27 subunit responsible for cell attachment and receptor binding. We systematically 28 analyzed and compared the NTD, CTD, SD1 and SD2 domains of the S1 subunit and 29 the S2 subunit of HKU2 spike with those of α-, β-, γ-, and δ-coronavirus spikes. The 30 results show that the NTD and CTD of HKU2/SADS-CoV are probably the most 31 ancestral in the evolution of spike. Although the S2 subunit mediating membrane fusion 32 is highly conserved, the connecting region after fusion peptide in HKU2/SADS-CoV 33 S2 subunit also adopts a conformation distinct from other coronaviruses. These results 34 structurally demonstrate a close evolutionary relationship between HKU2 /SADS-CoV 35 and β-coronavirus spikes and provide new insights into the evolution and cross-species 36 transmission of coronaviruses. : bioRxiv preprint 39 Coronaviruses, categorized into the order Nidovirales family Coronaviridae and 40 subfamily Coronavirinae, are a large group of viral pathogens with a wide host range 1 . 41 Their infections in humans, other mammals and birds can cause respiratory, hepatic, 42 enteric and neurological diseases with varying severity 2 . The severe acute respiratory 43 syndrome coronavirus (SARS-CoV) and Middle East respiratory syndrome coronavirus 44 (MERS-CoV) have posed severe threats to human health in the 21 st century 1,3 . In the 45 meantime, coronaviruses infecting domestic animals also bring substantial economic 46 losses 4 . For example, the swine acute diarrhea syndrome coronavirus (SADS-CoV) 47 (also known as SeACoV and PEAV) isolated in 2017 caused outbreaks of severe watery 48 diarrhea of suckling piglets with a mortality up to 90% in several commercial pig farms 49 in Guangdong Province of China 5-10 . SADS-CoV is an α-coronavirus and other 50 representative members in the α-genus are porcine epidemic diarrhea virus 51 (PEDV), porcine transmissible gastroenteritis coronavirus (TGEV), porcine respiratory 52 coronavirus (PRCV), human NL63 and 229E coronaviruses (HCoV-NL63 and HCoV-53 229E) 1 . Representative members in other three genera include mouse hepatitis 54 coronavirus (MHV), bovine coronavirus (BCoV), SARS-CoV, MERS-CoV, HCoV-55 OC43 and HCoV-HKU1 in the β-genus, avian infectious bronchitis virus (IBV) in the 56 γ-genus and porcine deltacoronavirus (PdCoV) in the δ-genus 1 . 57 Cross-species transmission promoted by genetic recombination and/or mutations 58 underlies the host range expansion of coronaviruses 3,11-13 . Bats are the natural reservoir 59 of more than 30 different α-and β-coronaviruses that have great potential for 60 interspecies transmission by recombination and/or mutation 12,14-16 . Data on genetic 61 evolution, receptor binding and pathogenesis have demonstrated that human SARS-62 CoV and MERS-CoV most likely originate from bats 1 . Palm civets and dromedary 63 camels are the intermediate hosts of SARS-CoV and MERS-CoV from bats to humans 1 , 64 respectively. The newly identified porcine SADS-CoV isolates are also found to share 65 ~95% sequence identity with Rhinolophus bat coronavirus HKU2, and this further 66 stressed the severe results of coronavirus spillover from bats to domestic animals 5-10 .

A: Cryo-EM structures of HKU2 and SADS-CoV spike 1 glycoproteins and insights into coronavirus evolution 2 3 4
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Q: Reproduction numbers, defined as averages of the number of people infected by a typical case, play a central role in tracking infectious disease outbreaks. The aim of this paper is to develop methods for estimating reproduction numbers which are simple enough that they could be applied with limited data or in real time during an outbreak. I present a new estimator for the individual reproduction number, which describes the state of the epidemic at a point in time rather than tracking individuals over time, and discuss some potential benefits. Then, to capture more of the detail that micro-simulations have shown is important in outbreak dynamics, I analyse a model of transmission within and between households, and develop a method to estimate the household reproduction number, defined as the number of households infected by each infected household. This method is validated by numerical simulations of the spread of influenza and measles using historical data, and estimates are obtained for would-be emerging epidemics of these viruses. I argue that the household reproduction number is useful in assessing the impact of measures that target the household for isolation, quarantine, vaccination or prophylactic treatment, and measures such as social distancing and school or workplace closures which limit between-household transmission, all of which play a key role in current thinking on future infectious disease mitigation.

A:
Estimating Individual and Household Reproduction Numbers in an Emerging Epidemic
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