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For the two-patch logistic model, we study the effect of dispersal intensity and dispersal asymmetry on the total population abundance and its distribution. Two complete classifications of the model parameter space are given: one concerning when dispersal causes smaller or larger total biomass than no dispersal, and the other addressing how the total biomass changes with dispersal intensity and dispersal asymmetry. The dependencies of the population abundance of each individual patch on dispersal intensity and dispersal asymmetry are also fully characterized. In addition, the maximal and minimal total population sizes induced by dispersal are determined for the logistic model with an arbitrary number of patches, and a weak order-preserving result correlated the local population abundances with and without dispersal is established.

Highlight • We compare the temporal trend of the COVID-19 pandemic and 1918-19 influenza pandemic in United Kingdom. • We found that the on-going COVID-19 wave of infection matched the major wave of the 1918-19 influenza pandemic surprisingly well for the past two months. • We discussed the similar characteristics between these two pandemics. • We also showed the years of life lost (YLL) due to 1918-19 pandemic. A comparison on YLL would be more appropriate., We compare the COVID-19 pandemic and 1918-19 influenza pandemic in United Kingdom. We found that the on-going COVID-19 wave of infection matched the major wave of the 1918-19 influenza pandemic surprisingly well, both reached similar magnitude (in term of estimated weekly new infections) and spent the same duration above 5 cases per 1000 inhabitants, for the past two months. We discussed the similar characteristics between these two pandemics.

Travel frequency of people varies widely with occupation, age, gender, ethnicity, income, climate and other factors. Meanwhile, the distribution of the numbers of times people in different regions or with different travel behaviors bitten by mosquitoes may be nonuniform. To reflect these two heterogeneities, we develop a multipatch model to study the impact of travel frequency and human biting rate on the spatial spread of mosquito-borne diseases. The human population in each patch is divided into four classes: susceptible unfrequent, infectious unfrequent, susceptible frequent, and infectious frequent. The basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} is defined. It is shown that the disease-free equilibrium is globally asymptotically stable if \begin{document}$ \mathcal{R}_0\leq 1 $\end{document} , and there is a unique endemic equilibrium that is globally asymptotically stable if \begin{document}$ \mathcal{R}_0>1 $\end{document} . A more detailed study is conducted on the single patch model. We use analytical and numerical methods to demonstrate that the model without considering the difference of humans in travel frequency mostly underestimates the risk of infection. Numerical simulations suggest that the greater the difference in travel frequency, the larger the underestimate of the transmission potential. In addition, the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} may decreasingly, or increasingly, or nonmonotonically vary when more people travel frequently.

In this paper, we propose a Rosenzweig–MacArthur predator-prey model with strong Allee effect and trigonometric functional response. The local and global stability of equilibria is studied, and the existence of bifurcation is determined in terms of the carrying capacity of the prey, the death rate of the predator and the Allee effect. An analytic expression is employed to determine the criticality and codimension of Hopf bifurcation. The existence of supercritical Hopf bifurcation and the non-existence of Bogdanov–Takens bifurcation at the positive equilibrium are proved. A point-to-point heteroclinic cycle is also found. Biologically speaking, such a heteroclinic cycle always indicates the collapse of the system after the invasion of the predator, i.e., overexploitation occurs. It is worth pointing out that heteroclinic relaxation cycles are driven by either the strong Allee effect or the high per capita death rate. In addition, numerical simulations are given to demonstrate the theoretical results.

A mathematical model is developed to investigate the impact of Trypanosoma cruzi and Trypanosoma rangeli co-infection and Trypanosoma rangeli-induced pathogenicity of triatomine bugs on the spread of Chagas disease. Due to the presence of two parasites, basic reproduction numbers of one parasite in the absence of the other parasite (\begin{document}$ \mathcal{R}_{10} $\end{document} and \begin{document}$ \mathcal{R}_{20} $\end{document}) and invasion reproduction numbers of one parasite invading the other parasite (\begin{document}$ \mathcal{R}_{12} $\end{document} and \begin{document}$ \mathcal{R}_{21} $\end{document}) are derived to determine the dynamics of the co-infection system. With a simple case of two parasites' independent transmission, we have found that both parasites go extinct if both \begin{document}$ \mathcal{R}_{i0}, thus no Chagas disease spread. Nevertheless, the condition of \begin{document}$ \mathcal{R}_{i0}>1\,(i=1,2) $\end{document} is not sufficient to cause Chagas disease persistence, the invasion reproduction number of Trypanosoma cruzi invading Trypanosoma rangeli transmission \begin{document}$ \mathcal{R}_{12} $\end{document} plays an important role. Specifically, Chagas disease could go extinct if \begin{document}$ \mathcal{R}_{12}, and uniformly persistent if \begin{document}$ \mathcal{R}_{12}>1 $\end{document}. Moreover, due to pathogenicity, oscillation pattern of Chagas disease is observed, which is different from other mechanisms such as maturation delay, seasonality and regular spraying with insecticides for vector control. In conclusion, we have found that the presence of Trypanosoma rangeli infection leads to the risk reduction of Chagas disease infection. Our findings are beneficial to the prevention and control of Chagas disease.

Zika virus disease is a viral disease primarily transmitted to humans through the bite of infected female mosquitoes. Recent evidence indicates that the virus can also be sexually transmitted in hosts and vertically transmitted in vectors. In this paper, we propose a Zika model with three transmission routes, that is, vector-borne transmission between humans and mosquitoes, sexual transmission within humans and vertical transmission within mosquitoes. The basic reproduction number [Formula: see text] is computed and shown to be a sharp threshold quantity. Namely, the disease-free equilibrium is globally asymptotically stable as [Formula: see text], whereas there exists a unique endemic equilibrium which is globally asymptotically stable as [Formula: see text]. The relative contributions of each transmission route on the reproduction number, and the short- and long-term host infections are analyzed. Numerical simulations confirm that vectorial transmission contributes the most to the initial and subsequent transmission. The role of sexual transmission in the early phase of a Zika outbreak is greater than the long term, while vertical transmission is the opposite. Reducing mosquito bites is the most effective measure in lowering the risk of Zika virus infection.

Based on a Ross-Macdonald type model with a number of identical patches, we study the role of the movement of humans and/or mosquitoes on the persistence of malaria and many other vector-borne diseases. By using a theorem on line-sum symmetric matrices, we establish an eigenvalue inequality on the product of a class of nonnegative matrices and then apply it to prove that the basic reproduction number of the multipatch model is always greater than or equal to that of the single patch model. Biologically, this means that habitat fragmentation or patchiness promotes disease outbreaks and intensifies disease persistence. The risk of infection is minimized when the distribution of mosquitoes is proportional to that of humans. Numerical examples for the two-patch submodel are given to investigate how the multipatch reproduction number varies with human and/or mosquito movement. The reproduction number can surpass any given value whenever an appropriate travel pattern is chosen. Fast human and/or mosquito movement decreases the infection risk, but may increase the total number of infected humans.

One of the key epidemiological characteristics that shape the transmission of coronavirus disease 2019 (COVID-19) is the serial interval (SI). Although SI is commonly considered following a probability distribution at a population scale, recent studies reported a slight shrinkage (or contraction) of the mean of effective SI across transmission generations or over time. Here, we develop a likelihood-based statistical inference framework with truncation to explore the change in SI across transmission generations after adjusting the impacts of case isolation. The COVID-19 contact tracing surveillance data in Hong Kong are used for exemplification. We find that for COVID-19, the mean of individual SI is likely to shrink with a factor at 0.72 per generation (95%CI: 0.54, 0.96) as the transmission generation increases, where a threshold may exist as the lower boundary of this shrinking process. We speculate that one of the probable explanations for the shrinkage in SI might be an outcome due to the competition among multiple candidate infectors within the same case cluster. Thus, the nonpharmaceutical interventive strategies are crucially important to block the transmission chains, and mitigate the COVID-19 epidemic.

The COVID-19 pandemic poses a serious threat to global health, and one of the key epidemiological factors that shape the transmission of COVID-19 is its serial interval (SI). Although SI is commonly considered following a probability distribution at a population scale, slight discrepancies in SI across different transmission generations are observed from the aggregated statistics in recent studies. To explore the change in SI across transmission generations, we develop a likelihood-based statistical inference framework to examine and quantify the change in SI. The COVID-19 contact tracing surveillance data in Hong Kong are used for exemplification. We find that the individual SI of COVID-19 is likely to shrink with a rate of 0.72 per generation and 95%CI: (0.54, 0.96) as the transmission generation increases. We speculate that the shrinkage in SI is an outcome of competition among multiple candidate infectors within a cluster of cases. The shrinkage in SI may speed up the transmission process, and thus the nonpharmaceutical interventive strategies are crucially important to mitigate the COVID-19 epidemic.

Based on a susceptible-infected-susceptible patch model, we study the influence of dispersal on the disease prevalence of an individual patch and all patches at the endemic equilibrium. Specifically, we estimate the disease prevalence of each patch and obtain a weak order-preserving result that correlated the patch reproduction number with the patch disease prevalence. Then we assume that dispersal rates of the susceptible and infected populations are proportional and derive the overall disease prevalence, or equivalently, the total infection size at no dispersal or infinite dispersal as well as the right derivative of the total infection size at no dispersal. Furthermore, for the two-patch submodel, two complete classifications of the model parameter space are given: one addressing when dispersal leads to higher or lower overall disease prevalence than no dispersal, and the other concerning how the overall disease prevalence varies with dispersal rate. Numerical simulations are performed to further investigate the effect of movement on disease prevalence.

Highlights • Since the first imported case was confirmed in Hong Kong on January 23, 2020, local public interventions for SARS-CoV-2 control and prevention have been enhanced. • A regression model is adopted to quantify the association between the confirmation delay and the calendar time, validated by a Cox proportional hazard model. • We estimate that the SARS-CoV-2 confirmation efficiency improved 5.40% (95%CI: 2.54% − 8.33%) per day in Hong Kong. • The sustaining enforcement in timely confirmation and other effective control measures are recommended to prevent the spreading of the SARS-CoV-2 infection., Backgrounds The emerging virus, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), caused a large outbreak of coronavirus disease, COVID-19, in Wuhan, China, since December 2019. COVID-19 soon spread to other regions of China and overseas. In Hong Kong, local mitigation measures were implemented since the first imported case was confirmed on January 23, 2020. Here we evaluated the temporal variation of detection delay from symptoms onset to laboratory confirmation of SARS-CoV-2 in Hong Kong. Methods A regression model is adopted to quantify the association between the SARS-CoV-2 detection delay and calendar time. The association is tested and further validated by a Cox proportional hazard model. Findings The estimated median detection delay was 9.5 days (95%CI: 6.5 − 11.5) in the second half of January, reduced to 6.0 days (95%CI: 5.5 − 9.5) in the first half of February 2020. We estimate that SARS-CoV-2 detection efficiency improved at a daily rate of 5.40% (95%CI: 2.54 − 8.33) in Hong Kong. Conclusions The detection efficiency of SARS-CoV-2 was likely being improved substantially in Hong Kong since the first imported case was detected. Sustaining enforcement in timely detection and other effective control measures are recommended to prevent the SARS-CoV-2 infection.

People infected with malaria may receive less mosquito bites when they are treated in well-equipped hospitals or follow doctors’ advice for reducing exposure to mosquitoes at home. This quarantine-like intervention measure is especially feasible in countries and areas approaching malaria elimination. Motivated by mathematical models with quarantine for directly transmitted diseases, we develop a mosquito-borne disease model where imperfect quarantine is considered to mitigate the disease transmission from infected humans to susceptible mosquitoes. The basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0$$\end{document}R0 is computed and the model equilibria and their stabilities are analyzed when the incidence rate is standard or bilinear. In particular, the model system may undergo a subcritical (backward) bifurcation at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0=1$$\end{document}R0=1 when standard incidence is adopted, whereas the disease-free equilibrium is globally asymptotically stable as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0\le 1$$\end{document}R0≤1 and the unique endemic equilibrium is locally asymptotically stable as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_0>1$$\end{document}R0>1 when the infection incidence is bilinear. Numerical simulations suggest that the quarantine strategy can play an important role in decreasing malaria transmission. The success of quarantine mainly relies on the reduction of bites on quarantined individuals.

The ongoing coronavirus disease 2019 (COVID-19) outbreak, emerged in Wuhan, China in the end of 2019, has claimed more than 2600 lives as of 24 February 2020 and posed a huge threat to global public health. The Chinese government has implemented control measures including setting up special hospitals and travel restriction to mitigate the spread. We propose conceptual models for the COVID-19 outbreak in Wuhan with the consideration of individual behavioural reaction and governmental actions, e.g., holiday extension, travel restriction, hospitalisation and quarantine. We employe the estimates of these two key components from the 1918 influenza pandemic in London, United Kingdom, incorporated zoonotic introductions and the emigration, and then compute future trends and the reporting ratio. The model is concise in structure, and it successfully captures the course of the COVID-19 outbreak, and thus sheds light on understanding the trends of the outbreak. Keywords: COVID-19, Epidemic, Mathematical modelling, Individual reaction, Governmental action, City lockdown

The novel coronavirus disease 2019 (COVID-19) outbreak on the Diamond Princess ship has caused over 634 cases as of February 20, 2020. We model the transmission process on the ship with a stochastic model and estimate the basic reproduction number at 2.2 (95%CI: 2.1−2.4). Through estimating the dispersion parameter which quantifies the superspreading events, we find that stochastic bursts were unlikely to occur if the reproduction number is less than 1. The epidemic doubling time is at 4.6 days (95%CI: 3.0−9.3), thus timely action were crucial. The lesson learn on the ship is generally applicable in other settings

Background: In December 2019, an outbreak of respiratory illness caused by a novel coronavirus (2019-nCoV) emerged in Wuhan, China and has swiftly spread to other parts of China and a number of foreign countries. The 2019-nCoV cases might have been under-reported roughly from 1 to 15 January 2020, and thus we estimated the number of unreported cases and the basic reproduction number, R0, of 2019-nCoV. Methods: We modelled the epidemic curve of 2019-nCoV cases, in mainland China from 1 December 2019 to 24 January 2020 through the exponential growth. The number of unreported cases was determined by the maximum likelihood estimation. We used the serial intervals (SI) of infection caused by two other well-known coronaviruses (CoV), Severe Acute Respiratory Syndrome (SARS) and Middle East Respiratory Syndrome (MERS) CoVs, as approximations of the unknown SI for 2019-nCoV to estimate R0. Results: We confirmed that the initial growth phase followed an exponential growth pattern. The under-reporting was likely to have resulted in 469 (95% CI: 403&minus, 540) unreported cases from 1 to 15 January 2020. The reporting rate after 17 January 2020 was likely to have increased 21-fold (95% CI: 18&minus, 25) in comparison to the situation from 1 to 17 January 2020 on average. We estimated the R0 of 2019-nCoV at 2.56 (95% CI: 2.49&minus, 2.63). Conclusion: The under-reporting was likely to have occurred during the first half of January 2020 and should be considered in future investigation.

BackgroundsAn ongoing outbreak of a novel coronavirus (2019-nCoV) pneumonia hit a major city of China, Wuhan, December 2019 and subsequently reached other provinces/regions of China and countries. We present estimates of the basic reproduction number, R0, of 2019-nCoV in the early phase of the outbreak.MethodsAccounting for the impact of the variations in disease reporting rate, we modelled the epidemic curve of 2019-nCoV cases time series, in mainland China from January 10 to January 24, 2020, through the exponential growth. With the estimated intrinsic growth rate (γ), we estimated R0 by using the serial intervals (SI) of two other well-known coronavirus diseases, MERS and SARS, as approximations for the true unknown SI.FindingsThe early outbreak data largely follows the exponential growth. We estimated that the mean R0 ranges from 2.24 (95%CI: 1.96-2.55) to 3.58 (95%CI: 2.89-4.39) associated with 8-fold to 2-fold increase in the reporting rate. We demonstrated that changes in reporting rate substantially affect estimates of R0.ConclusionThe mean estimate of R0 for the 2019-nCoV ranges from 2.24 to 3.58, and significantly larger than 1. Our findings indicate the potential of 2019-nCoV to cause outbreaks.

In this work, we showed that the classical theory of final size overestimated the infection attack rate in 2009 and 1918 influenza pandemic, if we only focus the first year.

In this work, we compare the case fatality rate (CFR), basic reproductive number and infection attack rate between COVID-19 and seasonal influenza (including 2009 pandemic influenza, which has a similar CFR as seasonal influenza with a larger infection attack rate). We conclude that in an ideal situation, the CFR of COVID-19 is about 3-4 folds of seasonal influenza, but the basic reproductive number is substantially higher for COVID-19 thus the infection attack rate is higher. Without control, the total deaths could be ten or twenty times that of seasonal influenza in a severe season.

We review the three mortality waves of 1918-19 influenza pandemic in England and Wales (EW), we also show all-cause and pneumonia and influenza associated deaths in pre- and post-pandemic periods. We argue that the current COVID-19 in EW is comparable to the major wave of 1918-19 influenza, thus the forecast of the trend in the next few month or years should consider that of the 1918-19 influenza.

The COVID-19 outbreak on the Diamond Princess shihas caused over 600 (still increasing) cases as of February 19, 2020 By using the growing process model, we

We propose two differential equation-based models to investigate the impact of awareness programs on cholera dynamics. The first model represents the disease transmission rates as decreasing functions of the number of awareness programs, whereas the second model divides the susceptible individuals into two distinct classes depending on their awareness/unawareness of the risk of infection. We study the essential dynamical properties of each model, using both analytical and numerical approaches. We find that the two models, though closely related, exhibit significantly different dynamical behaviors. Namely, the first model follows regular threshold dynamics while rich dynamical behaviors such as backward bifurcation may arise from the second one. Our results highlight the importance of validating key modeling assumptions in the development and selection of mathematical models toward practical application.

Ticks, including the Ixodes ricinus and Ixodes scapularis hard tick species, are regarded as the most common arthropod vectors of both human and animal diseases in Europe and the United States capable of transmitting a large number of bacteria, viruses and parasites. Since ticks in larval and nymphal stages share the same host community which can harbor multiple pathogens, they may be co-infected with two or more pathogens, with a subsequent high likelihood of co-transmission to humans or animals. This paper is devoted to the modeling of co-infection of tick-borne pathogens, with special focus on the co-infection of Borrelia burgdorferi (agent of Lyme disease) and Babesia microti (agent of human babesiosis). Considering the effect of co-infection, we illustrate that co-infection with B. burgdorferi increases the likelihood of B. microti transmission, by increasing the basic reproduction number of B. microti below the threshold smaller than one to be possibly above the threshold for persistence. The study confirms a mechanism of the ecological fitness paradox, the establishment of B. microti which has weak fitness (basic reproduction number less than one). Furthermore, co-infection could facilitate range expansion of both pathogens.

Trypanosoma rangeli (T. rangeli), a parasite, is not pathogenic to human but pathogenic to some vector species to induce the behavior changes of infected vectors and subsequently impact the transmission dynamics of other diseases such as Chagas disease which shares the same vector species. Here we develop a mathematical model and conduct qualitative analysis for the transmission dynamics of T. rangeli. We incorporate both systemic and co-feeding transmission routes, and account for the pathogenic effect using infection-induced fecundity and fertility change of the triatomine bugs. We derive two thresholds R v (the triatomine bug basic reproduction number) and R 0 (the T. rangeli basic reproduction number) to delineate the dynamical behaviors of the ecological and epidemiological systems. We show that when R v > 1 and R 0 > 1 , a unique parasite positive equilibrium E* appears. We find that E* can be unstable and periodic oscillations can be observed where the pathogenic effect plays a significant role. Implications of the qualitative analysis and numerical simulations suggest the need of an integrative vector-borne disease prevention and control strategy when multiple vector-borne diseases are transmitted by the same set of vector species.

Lassa fever, also known as Lassa hemorrhagic fever, is a virus that has generated recurrent outbreaks in West Africa. We use mechanistic modelling to study the Lassa fever epidemics in Nigeria from 2016-19. Our model describes the interaction between human and rodent populations with the consideration of quarantine, isolation and hospitalization processes. Our model supports the phenomenon of forward bifurcation where the stability between disease-free equilibrium and endemic equilibrium exchanges. Moreover, our model captures well the incidence curves from surveillance data. In particular, our model is able to reconstruct the periodic rodent and human forces of infection. Furthermore, we suggest that the three major epidemics from 2016-19 can be modelled by properly characterizing the rodent (or human) force of infection while the estimated human force of infection also present similar patterns across outbreaks. Our results suggest that the initial susceptibility likely increased across the three outbreaks from 2016-19. Our results highlight the similarity of the transmission dynamics driving three major Lassa fever outbreaks in the endemic areas.