Threshold behavior and exponential ergodicity of an sir epidemic model: the impact of random jamming and hospital capacity.

This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease's persistence or eradication depends on the basic reproduction number [Formula: see text]. If [Formula: see text], the disease is eradicated with a probability of 1, while [Formula: see text] results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if [Formula: see text], the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required.

Treatment strategies for the latent tuberculosis infections.

This study investigates the dynamics of tuberculosis transmission through mathematical modeling, incorporating exogenous reinfections and different treatment approaches for latent tuberculosis infections. We examine three types of treatment rates: saturated, unsaturated, and mass screening-then-treatment. Our results reveal that both saturated treatment and mass screening-then-treatment can lead to a backward bifurcation, while unsaturated treatment does not. To determine the global dynamics of the models, we employ a persistent approach that avoids classifying the steady mode. By applying the models to China, we demonstrate that the data favors the use of unsaturated treatment. If unsaturated treatment is not feasible, the optimal strategy is to screen high-risk groups, identify LTBIs, and administer unsaturated treatment. Saturated treatments are not recommended.

Basic reinfection number and backward bifurcation.

Some epidemiological models exhibit bi-stable dynamics even when the basic reproduction number $ {{{\cal R}_{0}}} $ is below $ 1 $, through a phenomenon known as a backward bifurcation. Causes for this phenomenon include exogenous reinfection, super-infection, relapse, vaccination exercises, heterogeneity among subpopulations, etc. To measure the reinfection forces, this paper defines a second threshold: the basic reinfection number. This number characterizes the type of bifurcation when the basic reproduction number is equal to one. If the basic reinfection number is greater than one, the bifurcation is backward. Otherwise it is forward. The basic reinfection number with the basic reproduction number together gives a complete measure for disease control whenever reinfections (or relapses) matter. We formulate the basic reinfection number for a variety of epidemiological models.

TIF-Reg: Point Cloud Registration with Transform-Invariant Features in SE(3).

Three-dimensional point cloud registration (PCReg) has a wide range of applications in computer vision, 3D reconstruction and medical fields. Although numerous advances have been achieved in the field of point cloud registration in recent years, large-scale rigid transformation is a problem that most algorithms still cannot effectively handle. To solve this problem, we propose a point cloud registration method based on learning and transform-invariant features (TIF-Reg). Our algorithm includes four modules, which are the transform-invariant feature extraction module, deep feature embedding module, corresponding point generation module and decoupled singular value decomposition (SVD) module. In the transform-invariant feature extraction module, we design TIF in SE(3) (which means the 3D rigid transformation space) which contains a triangular feature and local density feature for points. It fully exploits the transformation invariance of point clouds, making the algorithm highly robust to rigid transformation. The deep feature embedding module embeds TIF into a high-dimension space using a deep neural network, further improving the expression ability of features. The corresponding point cloud is generated using an attention mechanism in the corresponding point generation module, and the final transformation for registration is calculated in the decoupled SVD module. In an experiment, we first train and evaluate the TIF-Reg method on the ModelNet40 dataset. The results show that our method keeps the root mean squared error (RMSE) of rotation within 0.5∘ and the RMSE of translation error close to 0 m, even when the rotation is up to [-180∘, 180∘] or the translation is up to [-20 m, 20 m]. We also test the generalization of our method on the TUM3D dataset using the model trained on Modelnet40. The results show that our method's errors are close to the experimental results on Modelnet40, which verifies the good generalization ability of our method. All experiments prove that the proposed method is superior to state-of-the-art PCReg algorithms in terms of accuracy and complexity.

Estimating the quarantine failure rate for COVID-19.

Quarantine is a crucial control measure in reducing imported COVID-19 cases and community transmissions. However, some quarantined COVID-19 patients may show symptoms after finishing quarantine due to a long median incubation period, potentially causing community transmissions. To assess the recommended 14-day quarantine policy, we develop a formula to estimate the quarantine failure rate from the incubation period distribution and the epidemic curve. We found that the quarantine failure rate increases with the exponential growth rate of the epidemic curve. We apply our formula to United States, Canada, and Hubei Province, China. Before the lockdown of Wuhan City, the quarantine failure rate in Hubei Province is about 4.1%. If the epidemic curve flattens or slowly decreases, the failure rate is less than 2.8%. The failure rate in US may be as high as 8.3%-11.5% due to a shorter 10-day quarantine period, while the failure rate in Canada may be between 2.5% and 3.9%. A 21-day quarantine period may reduce the failure rate to 0.3%-0.5%.

Dynamics of a ratio-dependent population model for Green Sea Turtle with age structure.

The reproduction of the green sea turtles is characterized by the temperature dependent sex determination (TSD). Green sea turtle eggs are laid asexually. Temperature during hatching determines the sex of baby green sea turtles. In order to study the population dynamics of the green sea turtles and understand the dynamics of the sex ratio, in this paper we establish a stage-structured model by incorporating TSD and the ratio dependent Holling III functional response in the reproduction process of the green sea turtle population. The effects of incubation temperature and sex ratio deviation on persistence of the population are captured by the sole basic reproduction number. The persistent mode can be either a stable equilibrium or periodic oscillations. Numerical simulations and sensitive analysis help us to identify vital parameters in our model. Our research in the paper is in favor of elevating sexual encounter rates, reducing the searching time for males and increasing survival odds from baby state into adult in order to maintain sustainability of the green sea turtles.

Noise-induced transitions in a non-smooth SIS epidemic model with media alert.

We investigate a non-smooth stochastic epidemic model with consideration of the alerts from media and social network. Environmental uncertainty and political bias are the stochastic drivers in our mathematical model. We aim at the interfere measures assuming that a disease has already invaded into a population. Fundamental findings include that the media alert and social network alert are able to mitigate an infection. It is also shown that interfere measures and environmental noise can drive the stochastic trajectories frequently to switch between lower and higher level of infections. By constructing the confidence ellipse for each endemic equilibrium, we can estimate the tipping value of the noise intensity that causes the state switching.

A two-strain TB model with multiple latent stages.

A two-strain tuberculosis (TB) transmission model incorporating antibiotic-generated TB resistant strains and long and variable waiting periods within the latently infected class is introduced. The mathematical analysis is carried out when the waiting periods are modeled via parametrically friendly gamma distributions, a reasonable alternative to the use of exponential distributed waiting periods or to integral equations involving ``arbitrary'' distributions. The model supports a globally-asymptotically stable disease-free equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one. Conditions for the existence and maintenance of TB resistant strains are discussed. The possibility of exogenous re-infection is added and shown to be capable of supporting multiple equilibria; a situation that increases the challenges faced by public health experts. We show that exogenous re-infection may help established resilient communities of actively-TB infected individuals that cannot be eliminated using approaches based exclusively on the ability to bring the control reproductive number just below 1.

Different types of backward bifurcations due to density-dependent treatments.

A set of deterministic SIS models with density-dependent treatments are studied to understand the disease dynamics when different treatment strategies are applied. Qualitative analyses are carried out in terms of general treatment functions. It has become customary that a backward bifurcation leads to bistable dynamics. However, this study finds that finds that bistability may not be an option at all; the disease-free equilibrium could be globally stable when there is a backward bifurcation. Furthermore, when a backward bifurcation occurs, the fashion of bistability could be the coexistence of either dual stable equilibria or the disease-free equilibrium and a stable limit cycle. We also extend the formula for mean infection period from density-independent treatments to density-dependent ones. Finally, the modeling results are applied to the transmission of gonorrhea in China, suggesting that these gonorrhea patients may not seek medical treatments in a timely manner.

From the guest editors.

Carlos Castilo-Chavez is a Regents Professor, a Joaquin Bustoz Jr. Professor of Mathematical Biology, and a Distinguished Sustainability Scientist at Arizona State University. His research program is at the interface of the mathematical and natural and social sciences with emphasis on (i) the role of dynamic social landscapes on disease dispersal; (ii) the role of environmental and social structures on the dynamics of addiction and disease evolution, and (iii) Dynamics of complex systems at the interphase of ecology, epidemiology and the social sciences. Castillo-Chavez has co-authored over two hundred publications (see goggle scholar citations) that include journal articles and edited research volumes. Specifically, he co-authored a textbook in Mathematical Biology in 2001 (second edition in 2012); a volume (with Harvey Thomas Banks) on the use of mathematical models in homeland security published in SIAM's Frontiers in Applied Mathematics Series (2003); and co-edited volumes in the Series Contemporary Mathematics entitled '' Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges'' (American Mathematical Society, 2006) and Mathematical and Statistical Estimation Approaches in Epidemiology (Springer-Verlag, 2009) highlighting his interests in the applications of mathematics in emerging and re-emerging diseases. Castillo-Chavez is a member of the Santa Fe Institute's external faculty, adjunct professor at Cornell University, and contributor, as a member of the Steering Committee of the '' Committee for the Review of the Evaluation Data on the Effectiveness of NSF-Supported and Commercially Generated Mathematics Curriculum Materials,'' to a 2004 NRC report. The CBMS workshop '' Mathematical Epidemiology with Applications'' lectures delivered by C. Castillo-Chavez and F. Brauer in 2011 have been published by SIAM in 2013.

Mathematical modelling and control of echinococcus in Qinghai province, China.

In this paper, two mathematical models, the baseline model and the intervention model, are proposed to study the transmission dynamics of echinococcus. A global forward bifurcation completely characterizes the dynamical behavior of the baseline model. That is, when the basic reproductive number is less than one, the disease-free equilibrium is asymptotically globally stable; when the number is greater than one, the endemic equilibrium is asymptotically globally stable. For the intervention model, however, the basic reproduction number alone is not enough to describe the dynamics, particularly for the case where the basic reproductive number is less then one. The emergence of a backward bifurcation enriches the dynamical behavior of the model. Applying these mathematical models to Qinghai Province, China, we found that the infection of echinococcus is in an endemic state. Furthermore, the model appears to be supportive of human interventions in order to change the landscape of echinococcus infection in this region.

Epidemic spread of influenza viruses: The impact of transient populations on disease dynamics.

The recent H1N1 ("swine flu") pandemic and recent H5N1 ("avian flu") outbreaks have brought increased attention to the study of the role of animal populations as reservoirs for pathogens that could invade human populations. It is believed that pigs acquired flu strains from birds and humans, acting as a mixing vessel in generating new influenza viruses. Assessing the role of animal reservoirs, particularly reservoirs involving highly mobile populations (like migratory birds), on disease dispersal and persistence is of interests to a wide range of researchers including public health experts and evolutionary biologists. This paper studies the interactions between transient and resident bird populations and their role on dispersal and persistence. A metapopulation framework based on a system of nonlinear ordinary differential equations is used to study the transmission dynamics and control of avian diseases. Simplified versions of mathematical models involving a limited number of migratory and resident bird populations are analyzed. Epidemiological time scales and singular perturbation methods are used to reduce the dimensionality of the model. Our results show that mixing of bird populations (involving residents and migratory birds) play an important role on the patterns of disease spread.

Existence of multiple-stable equilibria for a multi-drug-resistant model of Mycobacterium tuberculosis.

The resurgence of multi-drug-resistant tuberculosis in some parts of Europe and North America calls for a mathematical study to assess the impact of the emergence and spread of such strain on the global effort to effectively control the burden of tuberculosis. This paper presents a deterministic compartmental model for the transmission dynamics of two strains of tuberculosis, a drug-sensitive (wild) one and a multi-drug-resistant strain. The model allows for the assessment of the treatment of people infected with the wild strain. The qualitative analysis of the model reveals the following. The model has a disease-free equilibrium, which is locally asymptotically stable if a certain threshold, known as the effective reproduction number, is less than unity. Further, the model undergoes a backward bifurcation, where the disease-free equilibrium coexists with a stable endemic equilibrium. One of the main novelties of this study is the numerical illustration of tri-stable equilibria, where the disease-free equilibrium coexists with two stable endemic equilibrium when the aforementioned threshold is less than unity, and a bi-stable setup, involving two stable endemic equilibria, when the effective reproduction number is greater than one. This, to our knowledge, is the first time such dynamical features have been observed in TB dynamics. Finally, it is shown that the backward bifurcation phenomenon in this model arises due to the exogenous re-infection property of tuberculosis.

Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment.

This paper addresses the synergistic interaction between HIV and mycobacterium tuberculosis using a deterministic model, which incorporates many of the essential biological and epidemiological features of the two dis- eases. In the absence of TB infection, the model (HIV-only model) is shown to have a globally asymptotically stable, disease-free equilibrium whenever the associated reproduction number is less than unity and has a unique endemic equilibrium whenever this number exceeds unity. On the other hand, the model with TB alone (TB-only model) undergoes the phenomenon of back- ward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. The analysis of the respective reproduction thresholds shows that the use of a targeted HIV treatment (using anti-retroviral drugs) strategy can lead to effective control of HIV provided it reduces the relative infectiousness of individuals treated (in comparison to untreated HIV-infected individuals) below a certain threshold. The full model, with both HIV and TB, is simulated to evaluate the impact of the various treatment strategies. It is shown that the HIV-only treatment strategy saves more cases of the mixed infection than the TB-only strategy. Further, for low treatment rates, the mixed-only strategy saves the least number of cases (of HIV, TB, and the mixed infection) in comparison to the other strategies. Thus, this study shows that if resources are limited, then targeting such resources to treating one of the diseases is more beneficial in reducing new cases of the mixed infection than targeting the mixed infection only diseases. Finally, the universal strategy saves more cases of the mixed infection than any of the other strategies.

Dynamics of starvation in humans.

A differential equation model describing the dynamics of stored energy in the form of fat mass, lean body mass and ketone body mass during prolonged starvation is developed. The parameters of the model are estimated using available data for 7 days into starvation. A simulation of energy stores for a normal individual with body mass index between 19 and 24 and an obese individual with body mass index over 30 are calculated. The length of time the obese subject can survive during prolonged starvation is estimated using the model.

Raves, clubs and ecstasy: the impact of peer pressure.

Ecstasy has gained popularity among young adults who frequent raves and nightclubs. The Drug Enforcement Administration reported a 500 percent increase in the use of ecstasy between 1993 and 1998. The number of ecstasy users kept growing until 2002, years after a national public education initiative against ecstasy use was launched. In this study, a system of differential equations is used to model the peer-driven dynamics of ecstasy use. It is found that backward bifurcations describe situations when sufficient peer pressure can cause an epidemic of ecstasy use. Furthermore, factors that have the greatest influence on ecstasy use as predicted by the model are highlighted. The effect of education is also explored, and the results of simulations are shown to illustrate some possible outcomes.

A simple epidemic model with surprising dynamics.

A simple model incorporating demographic and epidemiological processes is explored. Four re-parameterized quantities the basic demographic reproductive number (R(d)), the basic epidemiological reproductive number (R(0)), the ratio (v) between the average life spans of susceptible and infective class, and the relative fecundity of infectives (theta), are utilized in qualitative analysis. Mathematically, non-analytic vector fields are handled by blow-up transformations to carry out a complete and global dynamical analysis. A family of homoclinics is found, suggesting that a disease outbreak would be ignited by a tiny number of infectious individuals.

Dynamical models of tuberculosis and their applications.

The reemergence of tuberculosis (TB) from the 1980s to the early 1990s instigated extensive researches on the mechanisms behind the transmission dynamics of TB epidemics. This article provides a detailed review of the work on the dynamics and control of TB. The earliest mathematical models describing the TB dynamics appeared in the 1960s and focused on the prediction and control strategies using simulation approaches. Most recently developed models not only pay attention to simulations but also take care of dynamical analysis using modern knowledge of dynamical systems. Questions addressed by these models mainly concentrate on TB control strategies, optimal vaccination policies, approaches toward the elimination of TB in the U.S.A., TB co-infection with HIV/AIDS, drug-resistant TB, responses of the immune system, impacts of demography, the role of public transportation systems, and the impact of contact patterns. Model formulations involve a variety of mathematical areas, such as ODEs (Ordinary Differential Equations) (both autonomous and non-autonomous systems), PDEs (Partial Differential Equations), system of difference equations, system of integro-differential equations, Markov chain model, and simulation models.

Tuberculosis models with fast and slow dynamics: the role of close and casual contacts.

Models that incorporate local and individual interactions are introduced in the context of the transmission dynamics of tuberculosis (TB). The multi-level contact structure implicitly assumes that individuals are at risk of infection from close contacts in generalized household (clusters) as well as from casual (random) contacts in the general population. Epidemiological time scales are used to reduce the dimensionality of the model and singular perturbation methods are used to corroborate the results of time-scale approximations. The concept and impact of optimal average cluster or generalized household size on TB dynamics is discussed. We also discuss the potential impact of our results on the spread of TB.