A Mathematical Model of the Tuberculosis Epidemic

Tuberculosis has continued to retain its title as "the captain among these men of death". This is evident as it is the leading cause of death globally from a single infectious agent. TB as it is fondly called has become a major threat to the achievement of the sustainable development goals (SDG) and hence require inputs from different research disciplines. This work presents a mathematical model of tuberculosis. A compartmental model of seven classes was used in the model formulation comprising of the susceptible S, vaccinated V, exposed E, undiagnosed infectious I₁, diagnosed infectious I₂, treated T and recovered R. The stability analysis of the model was established as well as the condition for the model to undergo backward bifurcation. With the existence of backward bifurcation, keeping the basic reproduction number less than unity [Formula: see text] is no more sufficient to keep TB out of the community. Hence, it is shown by the analysis that vaccination program, diagnosis and treatment helps to control the TB dynamics. In furtherance to that, it is shown that preference should be given to diagnosis over treatment as diagnosis precedes treatment. It is as well shown that at lower vaccination rate (0-20%), TB would still be endemic in the population. As such, high vaccination rate is required to send TB out of the community.

Outcomes at 1 Year of Non-Left Main Trunk Bifurcation Lesions Treated With a 2-Stent Strategy Using Newer-Generation Everolimus-Eluting Stents

BACKGROUND

Percutaneous coronary intervention (PCI) for coronary bifurcation lesions using the 2-stent strategy remains a challenging procedure for interventionalists because of the higher incidence of in-stent restenosis (ISR) and adverse events. ISR predictors in patients treated with newer-generation everolimus-eluting stents (EES) and the 2-stent strategy remain unknown. Hence, we aimed to evaluate the 1-year clinical and angiographic outcomes of non-left main trunk (LMT) bifurcation lesions treated with the 2-stent strategy using newer-generation EES.Methods and Results:The study sample consisted of 262 non-LMT bifurcation lesions treated using culotte or T-stenting with EES between 2010 and 2018. One-year post-procedural angiographic and clinical examinations were conducted in 208 (79.4%) and 260 (99.2%) lesions, respectively. The primary outcome measure was the 1-year post-procedural ISR rate, which was found to be 15.9%. Independent predictors of 1-year post-procedural ISR were long side branch lesions (adjusted odds ratio [aOR] 2.31; 95% confidence interval [CI] 1.02-5.23; P=0.04) and 3-link EES implantation (aOR 2.45; 95% CI 1.07-5.61; P=0.03). The 1-year cumulative incidence of target lesion revascularization was 3.5%.

CONCLUSIONS

The 1-year clinical outcomes of non-LMT bifurcation lesions treated with the 2-stent strategy using EES were acceptable. Long side branch lesions and lesions treated with 3-link EES were independent predictors of 1-year post-procedural ISR.

Two dimensionless parameters and a mechanical analogue for the HKB model of motor coordination

The widely cited Haken-Kelso-Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself.

Dynamics of a stage-structured predator-prey model: cost and benefit of fear-induced group defense

In the predator-prey system, predators can affect the prey population (1) by direct killing and (2) by inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. However, the anti-predator strategy is not the same for all individual preys of different life stages. Also, anti-predator behavior has both cost and benefit, but most of the mathematical models observed the dynamics by incorporating its cost only. In the present study, we formulate a predator-prey model dividing the prey population into two stages: juvenile and adult. We assume that adult preys are only adapting group defense as an anti-predator strategy when they are sensitive to predation. Group defense plays a positive role for adult prey by reducing their predation, but, on the negative side, it simultaneously decreases their reproductive potential. A parameter, anti-predator sensitivity is introduced to interlink both the benefit and cost of group defense. Our result shows that when adult preys are not showing anti-predator behavior, with an increase of maturation rate, the system exhibits a population cycle of abruptly increasing amplitude, which may drive all species of the system to extinction. Anti-predator sensitivity may exclude oscillation through homoclinic bifurcation and avert the prey population for any possible random extinction. Anti-predator sensitivity also decreases the predator population density and produces bistable dynamics. Higher values of anti-predator sensitivity may lead to the extinction of the predator population and benefit adult preys to persist with large population density. Below a threshold value of anti-predator sensitivity, it may possible to retain the predator population in the system by increasing the fear level of the predator. We also observe our fear-induced stage-structured model exhibits interesting and rich dynamical behaviors, various types of bistabilities in different bi-parameter planes. Finally, we discuss the potential impact of our findings.

Chaos in memory function of sleep: A nonlinear dynamical analysis in thalamocortical study

Studying the dynamical behaviors of neuronal models may help in better understanding of real nervous system. In addition, it can help researchers to understand some specific phenomena in neuronal system. The thalamocortical network is made of neurons in the thalamus and cortex. In it, the memory function is consolidated in sleep by creating up and down state oscillations (1 Hz) and fast (13-17 Hz) - slow (8-12 Hz) spindles. Recently, a nonlinear biological model for up-down oscillations and fast-slow spindles of the thalamocortical network has been proposed. In this research, the power spectral for the fast-slow spindle of the model is extracted. Dynamical properties of the model, such as the bifurcation diagrams, and attractors are investigated. The results show that the variation of the synaptic power between the excitatory neurons of the cortex and the reticular neurons in the thalamus changes the spindles' activity. According to previous experimental findings, it is an essential rule for consolidating the memory function during sleep. It is also pointed out that when the fast-slow spindles of the brain increase, the dynamics of the thalamocortical system tend to chaos.

Safety, Efficacy, and Durability of Stent Plus Balloon-Assisted Coiling for the Treatment of Wide-Necked Intracranial Bifurcation Aneurysms

BACKGROUND

Wide-necked bifurcation aneurysms remain a challenge for endovascular surgeons. Dual-stent-assisted coiling techniques have been defined to treat bifurcation aneurysms with a complex neck morphology. However, there are still concerns about the safety of dual-stenting procedures. Stent plus balloon-assisted coiling is a recently described endovascular technique that enables the coiling of wide-necked complex bifurcation aneurysms by implanting only a single stent.

OBJECTIVE

To investigate the feasibility, efficacy, safety, and durability of this technique for the treatment of wide-necked bifurcation aneurysms.

METHODS

A retrospective review was performed of patients with wide-necked intracranial bifurcation aneurysms treated with stent plus balloon-assisted coiling. The initial and follow-up clinical and angiographic outcomes were assessed. Preprocedural and follow-up clinical statuses were assessed using modified Rankin scale.

RESULTS

A total of 61 patients (mean age: 54.6 ± 10.4 yr) were included in the study. The immediate postprocedural digital subtraction angiography revealed complete aneurysm occlusion in 86.9% of the cases. A periprocedural complication developed in 11.5% of the cases. We observed a delayed ischemic complication in 4.9%. There was no mortality in this study. The permanent morbidity rate was 3.3%. The follow-up angiography was performed in 55 of 61 patients (90.1%) (the mean follow-up period was 25.5 ± 27.3 mo). The rate of complete aneurysm occlusion at the final angiographic follow-up was 89.1%. The retreatment rate was 1.8%.

CONCLUSION

The results of this study showed that stent plus balloon-assisted coiling is a feasible, effective, and relatively safe endovascular technique for the treatment of wide-necked bifurcation aneurysms located in the posterior and anterior circulation.

[Bifid median nerve as a coincidental finding in traumatology : Case report and review of the literature]

The median nerve is an important sensory and motor nerve of the arm and is not infrequently associated with diagnostic and therapeutic misjudgements. A bifid median nerve as a coincidental finding in traumatology is another rarity and is mostly found during various elective procedures, such as carpal tunnel splitting. We report on a patient with a bifid median nerve who sustained a laceration wound near the carpal tunnel due to a domestic accident.

Transmission dynamics of SARS-CoV-2: A modeling analysis with high-and-moderate risk populations

Nigeria is second to South Africa with the highest reported cases of COVID-19 in sub-Saharan Africa. In this paper, we employ an SEIR-based compartmental model to study and analyze the transmission dynamics of SARS-CoV-2 outbreaks in Nigeria. The model incorporates different group of populations (that is, high- and- moderate risk populations) and is use to investigate the influence on each population on the overall transmission dynamics.The model, which is fitted well to the data, is qualitatively analyzed to evaluate the impacts of different schemes for controlstrategies. Mathematical analysis reveals that the model has two equilibria; i.e., disease-free equilibrium (DFE) which is local asymptotic stability (LAS) if the basic reproduction number ( R 0 ) is less than 1; and unstable for R 0 > 1 , and an endemic equilibrium (EE) which is globally asymptotic stability (LAS) whenever R 0 > 1 . Furthermore, we find that the model undergoes a phenomenon of backward bifurcation (BB, a coexistence of stable DFE and stable EE even if the R 0 < 1 ). We employ Partial Rank Correlation coefficients (PRCCs) for sensitivity analyses to evaluate the model's parameters. Our results highlight that proper surveillance, especially movement of individuals from high risk to moderate risk population, testing, as well as imposition of other NPIs measures are vital strategies for mitigating the COVID-19 epidemic in Nigeria. Besides, in the absence of an exact solution for the proposed model, we solve the model with the well-known ODE45 numerical solver and the effective numerical schemes such as Euler (EM), Runge-Kutta of order 2 (RK-2), and Runge-Kutta of order 4 (RK-4) in order to establish approximate solutions and to show the physical features of the model. It has been shown that these numerical schemes are very effective and efficient to establish superb approximate solutions for differential equations.

A model of the burglar alarm hypothesis of prey alarm calls

When approached by a predator many prey species will emit an "alarm call" as a form of anti-predator behavior. One hypothesis for the function of alarm calls is the "burglar alarm" hypothesis whereby upon attack, a prey renders itself dangerous to a predator by generating an alarm call that attracts a predator at higher trophic levels in the food chain; that is, attracts a predator to the prey's own predator. This paper concerns a model incorporating a mechanism to test the burglar alarm hypothesis. We prove in one space dimension global existence, of positive bounded classical solutions, and establish existence of non-constant equilibrium solutions and assess their stability. We provide some representative numerical simulations to emphasize the nature of pattern formation for this model and demonstrate the benefit achieved by a signal inducing prey species under the burglar alarm hypothesis.

Comparison of 2-Stenting Strategies Depending on Sequence or Technique for Bifurcation Lesions in the Second-Generation Drug-Eluting Stent Era - Analysis From the COBIS (Coronary Bifurcation Stenting) III Registry

BACKGROUND

It has not been determined which specific 2-stenting strategy is the best for bifurcation lesions. Our aim was to investigate the clinical outcomes of various 2-stenting strategies in the era of 2nd-generation drug-eluting stents (2G-DES).Methods and Results:We analyzed 454 patients who finally underwent 2-stenting for a bifurcation lesion, from among 2,648 patients enrolled in the COBIS III registry. The primary outcome was target lesion failure (TLF). Patients were analyzed according to stenting sequence (provisional [main vessel stenting first] vs. systemic [side branch stenting first]) and stenting technique (crush vs. T vs. culotte vs. kissing/V stenting). Overall, 4.4 years' TLF after 2-stenting treatment for bifurcation lesion was excellent: TLF 11.2% and stent thrombosis 1.3%. There was no difference in TLF according to 2-stenting strategy (11.1% vs. 10.5%, P=0.990 for provisional and systemic sequence; 8.6% vs. 14.4% vs. 12.9% vs. 12.2%, P=0.326 for crush, T, culotte, kissing/V technique, respectively). Only left main (LM) disease and a shorter duration of dual antiplatelet therapy (DAPT) were associated with TLF. The distribution of DAPT duration differed between patients with and without TLF, and the time-point of intersection was 2.5 years. Also, the side branch was the most common site of restenosis.

CONCLUSIONS

The stenting sequence or technique did not affect clinical outcomes, but LM disease and shorter DAPT were associated with TLF, in patients with bifurcation lesions undergoing 2-stenting with 2G-DES.

Bifurcation of the sciatic nerve: A descriptive study on a South African cadaver cohort

BACKGROUND

The sciatic nerve bifurcates into the tibial and common fibular nerves in six different regions: the pelvic region, gluteal region, proximal, middle, or distal third of the posterior thigh, or in the popliteal fossa. Inadequate knowledge of sciatic nerve anatomy could lead to failed nerve blocks, and damage during intramuscular injections. Limited studies have been done on African population groups and lack a quantifiable method to classify the levels of sciatic nerve bifurcation.

AIM

Thus, this descriptive cadaver study aimed to assess the bifurcation level of the sciatic nerve in a South African population group using a quantifiable method.

METHODS

Three hundred and thirty-eight formalin-fixed limbs from three universities were dissected between March and August 2018. Type A was classified if the sciatic nerve bifurcated within the pelvic region. A sciatic nerve bifurcation level index (SNBLI) was developed for this study to quantitatively classify the region of sciatic nerve bifurcation into types B to F.

RESULTS

Type F bifurcation was most common (79.6%), and types B and C were not observed. In 15 (0.04%) specimens, two separate nerves entered the gluteal region and converged to form the sciatic nerve distal to the piriformis, which could bifurcate either in the distal third of the posterior thigh or popliteal fossa.

CONCLUSION

This is the first study to provide information on the sciatic nerve bifurcation in a large South African cadaver cohort. Moreover, we created a quantifiable method that can be used to classify the sciatic nerve bifurcation level. This will be beneficial in ensuring accurate comparison between different population groups in future.

The European bifurcation club Left Main Coronary Stent study: a randomized comparison of stepwise provisional vs. systematic dual stenting strategies (EBC MAIN)

BACKGROUND

Patients with non-left-main coronary bifurcation lesions are usually best treated with a stepwise provisional approach. However, patients with true left main stem bifurcation lesions have been shown in one dedicated randomized study to benefit from systematic dual stent implantation.

METHODS AND RESULTS

Four hundred and sixty-seven patients with true left main stem bifurcation lesions requiring intervention were recruited to the EBC MAIN study in 11 European countries. Patients were aged 71 ± 10 years; 77% were male. Patients were randomly allocated to a stepwise layered provisional strategy (n = 230) or a systematic dual stent approach (n = 237). The primary endpoint (a composite of death, myocardial infarction, and target lesion revascularization at 12 months) occurred in 14.7% of the stepwise provisional group vs. 17.7% of the systematic dual stent group (hazard ratio 0.8, 95% confidence interval 0.5-1.3; P = 0.34). Secondary endpoints were death (3.0% vs. 4.2%, P = 0.48), myocardial infarction (10.0% vs. 10.1%, P = 0.91), target lesion revascularization (6.1% vs. 9.3%, P = 0.16), and stent thrombosis (1.7% vs. 1.3%, P = 0.90), respectively. Procedure time, X-ray dose and consumables favoured the stepwise provisional approach. Symptomatic improvement was excellent and equal in each group.

CONCLUSIONS

Among patients with true bifurcation left main stem stenosis requiring intervention, fewer major adverse cardiac events occurred with a stepwise layered provisional approach than with planned dual stenting, although the difference was not statistically significant. The stepwise provisional strategy should remain the default for distal left main stem bifurcation intervention.

STUDY REGISTRATION

http://clinicaltrials.gov NCT02497014.

Assessment of single and double coronary bifurcation stenting techniques using multimodal imaging and 3D modeling in reanimated swine hearts using Visible Heart® methodologies

Stent implantation in bifurcated coronary lesions is technically challenging so that procedural refinements are continuously investigated. Novel procedure modeling and intracoronary imaging techniques may offer critical insights on stent deformations and stent-wall interactions during bifurcation stenting procedures. Thus, we assessed coronary bifurcation stenting techniques using multimodal imaging and 3D modeling in reanimated swine hearts. Harvested swine hearts were reanimated using Visible Heart® methodologies and (under standard fluoroscopic guidance) used to test 1-stent (provisional and inverted provisional) and 2-stent (culotte, TAP and DK-crush) techniques on bifurcations within various coronary vessels using commercially available devices. Intracoronary angioscopy and frequency-domain optical-coherence-tomography (OCT) were obtained during the procedures. 3D OCT reconstruction and micro-computed tomography 3D modeling (post heart fixations) were used to assess stent deformations and stent-wall interactions. We conducted multiple stenting procedures and collected unique endoscopic and OCT images (and subsequent computational models from micro-CT) to assess stent deformations and device/wall interactions during different steps of bifurcation stenting procedures. Endoscopy, micro-CT and virtual reality processing documented that different 1- and 2-stent techniques, practiced according to experts’ recommended steps, achieve optimal post-intervention stent conformation. As compared with intra-procedural endoscopy, software-generated 3D OCT images accurately depicted stent deformations during 1-stent techniques. On the opposite, during more complex 2-stent techniques, some defects were appreciated at 3D OCT reconstruction despite optimal 2D OCT images. This study provided unique insights regarding both stent deformations occurring in the course of bifurcation stenting and the efficacy of OCT to visualize them.

Autophosphorylation and the Dynamics of the Activation of Lck

Lck (lymphocyte-specific protein tyrosine kinase) is an enzyme which plays a number of important roles in the function of immune cells. It belongs to the Src family of kinases which are known to undergo autophosphorylation. It turns out that this leads to a remarkable variety of dynamical behaviour which can occur during their activation. We prove that in the presence of autophosphorylation one phenomenon, bistability, already occurs in a mathematical model for a protein with a single phosphorylation site. We further show that a certain model of Lck exhibits oscillations. Finally, we discuss the relations of these results to models in the literature which involve Lck and describe specific biological processes, such as the early stages of T cell activation and the stimulation of T cell responses resulting from the suppression of PD-1 signalling which is important in immune checkpoint therapy for cancer.

Stability analysis in COVID-19 within-host model with immune response

The 2019 coronavirus disease (COVID-19) is now a global pandemic. Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) is the causative pathogen of COVID-19. Here, we study an in-host model that highlights the effector T cell response to SARS-CoV-2. The stability of a unique positive equilibrium point, with viral load V * , suggests that the virus may replicate fast enough to overcome T cell response and cause infection. This overcoming is the bifurcation point, near which the orders of magnitude for V * can be sensitive to numerical changes in the parameter values. Our work offers a mathematical insight into how SARS-CoV-2 causes the disease.

A Mathematical Model Supporting a Hyperpredation Effect in the Apparent Competition Between Invasive Eastern Cottontail and Native European Hare

In this work a mathematical model is built in order to validate on theoretical grounds field study results on a three-species system made of two prey, of which one is native and another one invasive, together with a native predator. Specifically, our results mathematically describe the negative effect on the native European hare after the introduction of the invasive Eastern cottontail, mediated by an increased predation rate by foxes. Two nonexclusive assumptions can be made: an increase in cottontail abundance would lead to a larger fox population, magnifying their predatory impact ("hyperpredation") on hares; alternatively, cottontails attract foxes in patches where they live, which are also important resting sites for hares and consequently the increased presence of foxes results in a higher predation rates on hares. The model results support hyperpredation of increasing fox populations on native hares.

Clinical Factors Associated with Long Fluoroscopy Time in Percutaneous Coronary Interventions to the Culprit Lesion of Non-ST-Segment Elevation Myocardial Infarction

The clinical outcomes of patients with non-ST-segment elevation myocardial infarction (NSTEMI) were comparable or even worse than those with ST-segment elevation myocardial infarction (STEMI). Although successful percutaneous coronary intervention (PCI) to the culprit lesions of NSTEMI would improve the clinical outcomes, some PCI require long fluoroscopy time, reflecting the difficulty of PCI. This study aims to find clinical factors associated with long fluoroscopy time in PCI to the culprit lesion of NSTEMI. We included 374 patients and divided those into the conventional fluoroscopy time (n = 302) and long fluoroscopy time (n = 72) groups according to the quintiles of fluoroscopy time. Clinical and angiographic parameters were compared between the two groups. Calcification and tortuosity were significantly more severe in the long fluoroscopy time group than in the conventional fluoroscopy time group. The prevalence of previous coronary artery bypass grafting (CABG) and bifurcation lesions was significantly greater in the long fluoroscopy time group than in the conventional fluoroscopy time group. In the multivariate stepwise logistic regression analysis, previous CABG (odds ratio [OR], 3.368; 95% confidence interval [CI], 1.407-8.064; P = 0.006), bifurcation lesion (OR, 2.407; 95% CI, 1.285-4.506; P = 0.006), excessive tortuosity (versus mild to moderate tortuosity; OR, 4.095; 95% CI, 1.159-14.476; P = 0.029), and moderate to severe calcification (versus none to mild; OR, 5.792; 95% CI, 3.254-10.310; P < 0.001) were significantly associated with long fluoroscopy time. In conclusion, previous CABG, bifurcation, excessive tortuosity, and moderate to severe calcification were associated with long fluoroscopy time. Our study provided a reference for PCI operators to identify the difficulties in PCI to the culprit lesion of NSTEMI.

Microsurgical Management of the Middle Cerebral Artery Bifurcation Aneurysms: An Anatomic Feasibility Study

BACKGROUND

The proper head positioning decreases the surgical complications by enabling a better surgical maneuverability. Middle cerebral artery (MCA) bifurcation aneurysms have been classified by Dashti et al. [Surg Neurol. 2007 May;67(5):441-56] as the intertruncal, inferior, lateral, insular, and complex types based on dome projection. Our aim was to identify the optimum head positions and to explain the anatomic variables, which may affect the surgical strategy of MCA bifurcation aneurysms.

METHODS

The lateral supraorbital approach bilaterally was performed in the 4 cadaveric heads. All steps of the dissection were recorded using digital camera.

RESULTS

The distal Sylvian fissure (SF) dissection may be preferred for insular type and the proximal SF dissection may be preferred for all other types. Fifteen degrees head rotation was found as the most suitable position for the intertruncal, lateral type and subtype of complex aneurysms related with superior trunk. Thirty degrees head rotation was found the most suitable position for the inferior type, insular type, and subtype of complex aneurysms related with inferior trunk.

CONCLUSIONS

The head positioning in middle cerebral bifurcation aneurysms surgery is a critical step. It should be tailored according to the projection and its relationship with the parent vessels of the middle cerebral bifurcation.

Mathematical computations on epidemiology: a case study of the novel coronavirus (SARS-CoV-2)

The outbreak of coronavirus COVID-19 is spreading at an unprecedented rate to the human populations and taking several thousands of life all over the world. Scientists are trying to map the pattern of the transmission of coronavirus (SARS-CoV-2). Many countries are in the phase of lockdown in the globe. In this paper we predict about the effect of coronavirus COVID-19 and give a sneak peak when it will reduce the transmission rate in the world via mathematical modelling. In this research work our study is based on extensions of the well-known susceptible-exposed-infected-recovered (SEIR) family of compartmental models and later we observe the new model changes into (SEIR) without changing its physical meanings. The stability analysis of the coronavirus depends on changing of its basic reproductive ratio. The progress rate of the virus in the critically infected cases and the recovery rate have major roles to control this epidemic. The impact of social distancing, lockdown of the country, self-isolation, home quarantine and the wariness of global public health system have significant influence on the parameters of the model system that can alter the effect of recovery rates, mortality rates and active contaminated cases with the progression of time in the real world. The prognostic ability of mathematical model is circumscribed as of the accuracy of the available data and its application to the problem.

Symmetry-breaking-induced rare fluctuations in a time-delay dynamic system

Inspired by the experimental and numerical findings, we study the dynamic instabilities of two coupled nonlinear delay differential equations that are used to describe the coherent oscillations between the top and bottom boundary layers in turbulent Rayleigh-Bénard convection. By introducing two sensitivity parameters for the instabilities of the top and bottom boundary layers, we find three different types of solutions, namely in-phase single-period oscillations, multi-period oscillations and chaos. The chaos solution contains rare but large amplitude fluctuations. The statistical properties of these fluctuations are consistent with those observed in the experiment for the massive eruption of thermal plumes, which causes random reversals of the large-scale circulation in turbulent Rayleigh-Bénard convection. Our study thus provides new insights into the origin of rare massive eruptions and sudden changes of large-scale flow pattern that are often observed in convection systems of geophysical and astrophysical scales.

Effects of Nonmonotonic Functional Responses on a Disease Transmission Model: Modeling and Simulation

In this article, a novel susceptible-infected-recovered epidemic model with nonmonotonic incidence and treatment rates is proposed and analyzed mathematically. The Monod-Haldane functional response is considered for nonmonotonic behavior of both incidence rate and treatment rate. The model analysis shows that the model has two equilibria which are named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The stability analysis has been performed for the local and global behavior of the DFE and EE. With the help of the basic reproduction number R 0 , we investigate that DFE is locally asymptotically stable when R 0 < 1 and unstable when R 0 > 1 . The local stability of DFE at R 0 = 1 has been analyzed, and it is obtained that DFE exhibits a forward transcritical bifurcation. Further, we identify conditions for the existence of EE and show the local stability of EE under certain conditions. Moreover, the global stability behavior of DFE and EE has been investigated. Lastly, numerical simulations have been done in the support of our theoretical findings.

Assessing the Effects of Holling Type-II Treatment Rate on HIV-TB Co-infection

In this paper, a HIV-TB co-infection model is explored which incorporates a non-linear treatment rate for TB. We begin with presenting a HIV-TB co-infection model and analyze both HIV and TB sub-models separately. The basic reproduction numbers corresponding to HIV-only, TB-only and the HIV-TB full model are computed. The disease-free equilibrium point of the HIV sub-model is shown to be locally as well as globally asymptotically stable when its corresponding reproduction number is less than unity. The HIV-only model exhibits a transcritical bifurcation. On the other hand, for the TB sub-model, the disease-free equilibrium point is locally asymptotically stable but may not be globally asymptotically stable. We have also analyzed the full HIV-TB co-infection model. Numerical simulations are performed to investigate the effect of treatment rate in the presence of resource limitation for TB infected individuals, which emphasize the fact that to reduce co-infection from the population programs to accelerate the treatment of TB should be implemented.

Contemporary approaches to bifurcation stenting

Bifurcation lesions are common and associated with higher risks of major cardiac events and restenosis after percutaneous coronary intervention (PCI). Treatment requires understanding of lesion characteristics, stent design and therapeutic options. We review the evidence for provisional vs 2-stent techniques. We conclude that provisional stenting is suitable for most bifurcation lesions. We detail situations where a 2-stent technique should be considered and the steps for performing each of the 2-step techniques. We review the importance of lesion preparation, intracoronary imaging, proximal optimization (POT) and kissing balloon inflation.

[Acute coronary lithotripsy in an 83-year-old man with syncope and acute coronary syndrome]

BACKGROUND

An 83-year-old male presented with syncope and acute coronary syndrome. An electrocardiogram (ECG) suggested left main (LM) coronary artery stenosis; angiography confirmed a heavily calcified tight LM stenosis.

CLINICAL COURSE

Percutaneous coronary intervention was performed using intravascular lithotripsy and implantation of drug eluting stents with an excellent angiographic result, resolution of symptoms and ECG changes.

CONCLUSION

Intravascular lithotripsy is a new and easily applicable method for heavily calcified coronary lesions. Cost-effectiveness and long-term results will define its future role.

Stability and bifurcation analysis of a diffusive modified Leslie-Gower prey-predator model with prey infection and Beddington DeAngelis functional response

In this paper, we present and analyze a spatio-temporal eco-epidemiological model of a prey predator system where prey population is infected with a disease. The prey population is divided into two categories, susceptible and infected. The susceptible prey is assumed to grow logistically in the absence of disease and predation. The predator population follows the modified Leslie-Gower dynamics and predates both the susceptible and infected prey population with Beddington-DeAngelis and Holling type II functional responses, respectively. The boundedness of solutions, existence and stability conditions of the biologically feasible equilibrium points of the system both in the absence and presence of diffusion are discussed. It is found that the disease can be eradicated if the rate of transmission of the disease is less than the death rate of the infected prey. The system undergoes a transcritical and pitchfork bifurcation at the Disease Free Equilibrium Point when the prey infection rate crosses a certain threshold value. Hopf bifurcation analysis is also carried out in the absence of diffusion, which shows the existence of periodic solution of the system around the Disease Free Equilibrium Point and the Endemic Equilibrium Point when the ratio of the rate of intrinsic growth rate of predator to prey crosses a certain threshold value. The system remains locally asymptotically stable in the presence of diffusion around the disease free equilibrium point once it is locally asymptotically stable in the absence of diffusion. The Analytical results show that the effect of diffusion can be managed by appropriately choosing conditions on the parameters of the local interaction of the system. Numerical simulations are carried out to validate our analytical findings.