Pharmacological interventions for preventing clotting of extracorporeal circuits during continuous renal replacement therapy.

BACKGROUND: Acute kidney injury (AKI) is a major comorbidity in hospitalised patients. Patients with severe AKI require continuous renal replacement therapy (CRRT) when they are haemodynamically unstable. CRRT is prescribed assuming it is delivered over 24 hours. However, it is interrupted when the extracorporeal circuits clot and the replacement is required. The interruption may impair the solute clearance as it causes under dosing of CRRT. To prevent the circuit clotting, anticoagulation drugs are frequently used. OBJECTIVES: To assess the benefits and harms of pharmacological interventions for preventing clotting in the extracorporeal circuits during CRRT. SEARCH METHODS: We searched the Cochrane Kidney and Transplant Register of Studies up to 12 September 2019 through contact with the Information Specialist using search terms relevant to this review. Studies in the Register are identified through searches of CENTRAL, MEDLINE, and EMBASE, conference proceedings, the International Clinical Trials Register (ICTRP) Search Portal and ClinicalTrials.gov. SELECTION CRITERIA: We selected randomised controlled trials (RCTs or cluster RCTs) and quasi-RCTs of pharmacological interventions to prevent clotting of extracorporeal circuits during CRRT. DATA COLLECTION AND ANALYSIS: Data were abstracted and assessed independently by two authors. Dichotomous outcomes were calculated as risk ratio (RR) with 95% confidence intervals (CI). The primary review outcomes were major bleeding, successful prevention of clotting (no need of circuit change in the first 24 hours for any reason), and death. Evidence certainty was determined using the Grading of Recommendation Assessment, Development, and Evaluation (GRADE) approach. MAIN RESULTS: A total of 34 completed studies (1960 participants) were included in this review. We identified seven ongoing studies which we plan to assess in a future update of this review. No included studies were free from risk of bias. We rated 30 studies for performance bias and detection bias as high risk of bias. We rated 18 studies for random sequence generation,ààsix studies for the allocation concealment, three studies for performance bias, three studies for detection bias,à nine studies for attrition bias,à14 studies for selective reporting and nine studies for the other potential source of bias, as having low risk of bias. We identified eight studies (581 participants) that compared citrate with unfractionated heparin (UFH). Compared to UFH, citrate probably reduces major bleeding (RR 0.22, 95% CI 0.08 to 0.62; moderate certainty evidence) and probably increases successful prevention of clotting (RR 1.44, 95% CI 1.10 to 1.87; moderate certainty evidence). Citrate may have little or no effect on death at 28 days (RR 1.06, 95% CI 0.86 to 1.30, moderate certainty evidence). Citrate versus UFH may reduce the number of participants who drop out of treatment due to adverse events (RR 0.47, 95% CI 0.15 to 1.49; low certainty evidence). Compared to UFH, citrate may make little or no difference to the recovery of kidney function (RR 1.04, 95% CI 0.89 to 1.21; low certainty evidence). Compared to UFH, citrate may reduceàthrombocytopenia (RR 0.39, 95% CI 0.14 to 1.03; low certainty evidence). It was uncertain whether citrate reduces a cost to health care services because of inadequate data. For low molecular weight heparin (LMWH) versus UFH, six studies (250 participants) were identified. Compared to LMWH, UFH may reduce major bleeding (0.58, 95% CI 0.13 to 2.58; low certainty evidence). It is uncertain whether UFH versus LMWH reduces death at 28 days or leads to successful prevention of clotting. Compared to LMWH, UFH may reduce the number of patient dropouts from adverse events (RR 0.29, 95% CI 0.02 to 3.53; low certainty evidence). It was uncertain whether UFH versus LMWH leads to the recovery of kidney function because no included studies reported this outcome. It was uncertain whether UFH versus LMWH leads to thrombocytopenia. It was uncertain whether UFH reduces a cost to health care services because of inadequate data. For the comparison of UFH to no anticoagulation, one study (10 participants) was identified. It is uncertain whether UFH compare to no anticoagulation leads to more major bleeding. It is uncertain whether UFH improves successful prevention of clotting in the first 24 hours, death at 28 days, the number of patient dropouts due to adverse events, recovery of kidney function, thrombocytopenia, or cost to health care services because no study reported these outcomes. For the comparison ofàcitrate to no anticoagulation,àno completed study was identified. AUTHORS' CONCLUSIONS: Currently,àavailable evidence does not support the overall superiority of any anticoagulant to another. Compared to UFH, citrate probably reduces major bleeding and prevents clotting and probably has little or no effect on death at 28 days. For other pharmacological anticoagulation methods, there is no available data showing overall superiority to citrate or no pharmacological anticoagulation. Further studies are needed to identify patient populations in which CRRT should commence with no pharmacological anticoagulation or with citrate.

The age distribution of mortality from novel coronavirus disease (COVID-19) suggests no large difference of susceptibility by age.

Among Italy, Spain, and Japan, the age distributions of COVID-19 mortality show only small variation even though the number of deaths per country shows large variation. To understand the determinant for this situation, we constructed a mathematical model describing the transmission dynamics and natural history of COVID-19 and analyzed the dataset of mortality in Italy, Spain, and Japan. We estimated the parameter which describes the age-dependency of susceptibility by fitting the model to reported data, including the effect of change in contact patterns during the epidemics of COVID-19, and the fraction of symptomatic infections. Our study revealed that if the mortality rate or the fraction of symptomatic infections among all COVID-19 cases does not depend on age, then unrealistically different age-dependencies of susceptibilities against COVID-19 infections between Italy, Japan, and Spain are required to explain the similar age distribution of mortality but different basic reproduction numbers (R0). Variation of susceptibility by age itself cannot explain the robust age distribution in mortality by COVID-19 infections in those three countries, however it does suggest that the age-dependencies of (i) the mortality rate and (ii) the fraction of symptomatic infections among all COVID-19 cases determine the age distribution of mortality by COVID-19.

Pharmacological interventions for preventing clotting of extracorporeal circuits during continuous renal replacement therapy.

BACKGROUND: Acute kidney injury (AKI) is a major comorbidity in hospitalised patients. Patients with severe AKI require continuous renal replacement therapy (CRRT) when they are haemodynamically unstable. CRRT is prescribed assuming it is delivered over 24 hours. However, it is interrupted when the extracorporeal circuits clot and the replacement is required. The interruption may impair the solute clearance as it causes under dosing of CRRT. To prevent the circuit clotting, anticoagulation drugs are frequently used. OBJECTIVES: To assess the benefits and harms of pharmacological interventions for preventing clotting in the extracorporeal circuits during CRRT. SEARCH METHODS: We searched the Cochrane Kidney and Transplant Register of Studies up to 12 September 2019 through contact with the Information Specialist using search terms relevant to this review. Studies in the Register are identified through searches of CENTRAL, MEDLINE, and EMBASE, conference proceedings, the International Clinical Trials Register (ICTRP) Search Portal and ClinicalTrials.gov. SELECTION CRITERIA: We selected randomised controlled trials (RCTs or cluster RCTs) and quasi-RCTs of pharmacological interventions to prevent clotting of extracorporeal circuits during CRRT. DATA COLLECTION AND ANALYSIS: Data were abstracted and assessed independently by two authors. Dichotomous outcomes were calculated as risk ratio (RR) with 95% confidence intervals (CI). The primary review outcomes were major bleeding, successful prevention of clotting (no need of circuit change in the first 24 hours for any reason), and death. Evidence certainty was determined using the Grading of Recommendation Assessment, Development, and Evaluation (GRADE) approach. MAIN RESULTS: A total of 34 completed studies (1960 participants) were included in this review. We identified seven ongoing studies which we plan to assess in a future update of this review. No included studies were free from risk of bias. We rated 30 studies for performance bias and detection bias as high risk of bias. We rated 18 studies for random sequence generation,  six studies for the allocation concealment, three studies for performance bias, three studies for detection bias,  nine studies for attrition bias, 14 studies for selective reporting and nine studies for the other potential source of bias, as having low risk of bias. We identified eight studies (581 participants) that compared citrate with unfractionated heparin (UFH). Compared to UFH, citrate probably reduces major bleeding (RR 0.22, 95% CI 0.08 to 0.62; moderate certainty evidence). Citrate may have little or no effect on death at 28 days (RR 1.06, 95% CI 0.86 to 1.30, moderate certainty evidence), while citrate versus UFH may have little or no effect on successful prevention of clotting (RR 1.01, 95% CI 0.77 to 1.32; moderate certainty evidence). Citrate versus UFH may reduce the number of participants who drop out of treatment due to adverse events (RR 0.47, 95% CI 0.15 to 1.49; low certainty evidence). Compared to UFH, citrate may make little or no difference to the recovery of kidney function (RR 0.95, 95% CI 0.66 to 1.36; low certainty evidence). Compared to UFH, citrate may reduce thrombocytopenia (RR 0.39, 95% CI 0.14 to 1.03; low certainty evidence). It was uncertain whether citrate reduces a cost to health care services because of inadequate data. For low molecular weight heparin (LMWH) versus UFH, six studies (250 participants) were identified. Compared to LMWH, UFH may reduce major bleeding (0.58, 95% CI 0.13 to 2.58; low certainty evidence). It is uncertain whether UFH versus LMWH reduces death at 28 days or leads to successful prevention of clotting. Compared to LMWH, UFH may reduce the number of patient dropouts from adverse events (RR 0.29, 95% CI 0.02 to 3.53; low certainty evidence). It was uncertain whether UFH versus LMWH leads to the recovery of kidney function because no included studies reported this outcome. It was uncertain whether UFH versus LMWH leads to thrombocytopenia. It was uncertain whether UFH reduces a cost to health care services because of inadequate data. For the comparison of UFH to no anticoagulation, one study (10 participants) was identified. It is uncertain whether UFH compare to no anticoagulation leads to more major bleeding. It is uncertain whether UFH improves successful prevention of clotting in the first 24 hours, death at 28 days, the number of patient dropouts due to adverse events, recovery of kidney function, thrombocytopenia, or cost to health care services because no study reported these outcomes. For the comparison of citrate to no anticoagulation, no completed study was identified. AUTHORS' CONCLUSIONS: Currently, available evidence does not support the overall superiority of any anticoagulant to another. Compared to UFH, citrate probably reduces major bleeding and probably has little or no effect on preventing clotting or death at 28 days. For other pharmacological anticoagulation methods, there is no available data showing overall superiority to citrate or no pharmacological anticoagulation. Further studies are needed to identify patient populations in which CRRT should commence with no pharmacological anticoagulation or with citrate.

Epidemic dynamics with a time-varying susceptibility due to repeated infections.

In this paper, we consider the impact of heterogeneous susceptibility on disease transmission dynamics, using a simple mathematical model formulated by a system of ordinary differential equations. Our model describes a time-varying immunity level, which is enhanced by reinfection and diminished by waning immunity. Through mathematical analysis, we show that unexpected outbreaks, called delayed outbreaks, occur even when the basic reproduction number is less than one. A reallocation of susceptibility at the individual level, by repeated infections, in the host population induces delayed outbreaks.

Dialysate temperature reduction for intradialytic hypotension for people with chronic kidney disease requiring haemodialysis.

BACKGROUND: Intradialytic hypotension (IDH) is a common complication of haemodialysis (HD), and a risk factor of cardiovascular morbidity and death. Several clinical studies suggested that reduction of dialysate temperature, such as fixed reduction of dialysate temperature or isothermal dialysate using a biofeedback system, might improve the IDH rate. OBJECTIVES: This review aimed to evaluate the benefits and harms of dialysate temperature reduction for IDH among patients with chronic kidney disease requiring HD, compared with standard dialysate temperature. SEARCH METHODS: We searched Cochrane Kidney and Transplant's Specialised Register up to 14 May 2019 through contact with the Information Specialist using search terms relevant to this review. Studies in the Register are identified through searches of CENTRAL, MEDLINE, EMBASE, conference proceedings, the International Clinical Trials Register (ICTRP) Search Portal, and ClinicalTrials.gov. SELECTION CRITERIA: All randomised controlled trials (RCTs), cross-over RCTs, cluster RCTs and quasi-RCTs were included in the review. DATA COLLECTION AND ANALYSIS: Two authors independently extracted information including participants, interventions, outcomes, methods of the study, and risks of bias. We used a random-effects model to perform quantitative synthesis of the evidence. We assessed the risks of bias for each study using the Cochrane 'Risk of bias' tool. We assessed the certainty of evidence using Grades of Recommendation, Assessment, Development and Evaluation (GRADE). MAIN RESULTS: We included 25 studies (712 participants). Three studies were parallel RCTs and the others were cross-over RCTs. Nineteen studies compared fixed reduction of dialysate temperature (below 36°C) and standard dialysate temperature (37°C to 37.5°C). Most studies were of unclear or high risk of bias. Compared with standard dialysate, it is uncertain whether fixed reduction of dialysate temperature improves IDH rate (8 studies, 153 participants: rate ratio 0.52, 95% CI 0.34 to 0.80; very low certainty evidence); however, it might increase the discomfort rate compared with standard dialysate (4 studies, 161 participants: rate ratio 8.31, 95% CI 1.86 to 37.12; very low certainty evidence). There were no reported dropouts due to adverse events. No study reported death, acute coronary syndrome or stroke.Three studies compared isothermal dialysate and thermoneutral dialysate. Isothermal dialysate might improve the IDH rate compared with thermoneutral dialysate (2 studies, 133 participants: rate ratio 0.68, 95% CI 0.60 to 0.76; I2 = 0%; very low certainty evidence). There were no reports of discomfort rate (1 study) or dropouts due to adverse events (2 studies). No study reported death, acute coronary syndrome or stroke. AUTHORS' CONCLUSIONS: Reduction of dialysate temperature may prevent IDH, but the conclusion is uncertain. Larger studies that measure important outcomes for HD patients are required to assess the effect of reduction of dialysate temperature. Six ongoing studies may provide much-needed high quality evidence in the future.

Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods.

We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.

The change of susceptibility following infection can induce failure to predict outbreak potential by R0.

Time-varying individual host susceptibility to a disease due to waning and boosting of immunity is known to induce rich long-term behavior of the disease transmission dynamics. Simultaneously, the impact of the time-varying heterogeneity of host susceptibility on the short-term behavior of epidemics is not well-studied despite the availability of a large amount of epidemiological data on short-term epidemics. This paper proposes a parsimonious mathematical model describing the short-term transmission dynamics by taking into account waning and enhancing susceptibility following the infection. In addition to the common classification in the standard SIR model, i.e., "no epidemic" as R0≤1 or normal epidemic as R0>1, the proposed model also shows the "delayed epidemic" class when an epidemic takes off after the negative slope of the epidemic curve at the initial phase of the epidemic. The condition for each of the three classes is derived based on the obtained explicit solution for the proposed model.

Dynamics of an ultra-discrete SIR epidemic model with time delay.

We propose an ultra-discretization for an SIR epidemic model with time delay. It is proven that the ultra-discrete model has a threshold property concerning global attractivity of equilibria as shown in differential and difference equation models. We also study an interesting convergence pattern of the solution, which is illustrated in a two-dimensional lattice.

Ultrasonography for confirmation of gastric tube placement.

BACKGROUND: Gastric tubes are commonly used for the administration of drugs and tube feeding for people who are unable to swallow. Feeding via a tube misplaced in the trachea can result in severe pneumonia. Therefore, the confirmation of tube placement in the stomach after tube insertion is important. Recent studies have reported that ultrasonography provides good diagnostic accuracy estimates in the confirmation of appropriate tube placement. Hence, ultrasound could provide a promising alternative to X-rays in the confirmation of tube placement, especially in settings where X-ray facilities are unavailable or difficult to access. OBJECTIVES: To assess the diagnostic accuracy of ultrasound for gastric tube placement confirmation. SEARCH METHODS: We searched the Cochrane Library (2016, Issue 3), MEDLINE (to March 2016), Embase (to March 2016), National Institute for Health Research (NIHR) PROSPERO Register (to May 2016), Aggressive Research Intelligence Facility Databases (to May 2016), ClinicalTrials.gov (to May 2016), ISRCTN registry (May 2016), World Health Organization International Clinical Trials Registry Platform (to May 2016) and reference lists of articles, and contacted study authors. SELECTION CRITERIA: We included studies that evaluated the diagnostic accuracy of naso- and orogastric tube placement confirmed by ultrasound visualization using X-ray visualization as the reference standard. We included cross-sectional studies, and case-control studies. We excluded case series or case reports. Studies were excluded if X-ray visualization was not the reference standard or if the tube being placed was a gastrostomy or enteric tube. DATA COLLECTION AND ANALYSIS: Two review authors independently assessed the risk of bias and extracted data from each of the included studies. We contacted authors of the included studies to obtain missing data. MAIN RESULTS: We identified 10 studies (545 participants and 560 tube insertions) which met our inclusion criteria.No study was assigned low risk of bias or low concern in every QUADAS-2 domain. We judged only three (30%) studies to have low risk of bias in the participant selection domain because they performed ultrasound after they confirmed correct position by other methods.Few data (43 participants) were available for misplacement detection (specificity) due to the low incidence of misplacement. We did not perform a meta-analysis because of considerable heterogeneity of the index test such as the difference of echo window, the combination of ultrasound with other confirmation methods (e.g. saline flush visualization by ultrasound) and ultrasound during the insertion of the tube. For all settings, sensitivity estimates for individual studies ranged from 0.50 to 1.00 and specificity estimates from 0.17 to 1.00. For settings where X-ray was not readily available and participants underwent gastric tube insertion for drainage (four studies, 305 participants), sensitivity estimates of ultrasound in combination with other confirmatory tests ranged from 0.86 to 0.98 and specificity estimates of 1.00 with wide confidence intervals.For the studies using ultrasound alone (four studies, 314 participants), sensitivity estimates ranged from 0.91 to 0.98 and specificity estimates from 0.67 to 1.00. AUTHORS' CONCLUSIONS: Of 10 studies that assessed the diagnostic accuracy of gastric tube placement, few studies had a low risk of bias. Based on limited evidence, ultrasound does not have sufficient accuracy as a single test to confirm gastric tube placement. However, in settings where X-ray is not readily available, ultrasound may be useful to detect misplaced gastric tubes. Larger studies are needed to determine the possibility of adverse events when ultrasound is used to confirm tube placement.

On the characteristic equation λ = α1 + (α2 + α3λ)e(-λ) and its use in the context of a cell population model.

In this paper we characterize the stability boundary in the (α1, α2)-plane, for fixed α3 with −1 < α3 < +1, for the characteristic equation from the title. Subsequently we describe a nonlinear cell population model involving quiescence and show that this characteristic equation governs the (in)stability of the nontrivial steady state. By relating the parameters of the cell model to the αi we are able to derive some biological conclusions.

The determinant of periodicity in Mycoplasma pneumoniae incidence: an insight from mathematical modelling.

Until the early 1990 s, incidences of Mycoplasma pneumoniae (MP) infection showed three to five year epidemic cycles in multiple countries, however, the mechanism for the MP epidemic cycle has not been understood. Here, we investigate the determinant of periodicity in MP incidence by employing a mathematical model describing MP transmission dynamics. Three candidates for the determinant of periodicity were evaluated: school-term forcing, minor variance in the duration of immunity, and epidemiological interference between MP serotypes. We find that minor variation in the duration of immunity at the population level must be considered essential for the MP epidemic cycle because the MP cyclic incidence pattern did not replicate without it. Minor variation, in this case, is a less dispersed distribution for the duration of immunity than an exponential distribution. Various lengths of epidemic cycles, including cycles typically found in nature, e.g. three to five year cycles, were also observed when there was minor variance in the duration of immunity. The cyclic incidence pattern is robust even if there is epidemiological interference due to cross-immune protection, which is observed in the epidemiological data as negative correlation between epidemics per MP serotype.

Transmission Dynamics and Final Epidemic Size of Ebola Virus Disease Outbreaks with Varying Interventions.

The 2014 Ebola Virus Disease (EVD) outbreak in West Africa was the largest and longest ever reported since the first identification of this disease. We propose a compartmental model for EVD dynamics, including virus transmission in the community, at hospitals, and at funerals. Using time-dependent parameters, we incorporate the increasing intensity of intervention efforts. Fitting the system to the early phase of the 2014 West Africa Ebola outbreak, we estimate the basic reproduction number as 1.44. We derive a final size relation which allows us to forecast the total number of cases during the outbreak when effective interventions are in place. Our model predictions show that, as long as cases are reported in any country, intervention strategies cannot be dismissed. Since the main driver in the current slowdown of the epidemic is not the depletion of susceptibles, future waves of infection might be possible, if control measures or population behavior are relaxed.

Global analysis for spread of infectious diseases via transportation networks.

We formulate an epidemic model for the spread of an infectious disease along with population dispersal over an arbitrary number of distinct regions. Structuring the population by the time elapsed since the start of travel, we describe the infectious disease dynamics during transportation as well as in the regions. As a result, we obtain a system of delay differential equations. We define the basic reproduction number R(0) as the spectral radius of a next generation matrix. For multi-regional systems with strongly connected transportation networks, we prove that if R(0) ≤ 1 then the disease will be eradicated from each region, while if R(0) > 1 there is a globally asymptotically stable equilibrium, which is endemic in every region. If the transportation network is not strongly connected, then the model analysis shows that numerous endemic patterns can exist by admitting a globally asymptotically stable equilibrium, which may be disease free in some regions while endemic in other regions. We provide a procedure to detect the disease free and the endemic regions according to the network topology and local reproduction numbers. The main ingredients of the mathematical proofs are the inductive applications of the theory of asymptotically autonomous semiflows and cooperative dynamical systems. We visualise stability boundaries of equilibria in a parameter plane to illustrate the influence of the transportation network on the disease dynamics. For a system consisting of two regions, we find that due to spatial heterogeneity characterised by different local reproduction numbers, R(0) may depend non-monotonically on the dispersal rates, thus travel restrictions are not always beneficial.

Global dynamics of two-compartment models for cell production systems with regulatory mechanisms.

We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators [25] to investigate the dynamics of the hematopoietic system. Using global stability analysis, we compare different regulatory mechanisms. For both models, we show that there exists a unique positive equilibrium that is globally asymptotically stable if and only if the respective reproduction numbers exceed one. The proof is based on constructing Lyapunov functions, which are appropriate to handle the specific nonlinearities of the model. Additionally, we propose a new model to test biological hypothesis on the regulation of the fraction of differentiating cells. We show that such regulatory mechanism is incapable of maintaining homeostasis and leads to unbounded cell growth. Potential biological implications are discussed.

Stability analysis of multi-compartment models for cell production systems.

We study two- and three-compartment models of a hierarchical cell production system with cell division regulated by the level of mature cells. We investigate the structure of equilibria with respect to parameters as well as local stability properties for the equilibria. To interpret the results we adapt the concept of reproduction numbers, which is well known in ecology, to stem cell population dynamics. In the two-compartment model, the positive equilibrium is stable wherever it exists. In the three-compartment model, we find that the intermediate stage of differentiation is responsible for the emergence of an instability region in the parameter plane. Moreover, we prove that this region shrinks as the mortality rate for mature cells increases and discuss this result.

On the global stability of a delayed epidemic model with transport-related infection.

We study the global dynamics of a time delayed epidemic model proposed by Liu et al. (2008) [J. Liu, J. Wu, Y. Zhou, Modeling disease spread via transport-related infection by a delay differential equation, Rocky Mountain J. Math. 38 (5) (2008) 1525-1540] describing disease transmission dynamics among two regions due to transport-related infection. We prove that if an endemic equilibrium exists then it is globally asymptotically stable for any length of time delay by constructing a Lyapunov functional. This suggests that the endemic steady state for both regions is globally asymptotically stable regardless of the length of the travel time when the disease is transferred between two regions by human transport.

Global stability for a class of discrete SIR epidemic models.

In this paper, we propose a class of discrete SIR epidemic models which are derived from SIR epidemic models with distributed delays by using a variation of the backward Euler method. Applying a Lyapunov functional technique, it is shown that the global dynamics of each discrete SIR epidemic model are fully determined by a single threshold parameter and the effect of discrete time delays are harmless for the global stability of the endemic equilibrium of the model.