Vector-borne Disease Model Research Articles

Hopf bifurcation and its stability for a vector-borne disease model with delay and reinfection

In this study, we investigated a delayed vector-borne disease model with partial immunity to reinfection. The equilibria and the threshold of the model were determined according to the basic reproductive number R0. The analysis showed that a time delay destabilized the system. Using the delay as a bifurcation parameter, we established the conditions for the stability of the equilibria and the existence of a Hopf bifurcation. We determined the properties of the Hopf bifurcation by applying the normal form theory and center manifold argument, and for the first time, we considered the global continuation of the local Hopf bifurcation for a delayed vector-borne disease epidemic model. Furthermore, we performed some numerical simulations to illustrate our theoretical analysis. Sensitivity analysis showed that preventive control to minimize vector–human contacts and using insecticide to control the vector are effective measures for reducing infections.

Stability Analysis of a Vector-borne Disease Model with Nonlinear and Bilinear Incidences

Background/Objectives: To formulate and analyze a vector host epidemic model with non-monotonic and bilinear incidences. Methods/Statistical Analysis: The stability conditions of disease free equilibrium and endemic equilibrium are investigated by constructing suitable Lyapunov functions. Numerical simulation is carried out to justify the theoretical results. Results/Findings: The disease becomes endemic when the basic reproduction number is greater than one and it fades out when it is less than one. Conclusion/Application: In endemic state of the disease, number of infective host decreases as awareness of vaccination and preventive measures increases and number of vectors approaches zero as the awareness of use of insecticides and cleanliness tends to infinity.

Models for the effects of host movement in vector-borne disease systems

Host and/or vector movement patterns have been shown to have significant effects in both empirical studies and mathematical models of vector-borne diseases. The processes of economic development and globalization seem likely to make host movement even more important in the future. This article is a brief survey of some of the approaches that have been used to study the effects of host movement in analytic mathematical models for vector-borne diseases. It describes the formulation and interpretation of various types of spatial models and describes a few of the conclusions that can be drawn from them. It is not intended to be comprehensive but rather to provide sufficient background material and references to the literature to serve as an entry point into this area of research for interested readers.

Understanding dengue fever dynamics: a study of seasonality in vector-borne disease models

Dengue fever dynamics show seasonality, with the disease transmission being higher during the warmer seasons. In this paper, we analyse seasonally forced epidemic models with and without vector dynamics. We assume small seasonal effects and obtain approximations for the real response of each state variable and also for the corresponding amplitude and phase via decomposition of the sinusoidal forcing into imaginary exponential functions. The analysis begins with the simplest susceptible-infected-susceptible (SIS) model, followed by the simplest model with vector dynamics, susceptible-infected-susceptible for hosts and uninfected-vector (SISUV). Finally, we compare the more complex susceptible-infected-recovered (SIR) and susceptible-infected-recovered for hosts and uninfected-vector (SIRUV) models and conclude that the models give basically the same information when we replace, in the SIR model, the human infectivity by a function of both human and mosquito infectivities.

Dynamic analysis of a delayed model for vector-borne diseases on bipartite networks

In this paper, to study the spread of vector-borne diseases in human population, we build two coupled models for human population and vector population respectively on bipartite networks. By taking approximate expression for the density of infective vectors, we reduce the coupled models to a delayed SIS model describing the spread of diseases in human population. For the delayed dynamic model, we analyze its dynamic behavior. The basic reproduction number R0 is given. And based on the Lyapunov–LaSalle invariance principle, we prove the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium. Finally we carry out simulations to verify the conclusions and reveal the effect of the topology structure of networks and the time delay on the transmission process. Our results show that the basic reproduction number depends on the topology structure of bipartite networks and the time delay. It is also pointed out that the time delay can reduce the basic reproduction number. Furthermore, when the disease will disappear, the delay speeds up the disappearing process; when disease will become endemic, the delay slows the disease spreading down and reduces the density of infective humans.

Age- and bite-structured models for vector-borne diseases

The biology and behaviour of biting insects is a vitally important aspect in the spread of vector-borne diseases. This paper aims to determine, through the use of mathematical models, what effect incorporating vector senescence and realistic feeding patterns has on disease. A novel model is developed to enable the effects of age- and bite-structure to be examined in detail. This original PDE framework extends previous age-structured models into a further dimension to give a new insight into the role of vector biting and its interaction with vector mortality and spread of disease. Through the PDE model, the roles of the vector death and bite rates are examined in a way which is impossible under the traditional ODE formulation. It is demonstrated that incorporating more realistic functions for vector biting and mortality in a model may give rise to different dynamics than those seen under a more simple ODE formulation. The numerical results indicate that the efficacy of control methods that increase vector mortality may not be as great as predicted under a standard host–vector model, whereas other controls including treatment of humans may be more effective than previously thought.

Volterra-type Lyapunov functions for fractional-order epidemic systems

In this paper we prove an elementary lemma which estimates fractional derivatives of Volterra-type Lyapunov functions in the sense Caputo when α∈(0,1). Moreover, by using this result, we study the uniform asymptotic stability of some Caputo-type epidemic systems with a pair of fractional-order differential equations. These epidemic systems are the Susceptible–Infected–Susceptible (SIS), Susceptible–Infected–Recovered (SIR) and Susceptible–Infected–Recovered–Susceptible (SIRS) models and Ross–Macdonald model for vector-borne diseases. We show that the unique endemic equilibrium is uniformly asymptotically stable if the basic reproductive number is greater than one. We illustrate our theoretical results with numerical simulations using the Adams–Bashforth–Moulton scheme implemented in the fde12 Matlab function.

Modelling the potential spatial distribution of mosquito species using three different techniques.

BackgroundModels for the spatial distribution of vector species are important tools in the assessment of the risk of establishment and subsequent spread of vector-borne diseases. The aims of this study are to define the environmental conditions suitable for several mosquito species through species distribution modelling techniques, and to compare the results produced with the different techniques.MethodsThree different modelling techniques, i.e., non-linear discriminant analysis, random forest and generalised linear model, were used to investigate the environmental suitability in the Netherlands for three indigenous mosquito species (Culiseta annulata, Anopheles claviger and Ochlerotatus punctor). Results obtained with the three statistical models were compared with regard to: (i) environmental suitability maps, (ii) environmental variables associated with occurrence, (iii) model evaluation.ResultsThe models indicated that precipitation, temperature and population density were associated with the occurrence of Cs. annulata and An. claviger, whereas land surface temperature and vegetation indices were associated with the presence of Oc. punctor. The maps produced with the three different modelling techniques showed consistent spatial patterns for each species, but differences in the ranges of the predictions. Non-linear discriminant analysis had lower predictions than other methods. The model with the best classification skills for all the species was the random forest model, with specificity values ranging from 0.89 to 0.91, and sensitivity values ranging from 0.64 to 0.95.ConclusionsWe mapped the environmental suitability for three mosquito species with three different modelling techniques. For each species, the maps showed consistent spatial patterns, but the level of predicted environmental suitability differed; NLDA gave lower predicted probabilities of presence than the other two methods. The variables selected as important in the models were in agreement with the existing knowledge about these species. All model predictions had a satisfactory to excellent accuracy; best accuracy was obtained with random forest. The insights obtained can be used to gain more knowledge on vector and non-vector mosquito species. The output of this type of distribution modelling methods can, for example, be used as input for epidemiological models of vector-borne diseases.

Stability, bifurcation and chaos analysis of vector-borne disease model with application to Rift Valley fever.

This paper investigates a RVF epidemic model by qualitative analysis and numerical simulations. Qualitative analysis have been used to explore the stability dynamics of the equilibrium points while visualization techniques such as bifurcation diagrams, Poincaré maps, maxima return maps and largest Lyapunov exponents are numerically computed to confirm further complexity of these dynamics induced by the seasonal forcing on the mosquitoes oviposition rates. The obtained results show that ordinary differential equation models with external forcing can have rich dynamic behaviour, ranging from bifurcation to strange attractors which may explain the observed fluctuations found in RVF empiric outbreak data, as well as the non deterministic nature of RVF inter-epidemic activities. Furthermore, the coexistence of the endemic equilibrium is subjected to existence of certain number of infected Aedes mosquitoes, suggesting that Aedes have potential to initiate RVF epidemics through transovarial transmission and to sustain low levels of the disease during post epidemic periods. Therefore we argue that locations that may serve as RVF virus reservoirs should be eliminated or kept under control to prevent multi-periodic outbreaks and consequent chains of infections. The epidemiological significance of this study is: (1) low levels of birth rate (in both Aedes and Culex) can trigger unpredictable outbreaks; (2) Aedes mosquitoes are more likely capable of inducing unpredictable behaviour compared to the Culex; (3) higher oviposition rates on mosquitoes do not in general imply manifestation of irregular behaviour on the dynamics of the disease. Finally, our model with external seasonal forcing on vector oviposition rates is able to mimic the linear increase in livestock seroprevalence during inter-epidemic period showing a constant exposure and presence of active transmission foci. This suggests that RVF outbreaks partly build upon RVF inter-epidemic activities. Therefore, active RVF surveillance in livestock is recommended.

Global dynamics of a vector-borne disease model with latency and saturating incidence rate

This paper deals with a vector-borne disease model containing latency and nonlinear incidence rates. Global analysis is completely determined by suitable Lyapunov functionals. We explicitely determine the basic reproduction number and flnd that if this number is less than one then disease dies out, but if the number is larger than one, the disease causing strain become endemic. The study shows that the latency delay explicitely in∞uence the disease persistence.

Global dynamic analysis of a vector-borne plant disease model

An epidemic model which describes vector-borne plant diseases is proposed with the aim to investigate the effect of insect vectors on the spread of plant diseases. Firstly, the analytical formula for the basic reproduction number R0 is obtained by using the next generation matrix method, and then the existence of disease-free equilibrium and endemic equilibrium is discussed. Secondly, by constructing a suitable Lyapunov function and employing the theory of additive compound matrices, the threshold for the dynamics is obtained. If R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable, which means that the plant disease will disappear eventually; if R0 > 1, then the endemic equilibrium is globally asymptotically stable, which indicates that the plant disease will persist for all time. Finally some numerical investigations are provided to verify our theoretical results, and the biological implications of the main results are briefly discussed in the last section.

Bifurcation analysis in models for vector-borne diseases with logistic growth.

We establish and study vector-borne models with logistic and exponential growth of vector and host populations, respectively. We discuss and analyses the existence and stability of equilibria. The model has backward bifurcation and may have no, one, or two positive equilibria when the basic reproduction number R 0 is less than one and one, two, or three endemic equilibria when R 0 is greater than one under different conditions. Furthermore, we prove that the disease-free equilibrium is stable if R 0 is less than 1, it is unstable otherwise. At last, by numerical simulation, we find rich dynamical behaviors in the model. By taking the natural death rate of host population as a bifurcation parameter, we find that the system may undergo a backward bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, and cusp bifurcation with the saturation parameter varying. The natural death rate of host population is a crucial parameter. If the natural death rate is higher, then the host population and the disease will die out. If it is smaller, then the host and vector population will coexist. If it is middle, the period solution will occur. Thus, with the parameter varying, the disease will spread, occur periodically, and finally become extinct.

Modelling vertical transmission in vector-borne diseases with applications to Rift Valley fever

We present two ordinary differential equation models for Rift Valley fever (RVF) transmission in cattle and mosquitoes. We extend existing models for vector-borne diseases to include an asymptomatic host class and vertical transmission in vectors. We define the basic reproductive number, 0, and analyse the existence and stability of equilibrium points. We compute sensitivity indices of 0 and a reactivity index (that measures epidemicity) to parameters for baseline wet and dry season values. 0 is most sensitive to the mosquito biting and death rates. The reactivity index is most sensitive to the mosquito biting rate and the infectivity of hosts to vectors. Numerical simulations show that even with low equilibrium prevalence, increases in mosquito densities through higher rainfall, in the presence of vertical transmission, can result in large epidemics. This suggests that vertical transmission is an important factor in the size and persistence of RVF epidemics.

Application of three controls optimally in a vector-borne disease – a mathematical study

We have proposed and analyzed a vector-borne disease model with three types of controls for the eradication of the disease. Four different classes for the human population namely susceptible, infected, recovered and vaccinated and two different classes for the vector populations namely susceptible and infected are considered. In the first part of our analysis the disease dynamics are described for fixed controls and some inferences have been drawn regarding the spread of the disease. Next the optimal control problem is formulated and solved considering control parameters as time dependent. Different possible combination of controls are used and their effectiveness are compared by numerical simulation.

An ecohydrological model of malaria outbreaks

Abstract. Malaria is a geographically widespread infectious disease that is well known to be affected by climate variability at both seasonal and interannual timescales. In an effort to identify climatic factors that impact malaria dynamics, there has been considerable research focused on the development of appropriate disease models for malaria transmission driven by climatic time series. These analyses have focused largely on variation in temperature and rainfall as direct climatic drivers of malaria dynamics. Here, we further these efforts by considering additionally the role that soil water content may play in driving malaria incidence. Specifically, we hypothesize that hydro-climatic variability should be an important factor in controlling the availability of mosquito habitats, thereby governing mosquito growth rates. To test this hypothesis, we reduce a nonlinear ecohydrological model to a simple linear model through a series of consecutive assumptions and apply this model to malaria incidence data from three South African provinces. Despite the assumptions made in the reduction of the model, we show that soil water content can account for a significant portion of malaria's case variability beyond its seasonal patterns, whereas neither temperature nor rainfall alone can do so. Future work should therefore consider soil water content as a simple and computable variable for incorporation into climate-driven disease models of malaria and other vector-borne infectious diseases.

Transmission Dynamics of Carbapenemase-Producing Klebsiella Pneumoniae and Anticipated Impact of Infection Control Strategies in a Surgical Unit

BackgroundCarbapenemase-producing Klebsiella pneumoniae (CPKP) has been established as important nosocomial pathogen in many geographic regions. Transmission from patient to patient via the hands of healthcare workers is the main route of spread in the acute-care setting.Methodology/Principal FindingsEpidemiological and infection control data were recorded during a prospective observational study conducted in a surgical unit of a tertiary-care hospital in Greece. Surveillance culture for CPKP were obtained from all patients upon admission and weekly thereafter. The Ross-Macdonald model for vector-borne diseases was applied to obtain estimates for the basic reproduction number R0 (average number of secondary cases per primary case in the absence of infection control) and assess the impact of infection control measures on CPKP containment in endemic and hyperendemic settings. Eighteen of 850 patients were colonized with CPKP on admission and 51 acquired CPKP during hospilazation. R0 reached 2 and exceeded unity for long periods of time under the observed hand hygiene compliance (21%). The minimum hand hygiene compliance level necessary to control transmission was 50%. Reduction of 60% to 90% in colonized patients on admission, through active surveillance culture, contact precautions and isolation/cohorting, in combination with 60% compliance in hand hygiene would result in rapid decline in CPKP prevalence within 8–12 weeks. Antibiotics restrictions did not have a substantial benefit when an aggressive control strategy was implemented.Conclusions/SignificanceSurveillance culture on admission and isolation/cohorting of colonized patients coupled with moderate hand hygiene compliance and contact precautions may lead to rapid control of CPKP in endemic and hyperendemic healthcare settings.

Web-based GIS: the vector-borne disease airline importation risk (VBD-AIR) tool

BackgroundOver the past century, the size and complexity of the air travel network has increased dramatically. Nowadays, there are 29.6 million scheduled flights per year and around 2.7 billion passengers are transported annually. The rapid expansion of the network increasingly connects regions of endemic vector-borne disease with the rest of the world, resulting in challenges to health systems worldwide in terms of vector-borne pathogen importation and disease vector invasion events. Here we describe the development of a user-friendly Web-based GIS tool: the Vector-Borne Disease Airline Importation Risk Tool (VBD-AIR), to help better define the roles of airports and airlines in the transmission and spread of vector-borne diseases.MethodsSpatial datasets on modeled global disease and vector distributions, as well as climatic and air network traffic data were assembled. These were combined to derive relative risk metrics via air travel for imported infections, imported vectors and onward transmission, and incorporated into a three-tier server architecture in a Model-View-Controller framework with distributed GIS components. A user-friendly web-portal was built that enables dynamic querying of the spatial databases to provide relevant information.ResultsThe VBD-AIR tool constructed enables the user to explore the interrelationships among modeled global distributions of vector-borne infectious diseases (malaria. dengue, yellow fever and chikungunya) and international air service routes to quantify seasonally changing risks of vector and vector-borne disease importation and spread by air travel, forming an evidence base to help plan mitigation strategies. The VBD-AIR tool is available at http://www.vbd-air.com.ConclusionsVBD-AIR supports a data flow that generates analytical results from disparate but complementary datasets into an organized cartographical presentation on a web map for the assessment of vector-borne disease movements on the air travel network. The framework built provides a flexible and robust informatics infrastructure by separating the modules of functionality through an ontological model for vector-borne disease. The VBD‒AIR tool is designed as an evidence base for visualizing the risks of vector-borne disease by air travel for a wide range of users, including planners and decisions makers based in state and local government, and in particular, those at international and domestic airports tasked with planning for health risks and allocating limited resources.

Stochastically perturbed vector-borne disease models with direct transmission

In this paper we study a stochastic epidemic model of vector-borne diseases with direct mode of transmission and its delay modification. More precisely, we extend the deterministic epidemic models by introducing random perturbations around the endemic equilibrium state. By using suitable Lyapunov functions and functionals, we obtain stability conditions for the considered models and study the effect of the delay on the stability of the endemic equilibrium. Finally, numerical simulations for the stochastic model of malaria disease transmission are presented to illustrate our mathematical findings.

Stochastically perturbed vector-borne disease models with direct transmission

The subject of this paper is the stochastic epidemic malaria model with time delay, described by the system of the It o ˆ stochastic functional delay equations. We center such a system around the endemic equilibrium state and, by the Lyapunov functional method, we obtain sufficient conditions for model parameters, as well as for time delays within which we can claim the asymptotical mean square stability and stability in probability. Finally, we present an example to show the compatibility of our mathematical results with the stochastic delay malaria model with quantities which are reliable data, as well as an example which shows that introduction of environmental noise annuls Hopf Bifurcation of the corresponding deterministic model.

Parameterization and Sensitivity Analysis of a Complex Simulation Model for Mosquito Population Dynamics, Dengue Transmission, and Their Control

Models can be useful tools for understanding the dynamics and control of mosquito-borne disease. More detailed models may be more realistic and better suited for understanding local disease dynamics; however, evaluating model suitability, accuracy, and performance becomes increasingly difficult with greater model complexity. Sensitivity analysis is a technique that permits exploration of complex models by evaluating the sensitivity of the model to changes in parameters. Here, we present results of sensitivity analyses of two interrelated complex simulation models of mosquito population dynamics and dengue transmission. We found that dengue transmission may be influenced most by survival in each life stage of the mosquito, mosquito biting behavior, and duration of the infectious period in humans. The importance of these biological processes for vector-borne disease models and the overwhelming lack of knowledge about them make acquisition of relevant field data on these biological processes a top research priority.