Global Dynamics of a Kawasaki Disease Vascular Endothelial Cell Injury Model with Backward Bifurcation and Hopf Bifurcation

Kawasaki disease(KD)is an acute,febrile,systemic vasculitis that mainly affects children under five years of age.In this paper,we propose and study a class of 5-dimensional ordinary differential equation model describing the vascular endothelial cell injury in the lesion area of KD.This model exhibits forward/backward bifurcation.It is shown that the vascular injury-free equilibrium is locally asymptotically stable if the basic reproduction number R_(0)<1.Further,we obtain two types of sufficient conditions for the global asymptotic stability of the vascular injury-free equilibrium,which can be applied to both the forward and backward bifurcation cases.In addition,the local and global asymptotic stability of the vascular injury equilibria and the presence of Hopf bifurcation are studied.It is also shown that the model is permanent if the basic reproduction number R_(0)>1,and some explicit analytic expressions of ultimate lower bounds of the solutions of the model are given.Our results suggest that the control of vascular injury in the lesion area of KD is not only correlated with the basic reproduction number Ro,but also with the growth rate of normal vascular endothelial cells promoted by the vascular endothelial growth factor.

Backward Bifurcation in Simple SIS Model

We describe and analyze a simple SIS model with treatment. In particular, we give a completely qualitative analysis by means of the theory of asymptotically autonomous system. It is found that a backward bifurcation occurs if the adequate contact rate or the capacity is small. It is also found that there exists bistable endemic equilibria. In the case of disease-induced death, it is shown that the backward bifurcation also occurs. Moreover, there is no limit cycle under some conditions, and the subcritical Hopf bifurcation occurs under another conditions.

Backward bifurcation in a fractional-order and two-patch model of tuberculosis epidemic with incomplete treatment

In this paper,we consider a tuberculosis model with incomplete treatment and extend the model to a Caputo fractional-order and two-patch version with exogenous re-infection among the treated individuals,in which only susceptible individuals can travel freely between the patches.The model has multiple equilibria.We determine conditions that lead to the appearance of a backward bifurcation.The results show that the TB model can have exogenous reinfection among the treated individuals and,at the same time,does not exhibit backward bifurcation.Also,conditions that lead to the global asymptotic stability of the disease-free equilibrium are obtained.In case without reinfection,the model has four equilibria.In this case,the global asymptotic stability of the equilibria is established using the Lyapunov function theory together with the LaSalle invariance principle for fractional differential equations(FDEs).Numerical simulations confirm the validity of the theoretical results.

A nonlinear relapse model with disaggregated contact rates:Analysis of a forward-backward bifurcation

Throughout the progress of epidemic scenarios,individuals in different health classes are expected to have different average daily contact behavior.This contact heterogeneity has been studied in recent adaptive models and allows us to capture the inherent differences across health statuses better.Diseases with reinfection bring out more complex scenarios and offer an important application to consider contact disaggregation.Therefore,we developed a nonlinear differential equation model to explore the dynamics of relapse phenomena and contact differences across health statuses.Our incidence rate function is formulated,taking inspiration from recent adaptive algorithms.It incorporates contact behavior for individuals in each health class.We use constant contact rates at each health status for our analytical results and prove conditions for different forward-backward bifurcation scenarios.The relationship between the different contact rates heavily in-fluences these conditions.Numerical examples highlight the effect of temporarily recov-ered individuals and initial conditions on infected population persistence.

Backward bifurcation,basic reinfection number and robustness of an SEIRE epidemic model with reinfection

Recent evidences show that individuals who recovered from COVID-19 can be reinfected.However,this phenomenon has rarely been studied using mathematical models.In this paper,we propose an SEIRE epidemic model to describe the spread of the epidemic with reinfection.We obtain the important thresholds R_(0)(the basic reproduction number)and R_(c)(a threshold less than one).Our investigations show that when R_(0)>1,the system has an endemic equilibrium,which is globally asymptotically stable.When R_(c)<R_(0)<1,the epidemic system exhibits bistable dynamics.That is,the system has backward bifurcation and the disease cannot be eradicated.In order to eradicate the disease,we must ensure that the basic reproduction number R_(0) is less than R_(c).The basic reinfection number is obtained to measure the reinfection force,which turns out to be a new tipping point for disease dynamics.We also give definition of robustness,a new concept to measure the dificulty of completely eliminating the disease for a bistable epidemic system.Numerical simulations are carried out to verify the conclusions.

Backward,Hopf bifurcation in a heroin epidemic model with treat age

Heroin is considered as damage on the health of the individuals.In this paper,we consider a heroin epidemic model with treat age,where the inner rivalry in between the drug users for a small dose of drugs is investigated.The influence of treatment period for a drug consumer before quitting treatment is investigated,where it is obtained that the considered model can undergo backward bifurcation,which shows the possibility of having two endemic equilibriums,this type of bifurcation is discussed in terms of the basic reproduction number R0.The bifurcation diagram is drown in the case of an age structured model.Also,we show the existence of Hopf bifurcation is shown under suitable conditions on the model parameters.The obtained mathematical results are confirmed numerically.

GLOBAL ANALYSIS OF SIS EPIDEMIC MODEL WITH A SIMPLE VACCINATION AND MULTIPLE ENDEMIC EQUILIBRIA

An SIS epidemic model with a simple vaccination is investigated in this article, The efficiency of vaccine, the disease-related death rate and population dynamics are also considered in this model. The authors find two threshold R0 and Rc （Rc may not exist）. There is a unique endemic equilibrium for R0 〉 1 or Rc = R0; there are two endemic equilibria for Rc 〈 R0 〈 1; and there is no endemic equilibrium for Rn 〈 Rc 〈 1. When Rc exists, there is a backward bifurcation from the disease-free equilibrium for R0 = 1. They analyze the stability of equilibria and obtain the globally dynamic behaviors of the model. The results acquired in this article show that an accurate estimation of the efficiency of vaccine is necessary to prevent and controll the spread of disease.

Bifurcation analysis and optimal control of COVID-19 with exogenous reinfection and media coverages

In this paper,a SEIR epidemic model related to media coverage and exogenous reinfections is established to explore the transmission dynamics of COVID-19.The basic reproduction number is calculated using the next generation matrix method.First,the existence of equilibrium points is investigated,and different kinds of equilibrium points indicate that the disease may disappear,or exist that result in different quantity of susceptible individuals,pre-symptomatic infected individuals and symptomatic infected individuals.The stability of the equilibria is discussed by a geometric approach,and it is found that controlling reproduction number to be lower than 1 is not suficient for eradication of COVID-19.Second,transcritical bifurcation is explored,and it is found that improving the ratio of exogenous reinfection may lead to backward bifurcation under poor medical conditions,which indicates that two endemic equilibrium points appear.Third,to investigate the infuence of parameters on the basic reproduction,sensitivity analysis is done to choose relatively sensitive parameters,and the parameters for treatment and media coverage are selected.An optimal control model is established to balance the treatment and media awareness.By exploring the existence and the uniqueness of the optimal control solution,the optimal control strategies are given.Finally,we run numerical simulations to verify the theoretical analysis on actual data of China,and the data from the four different states of India is used for forecasting the situation of infected individuals in a short period.It is found by the simulation that the co-function of treatment and media coverage results in the reduced number of infectious individuals.

Tuberculosis transmission with multiple saturated exogenous reinfections

In this paper,a nonlinear mathematical model for tuberculosis transmission,which incorporates multiple saturated exogenous reinfections,is proposed and explored.The existence of disease-free and endemic steady states is investigated.Disease-free equilibrium(DFE)is locally asymptotically stable(LAS)but not globally asymptotically stable(GAS)when the basic reproduction number,R_(0)<1.However,it is GAS only when there is no exogenous reinfection.The local asymptotic stability and global asymptotic stability of the unique endemic equilibrium point(EEP)are established under certain conditions when R_(0)>1.Further,the EEP is GAS when R_(0)>1,provided there is no exogenous reinfection.When Ro is below unity,the presence of multiple endemic equilibria is found which leads to backward bifurcation.It is demonstrated that the system encounters a Hopf-bifurcation when the transmission rate β crosses a critical value,resulting in the formation of limit cycles,i.e.periodic solutions bifurcate around the EEP whenβpasses a critical value.The stability and direction of Hopf-bifurcation are also studied.The results of the analytical work are validated through numerical simulations.A numerical simulation illustrates that EEP losses its stability via Hopf-bifurcation for specific parameters.However,when the bifurcation parameterβis increased further,the EEP regains its stability.In addition,Hopf-bifurcation occurs due to exogenous reinfection rates p and o.Thus,our model shows some important nonlinear dynamical behaviors.

Assessing the role of active case detection on visceral leishmaniasis control:A case study

In this paper,we formulate and analyze a compartmental model of visceral leishmaniasis(VL).We validate our model by calibrating it to the yearly VL incidence data for India and Bangladesh.The proposed model's basic reproduction number(R_(0))has been derived and estimated.We have proved the existence of backward bifurcation in our system.The phenomenon of backward bifurcation has public health implications because the classical requirement of R_(0)<1,while necessary,is no longer sufficient for effective disease control(or elimination).In such a(backward bifurcation)situation,the initial sizes of the model's subpopulations(state variables)would determine the effectiveness of disease control or elimination.As a result,it is the first attempt to represent and study VL disease dynamics using active case detection(ACD)as a control strategy,although a few experimental studies have been conducted to evaluate ACD in disease transmission.We use sensitivity analysis to investigate the effects of the model's controllable parameters for the basic reproduction number.We found that σ1,σ2,σ3(monitoring rate to infected individuals)have negative impacts on R_(0).The numerical result suggests that the ACD strategy can be useful for the VL elimination program.Due to a small increment in the monitoring rate,the dynamic behavior of infected individuals dramatically decreased.Successful employment of ACD strategy may reduce more than 60-80%symptomatic and post kala-azar dermal leishmaniasis(PKDL)individuals.We found that healthcare organizations should prioritize ACD in symptomatic and PKDL individuals over asymptomatic individuals.We also observed that the use of only culling effect to the reservoirs is not beneficial to society in the control of VL,but spraying of insecticides and the use of treated bednets can be effective control strategies to curtail the outbreak of the disease.

Mathematical model of zika virus dynamics with vector control and sensitivity analysis

In this paper,we have developed and analyzed a deterministic Zika model considering both vector and sexual transmission route with the effect of human awareness and vector control in the absence of disease induce death.To formulate the model,we assume that the Zika virus is being first transmitted to human by mosquito bite,and then it is being transmitted to his or her sexual partner.The system contains at most three equilibrium points among them one is the disease free and other two are endemic equilibrium points,exists under certain conditions.The theoretical analysis shows that the diseases-free equilibrium is locally and globally asymptotically stable if the basic reproduction number is less than one.Theatrically we have established that endemic equilibrium point which is locally asymptotically stable if the basic reproduction number is greater than one.The system exhibits backward bifurcation when the transmission probability per biting of susceptible mosquito with infected humans crosses the critical value.We estimate the model parameters and validate the model by fitting the model with the reported Zika infected human data from 1 to 36 week of 2016 Zika outbreak in Colombia.Furthermore,using the normalised forward sensitivity index method we have established that the model parameter mosquito biting rate,recruitment rate of mosquito,transmission probability per biting of Susceptible(infected)humans with infected(susceptible)mosquito,rate of awareness in host population,recovery rates of infected human are most sensitive parameters of the considered Zika model.Lastly,some conclusions are given to control the spreading of the Zika disease.

Mathematical analysis of a model for zoonotic visceral leishmaniasis

Zoonotic visceral leishmaniasis(ZVL),caused by the protozoan parasite Leishmania infantum and transmitted to humans and reservoirhosts by female sandflies,is endemic inmany parts of the world(notably in Africa,Asia and the Mediterranean).This study presents a new mathematical model for assessing the transmission dynamics of ZVL in human and non-human animal reservoir populations.The model undergoes the usual phenomenon of backward bifurcation exhibited by similar vector-borne disease transmission models.In the absence of such phenomenon(which is shown to arise due to the disease-induced mortality in the host populations),the nontrivial disease-free equilibrium of the model is shown to be globallyasymptotically stable when the associated reproduction number of the model is less than unity.Using case and demographic data relevant to ZVL dynamics in Arac atubamunicipality of Brazil,it is shown,for the default case when systemic insecticide-based drugs are not used to treat infected reservoir hosts,that the associated reproduction number of the model (ℛ0) ranges from 0.3 to 1.4,with a mean of ℛ0=0:85.Furthermore,when the effect of such drug treatment is explicitly incorporated in the model(i.e.,accounting for the additional larval and sandfly mortality,following feeding on the treated reservoirs),the range ofℛ0 decreases toℛ02∈[0.1,0.6],with a mean ofℛ0=0:35(this significantly increases the prospect of the effective control or eliminationof the disease).Thus,ZVL transmissionmodels(in communities where such treatment strategy is implemented)that do not explicitly incorporate the effect of such treatmentmay be over-estimating the disease burden(asmeasured in terms ofℛ0)in the community.It is shown thatℛ0 ismore sensitive to increases in sandflylifespanthan thatof the animal reservoir(so,a strategy that focuses on reducing sandflies,rather than the animal reservoir(e.g.,via culling),may be more effective in reducing ZVL burden in the community).Further sensitivity analysis of themodel ranks the sandfly removal rate(by natural death or by feeding from insecticide-treated reservoir hosts),the biting rate of sandflies on the reservoir hosts and the progression rate of exposed reservoirs to active ZVL as the three parameters with themost effect on the disease dynamics or burden(as measured in terms of the reproduction number ℛ0).Hence,this study identifies the key parameters that play a key role on the disease dynamics,and thereby contributing in the design of effective control strategies(that target the identified parameters).

Role of precautionary measures in HIV epidemics： A mathematical assessment

United Nations Political Declaration 2011 on HIV and AIDS calls to reduce the sexual transmission and the transmission of HIV among people, who inject drugs by 50% by 2015, through different control strategies and precautionary measures. In this paper, we propose and study a simple SI type model that considers the effect of various precaution- ary measures to control HIV epidemic. We show, unlike conventional epidemic models, that the basic reproduction number which essentially considered as the disease eradica- tion condition is no longer sufficient to eliminate HIV infection. In particular, we show that even when the basic reproduction number is made less than unity, the disease may persist if the initial outbreak is not low. Eradication of disease is however guaranteed if the ensemble control measure exceeds some upper critical value. It is also shown that an epidemic model with mass action incidence may exhibit backward bifurcation and bistability if density-dependent demography is considered. Our theoretical study thus indicates that extra attention should be given in controlling HIV epidemic to achieve the desired result.

DYNAMICS OF SIS EPIDEMIC MODEL WITH THE STANDARD INCIDENCE RATE AND SATURATED TREATMENT FUNCTION

An SIS epidemic model with the standard incidence rate and saturated treatment func- tion is proposed. The dynamics of the system are discussed, and the effect of the capacity for treatment and the recruitment of the population are also studied. We find that the effect of the maximum recovery per unit of time and the recruitment rate of the popula- tion over some level are two factors which lead to the backward bifurcation, and in some cases, the model may undergo the saddle-node bifurcation or Bogdanov-Takens bifurca- tion. It is shown that the disease-free equilibrium is globally asymptotically stable under some conditions, Numerical simulations are consistent with our obtained results in the- orems, which show that improving the efficiency and capacity of treatment is important for control of disease.

Analysis of a malaria model with mosquito host choice and bed-net control

A malaria model is formulated which includes the enhanced attractiveness of infectious humans to mosquitoes, as result of host manipulation by malaria parasite, and the human behavior, represented by insecticidetreated bed-nets usage. The occurrence of a backward bifurcation at R0 = 1 is shown to be possible, which implies that multiple endemic equilibria co-exist with a stable disease-free equilibrium when the basic repro- duction number is less than unity. This phenomenon is found to be caused by disease- induced human mortality. The global asymptotic stability of the endemic equilibrium for R0 〉1 is proved, by using the geometric method for global stability. Therefore, the disease becomes endemic for R0〉 1 regardless of the number of initial cases in both the human and vector populations. Finally, the impact on system dynamics of vector＇s host preferences and bed-net usage behavior is investigated.

Global dynamics in a model for anthrax transmission in animal populations

In this paper,we propose a deterministic model to study the transmission dynamics of anthrax disease,which includes live animals,carcasses,spores in the environment and vectors.We derive three biologically plausible and insightful quantities(reproduction numbers)that determine the stability of the equilibria.We carry out rigorous mathematical analysis on the model dynamics,the global stability of the disease-free and vector-free equilibrium,the disease-free equilibrium and the vector-free disease equilibrium is proved.The global stability of the endemic equilibrium as the basic reproduction number is greater than one is derived in the special case in which the disease-related death rate is zero.The possibility of backward bifurcation is briefly discussed.Numerical analyses are carried out to understand the transmission dynamics of anthrax and investigate effective control strategies for the outbreaks of the disease.Our studies suggest that the larval vector control measure should be taken as early as possible to control the vector population size,a vaccination policy and an animal carcass removal policy are useful methods to control the prevalence of the diseases in infected animal populations,the adult vector control measure is also necessary to prevent the transmission of anthrax.

A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection

In this paper,we propose a mathematical model of tuberculosis with two treatments and exogenous re-infection,in which the treatment is effective for number of infectious individuals and it fails for some other infectious individuals who are being treated.We show that the model exhibits the phenomenon of backward bifurcation,where a stable disease-free equilibrium coexists with a stable endemic equilibria when the related basic reproduction number is less than unity.Also,it is shown that under certain conditions the model cannot exhibit backward bifurcation.Furthermore,it is shown in the absence of re-infection,the backward bifurcation phenomenon does not exist,in which the disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity.The global asymptotic stability of the endemic equilibrium,when the associated reproduction number is greater than unity,is established using the geometric approach.Numerical simulations are presented to illustrate our main results.

Impact of predation in the spread of an infectious disease with time fractional derivative and social behavior

The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior.The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction.This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models.For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devotedto study the effect of a non-selective hunting on the spread of the disease, where the localstability of the equilibria is investigated. Further the backward bifurcation is obtainedconcerning basic reproduction rate of the infection. The second case is for explaining theimpact of selecting the weakest infected prey on the edge of the herd by a predator onthe prevalence of the infection, where the local behavior is scrutinized. Moreover, for thegraphical representation part, a numerical simulation scheme has been achieved usingthe Caputo fractional derivative operator.

A threshold delay model of HIV infection of newborn infants through breastfeeding

The breast milk of HIV infected women contains HIV virus particles,therefore children can become infected through breastfeeding.We develop a mathematical epidemiological model of HIV infection in infants,infected children and infected women that represents infection of an infant/child as a series of exposures,by incorporating within-host virus dynamics in the individuals exposed to the virus through breastfeeding.We show that repeated exposures of the infant/child via breastfeeding can cause bi-stability dynamics and,subsequently,infection persistence even when the epidemiological basic reproduction number R0 is less than unity.This feature of the model,due to a backward bifurcation,gives new insight into the control mechanisms of HIV disease through breastfeeding.

Analysis of a fractional-order SIS epidemic model with saturated treatment

In this paper,we propose and analyze a fractional-order SIS epidemic model with the saturated treatment and disease transmission.The existence and uniqueness,nonnegativity and finiteness of solutions for our suggested model have been studied.Different equilibria of the model are found and their local and global stability analyses are also examined.Furthermore,the conditions for fractional backward and fractional Hopf bifurcation are also analyzed in both the commensurate and incommensurate fractional-order model.We study how the control parameter and the order of the fractional derivative play role in local as well as global stability of equilibrium points and Hopf bifurcation.We have demonstrated the analytical results of our proposed model system through several numerical simulations.