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 describ...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 R_(0),but also with the growth rate of normal vascular endothelial cells promoted by the vascular endothelial growth factor.展开更多
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 o...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.展开更多
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 ...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.展开更多
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...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.展开更多
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 descri...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.展开更多
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 investiga...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.展开更多
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 tw...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.展开更多
Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect...Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect of nonlinear harvesting in this research paper.Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system.We have examined the local stability analysis of all the equilibrium points.Besides these various types,rich and complex dynamical behaviors such as backward,saddle-node,Hopf and Bogdanov-Takens(BT)bifurcations,homoclinic loop and limit cycles appear in this model.Furthermore,interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points.Also,we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species.We have observed that the extinction of predator species may not happen due to backward bifurcation,although a stable predator-free equilibrium(PFE)exists.Finally,numerical simulations are discussed using MATLAB to verify all the theoretical results.展开更多
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 gener...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.展开更多
This paper formulates an SEIRSHM epidemic model with general birth rate,media report and limited medical resources.Firstly,the well-posedness of the solutions and the extinction of the disease are discussed.Then,the e...This paper formulates an SEIRSHM epidemic model with general birth rate,media report and limited medical resources.Firstly,the well-posedness of the solutions and the extinction of the disease are discussed.Then,the existence of the endemic equilibrium is discussed and we find when R^(*)>1 and R0=1,there exhibits a backward bifurcation,if R^(*)<1 and R0=1,there exhibits a forward bifurcation.Finally,numerical simulations are carried out to illustrate the main results and show that media report and limited medical resources have a great impact on disease transmission.展开更多
Tuberculosis(TB)continues to be a major global health challenge,with millions of new cases and deaths each year despite the massive efforts and funding put in the fight against the disease.In this paper,we develop a m...Tuberculosis(TB)continues to be a major global health challenge,with millions of new cases and deaths each year despite the massive efforts and funding put in the fight against the disease.In this paper,we develop a mathematical model to evaluate the impact of TB-funded prevention programs on the transmission dynamics of TB.The model incorporates stages of TB infection(latent and active),and accounts for the effects of treatment,funding and TB-funded prevention programs.Our analysis shows that increased funding and enhanced prevention programs reduce the number of active TB cases,thereby decreasing the reproduction number and TB endemicity.Specifically,higher funding rates lead to improved prevention and treatment outcomes,resulting in the lowering of the effective reproduction number(R0)and reduced transmission.The model's steady states are determined and it is shown that the model has a disease-free equilibrium that is locally asymptotically stable whenever R0<1 and multiple endemic equilibria for Rc0<R0<1 and a unique endemic equilibrium for R0>1.The model is shown to exhibit a backward bifurcation that vanishes as the funding for TB is increased.The paper also highlights that treatment alone,while beneficial,is less effective than a combined strategy involving funding and prevention.Numerical simulations are carried out and the influences of various parameters on the effective reproduction number are investigated.The implications of TB-funded prevention programs on TB dynamics and control of TB are discussed and valuable insights for policymakers in designing effective TB control programs are highlighted.展开更多
Whenever a disease spreads in the population,people have a tendency to alter their behavior due to the availability of knowledge concerning disease prevalence.Therefore,the incidence term of the model must be suitably...Whenever a disease spreads in the population,people have a tendency to alter their behavior due to the availability of knowledge concerning disease prevalence.Therefore,the incidence term of the model must be suitably changed to reflect the impact of information.Furthermore,a lack of medical resources affects the dynamics of disease.In this paper,a mathematical model of malaria of type ShIn Rh-Su Iu with media information and saturated treatment is considered.The analysis of the model is performed and it is established that when the basic reproduction number,R_(0),is less than unity,the disease may or may not die out due to saturated treatment.Furthermore,it is pointed out that if medical resources are accessible to everyone,disease elimination in this situation is achievable.The global asymptotic stability of the unique endemic equilibrium point(EEP)is established using the geometric approach under parametric restriction.The sensitivity analysis is also carried out using the normalized forward sensitivity index(NFSI).It is dificult to derive the analytical solution for the governing model due to it being a system of highly nonlinear ordinary differential equations.To overcome this challenge,a specialized numerical scheme known as the non-standard finite difference(NSFD)approach has been applied.The suggested numerical method is subjected to an elaborate theoretical analysis and it is determined that the NSFD scheme maintains the positivity and conservation principles of the solutions.It is also established that the disease-free equilibrium(DFE)point has the same local stability criteria as that of continuous model.Our proposed NSFD scheme also captures the backward bifurcation phenomena.The outcomes of the NSFD scheme are compared to two well-known standard numerical techniques,namely the fourth-order Runge-Kutta(RK4)method and theforward Euler method.展开更多
In this paper,we introduce a partial differential equation(PDE)model to describe the transmission dynamics of dengue with two viral strains and possible secondary infection for humans.The model features the variable i...In this paper,we introduce a partial differential equation(PDE)model to describe the transmission dynamics of dengue with two viral strains and possible secondary infection for humans.The model features the variable infectiousness during the infectious period,which we call the infection age of the infectious host.We define two thresholds R_(1)^(j) and R_(2)^(j),j=1,2,and show that the strain j can not invade the system if R_(1)^(j)+R_(2)^(j)<1.Further,the disease-free equilibrium of the system is globally asymptotically stable if maxj{R_(1)^(j)+R_(2)^(j)}<1.When R_(1)^(j)>1,strain j dominance equilibrium 6,exists,and is locally asymptotically stable when R_(1)^(j)>1,R_(1)^(j)i<■R_(1)^(j),i,j=1,2,i ≠j,■∈ E(0,1).Then,by applying Lyapunov-LaSalle techniques,we establish the global asymptotical stability of the dominance equilibrium corresponding to some strain j.This implies strain j eliminates the other strain as long as R_(1)^(j)/R_(1)^(j)<b_(i)/b_(j)<1,i≠j,where bj denotes the probability of a given susceptible mosquito being transmitted by a primarily infected human with strain j.Finally,we study the existence of the coexistence equilibria under some conditions.展开更多
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 an...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).展开更多
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 deat...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.展开更多
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 ...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.展开更多
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 pre...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.展开更多
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 insecticidetreate...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.展开更多
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 a...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.展开更多
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 infecti...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.展开更多
基金supported by the National Natural Science Foundation of China(12201038)the Project funded by China Postdoctoral Science Foundation(2022TQ0026)+1 种基金the Fundamental Research Funds for the Central Universities(FRF-TP-22-102A1)the Beijing Natural Science Foundation(1202019).
文摘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 R_(0),but also with the growth rate of normal vascular endothelial cells promoted by the vascular endothelial growth factor.
基金Supported by the National Natural Science Foundation of China(No.10571143,30770555)
文摘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.
文摘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.
基金support from the Research Center in Pure and Applied Mathematics and the Department of Mathematics at Universidad de Costa Rica.
文摘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.
基金supported by the National Natural Science Foundation of China(U21A20206)Natural Science Foundations of Henan(192102310089,202300410045).
文摘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.
基金S.Bentout and S.Djilali are partially supported by the DGRSTD of Algeria No.C00L03UN130120200004.
文摘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.
基金Supported by the Nature Science Foundation of China(19971066)Postdoctoral Foundation of China(2005037785)
文摘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.
文摘Due to the random search of species and from the economic point of view,combined harvesting is more suitable than selective harvesting.Thus,we have developed and analyzed a prey-predator model with the combined effect of nonlinear harvesting in this research paper.Nonlinear harvesting possesses multiple predator-free and interior equilibrium points in the dynamical system.We have examined the local stability analysis of all the equilibrium points.Besides these various types,rich and complex dynamical behaviors such as backward,saddle-node,Hopf and Bogdanov-Takens(BT)bifurcations,homoclinic loop and limit cycles appear in this model.Furthermore,interesting phenomena like bi-stability and tri-stability occur in our model between the different equilibrium points.Also,we have derived different threshold values of predator harvesting parameters and prey environmental carrying capacity from these bifurcations to obtain the different harvesting strategies for both species.We have observed that the extinction of predator species may not happen due to backward bifurcation,although a stable predator-free equilibrium(PFE)exists.Finally,numerical simulations are discussed using MATLAB to verify all the theoretical results.
基金This research is supported by the Natural Science Foundation of Beijing Municipality(No.4202025).
文摘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.
基金supported by the National Natural Science Foundation(12201540)the Natural Science Foundation of Xinjiang Uygur Autonomous Region,China(2022D01C64)the Doctoral Research Initiation Fund of Xinjiang University,China(620320024).
文摘This paper formulates an SEIRSHM epidemic model with general birth rate,media report and limited medical resources.Firstly,the well-posedness of the solutions and the extinction of the disease are discussed.Then,the existence of the endemic equilibrium is discussed and we find when R^(*)>1 and R0=1,there exhibits a backward bifurcation,if R^(*)<1 and R0=1,there exhibits a forward bifurcation.Finally,numerical simulations are carried out to illustrate the main results and show that media report and limited medical resources have a great impact on disease transmission.
基金support of their respective institutions in the production of the manuscript.
文摘Tuberculosis(TB)continues to be a major global health challenge,with millions of new cases and deaths each year despite the massive efforts and funding put in the fight against the disease.In this paper,we develop a mathematical model to evaluate the impact of TB-funded prevention programs on the transmission dynamics of TB.The model incorporates stages of TB infection(latent and active),and accounts for the effects of treatment,funding and TB-funded prevention programs.Our analysis shows that increased funding and enhanced prevention programs reduce the number of active TB cases,thereby decreasing the reproduction number and TB endemicity.Specifically,higher funding rates lead to improved prevention and treatment outcomes,resulting in the lowering of the effective reproduction number(R0)and reduced transmission.The model's steady states are determined and it is shown that the model has a disease-free equilibrium that is locally asymptotically stable whenever R0<1 and multiple endemic equilibria for Rc0<R0<1 and a unique endemic equilibrium for R0>1.The model is shown to exhibit a backward bifurcation that vanishes as the funding for TB is increased.The paper also highlights that treatment alone,while beneficial,is less effective than a combined strategy involving funding and prevention.Numerical simulations are carried out and the influences of various parameters on the effective reproduction number are investigated.The implications of TB-funded prevention programs on TB dynamics and control of TB are discussed and valuable insights for policymakers in designing effective TB control programs are highlighted.
文摘Whenever a disease spreads in the population,people have a tendency to alter their behavior due to the availability of knowledge concerning disease prevalence.Therefore,the incidence term of the model must be suitably changed to reflect the impact of information.Furthermore,a lack of medical resources affects the dynamics of disease.In this paper,a mathematical model of malaria of type ShIn Rh-Su Iu with media information and saturated treatment is considered.The analysis of the model is performed and it is established that when the basic reproduction number,R_(0),is less than unity,the disease may or may not die out due to saturated treatment.Furthermore,it is pointed out that if medical resources are accessible to everyone,disease elimination in this situation is achievable.The global asymptotic stability of the unique endemic equilibrium point(EEP)is established using the geometric approach under parametric restriction.The sensitivity analysis is also carried out using the normalized forward sensitivity index(NFSI).It is dificult to derive the analytical solution for the governing model due to it being a system of highly nonlinear ordinary differential equations.To overcome this challenge,a specialized numerical scheme known as the non-standard finite difference(NSFD)approach has been applied.The suggested numerical method is subjected to an elaborate theoretical analysis and it is determined that the NSFD scheme maintains the positivity and conservation principles of the solutions.It is also established that the disease-free equilibrium(DFE)point has the same local stability criteria as that of continuous model.Our proposed NSFD scheme also captures the backward bifurcation phenomena.The outcomes of the NSFD scheme are compared to two well-known standard numerical techniques,namely the fourth-order Runge-Kutta(RK4)method and theforward Euler method.
基金supported by the National Natural Science Foundation of China(grant numbers 11871415 and 12271466)the Henan Province Distinguished Professor program.
文摘In this paper,we introduce a partial differential equation(PDE)model to describe the transmission dynamics of dengue with two viral strains and possible secondary infection for humans.The model features the variable infectiousness during the infectious period,which we call the infection age of the infectious host.We define two thresholds R_(1)^(j) and R_(2)^(j),j=1,2,and show that the strain j can not invade the system if R_(1)^(j)+R_(2)^(j)<1.Further,the disease-free equilibrium of the system is globally asymptotically stable if maxj{R_(1)^(j)+R_(2)^(j)}<1.When R_(1)^(j)>1,strain j dominance equilibrium 6,exists,and is locally asymptotically stable when R_(1)^(j)>1,R_(1)^(j)i<■R_(1)^(j),i,j=1,2,i ≠j,■∈ E(0,1).Then,by applying Lyapunov-LaSalle techniques,we establish the global asymptotical stability of the dominance equilibrium corresponding to some strain j.This implies strain j eliminates the other strain as long as R_(1)^(j)/R_(1)^(j)<b_(i)/b_(j)<1,i≠j,where bj denotes the probability of a given susceptible mosquito being transmitted by a primarily infected human with strain j.Finally,we study the existence of the coexistence equilibria under some conditions.
文摘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).
文摘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.
文摘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.
文摘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.
文摘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.
基金This work was supported by the National Natural Science Foundation of China(11801431)the Natural Science Basic Research Plan in Shaanxi Province of China(2021JM-445,2022JM-023).
文摘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.
文摘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.
