Existence Of Backward Bifurcation Research Articles

Optimal control analysis of a malaria transmission model with applications to Democratic Republic of Congo

In this paper, a dynamical model of malaria transmission with vector-bias and timedependent controls is investigated. The controls include the RTS,S malaria vaccine, using insecticide-treated mosquito net, treatment of infectious human, and indoor spraying. For constant controls, the existence and stability of equilibrium, as well as the existence of backward bifurcation, are obtained. The sensitivity analysis quantifies the impact of parameters and controls on the basic reproduction number. For time-dependent controls, by using the Pontryagin’s maximum principle the existence and expression of optimal controls are established. As an application of the model and control strategies, the malaria transmission and controls in Democratic Republic of Congo are discussed. To be specific, we simulate the reported cases of Democratic Republic of Congo by World Health Organization and predict the trends. Cost-effectiveness analysis and numerical simulations show that combining all controls can minimize the number of infected humans to the full extent, using insecticide-treated mosquito net is the most cost-effectiveness strategy, combining RTS,S malaria vaccine with using insecticide-treated mosquito net and treatment of infectious human is also cost-effective among all the strategies with good effect.

A mathematical model on the transmission dynamics of typhoid fever with treatment and booster vaccination

Typhoid fever is a potentially fatal illness that is caused by the bacteria Salmonella typhi. In this study, a deterministic mathematical model was formulated to look into transmission dynamics of typhoid fever with treatment and booster vaccination. The reproduction number R0 is calculated using the next-generation matrix approach. Then, a stability analysis on the equilibrium points was performed using Routh–Hurwitz criteria. It was revealed that the disease-free equilibrium point is locally asymptotically stable whenever R0 is less than 1 together with other conditions. We also showed that R0≤1 does not guarantee global stability of the typhoid-free equilibrium point and corroborated the result by showing the possible existence of backward bifurcation at R0=1. The model parameters in R0 were also subjected to sensitivity analysis, which revealed that the transmission rate, infection through an exposed person, and bacteria are the most influential parameters of the reproduction number R0. Numerical simulations were run to determine the impact of various parameters on the dynamics of typhoid.

A Mathematical Model on the Dynamics of In-Host Infection Cholera Disease with Vaccination

In this paper, a within-host cholera mathematical model has been developed using a system of ordinary differential equations incorporating vaccine efficacy. The formulated model considers cells in an already vaccinated individual with a vaccine whose efficacy is γ . The solutions of the model have been shown to be both positive and bounded hence well-posed. The vaccine basic reproduction number has been carried out using the next generation matrix approach and is given by R 0 V = γ / d + μ 2 and R 0 V < 1 if γ < d + μ 2 . Analysis of the model shows that i n f e c t i o n f r e e e q u i l i b r i u m I F E point is both locally and globally asymptotically stable when R 0 V < 1 and i n f e c t i o n e q u i l i b r i u m I E point is locally asymptotically stable when R 0 V > 1 . Furthermore, analysis of the model shows that R 0 V < 1 is not sufficient enough to eradicate in-host cholera disease, hence the existence of backward bifurcation which is an indication as to why cholera disease is persistent. To highlight the relevance of vaccine efficacy, a numerical simulation of the model with respect to vaccination is carried out and shows that when the vaccine efficacy γ is high, there will be a lower infection rate of cells, hence the need to improve cholera vaccine efficacy.

Mathematical Modeling of the Population Dynamics of Age-Structured Criminal Gangs with Correctional Intervention Measures

The prevalence of criminal gang activities in the society has continued to be a huge burden to mankind. Despite several attempts by the government and non-governmental organizations in curtailing the activities of criminal gangs, new criminal gangs are beginning to emerge with associated criminal cases on the high side, thereby causing social unrest. This paper examines the impact of correctional intervention measures on the population dynamics of the criminal gangs in Nigeria, following an age-structured paradigm. The existence of backward bifurcation is presented sequel to the coexistence of gang free equilibrium and gang endemic equilibrium when the reproduction number is less than unity. Also, the model exhibits periodic trajectories (Hopf bifurcation) as a result of coexistence among the criminal gang classes. Using the Nigeria crime data (from 1993 to 2013) as obtained from Elsevier Data in Brief, we analyze the criminal gang model quantitatively and provide the sociological implications of the results. The numerical results obtained from this work show that if at least 60% of adolescents between 8–17 years who commit crimes are apprehended with prompt prosecution and correction, we can potentially eradicate adolescent delinquencies from the population and thus control the entire criminal gangs’ population.

Linking the disease transmission to information dissemination dynamics: An insight from a multi-scale model study

During the outbreak of emerging infectious diseases, information dissemination dynamics significantly affects the individuals’ psychological and behavioral changes, and consequently influences on the disease transmission. To investigate the interaction of disease transmission and information dissemination dynamics, we proposed a multi-scale model which explicitly models both the disease transmission with saturated recovery rate and information transmission to evaluate the effect of information transmission on dynamic behaviors. Considering time variation between information dissemination, epidemiological and demographic processes, we obtained a slow-fast system by reasonably introducing a sufficiently small quantity. We carefully examined the dynamics of proposed system, including existence and stability of possible equilibria and existence of backward bifurcation, by using the fast-slow theory and directly investigating the full system. We then compared the dynamics of the proposed system and the essential thresholds based on two methods, and obtained the similarity between the basic dynamical behaviors of the slow system and that of the full system. Finally, we parameterized the proposed model on the basis of the COVID-19 case data in mainland China and data related to news items, and estimated the basic reproduction number to be 3.25. Numerical analysis suggested that information transmission about COVID-19 pandemic caused by media coverage can reduce the peak size, which mitigates the transmission dynamics during the early stage of the COVID-19 pandemic.

A Mathematical Model of the Tuberculosis Epidemic.

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

Dynamical behaviour in discrete coupled within-host and between-host epidemic model with environmentally driven and saturation incidence

In this paper, a discrete-time coupled within-host and between-host epidemic model with environmentally driven and saturation incidence is discussed. The model is divided into an isolated fast model and an isolated coupled slow time model by using the way of limit equations. The positivity, boundedness and existence of equilibria for isolated fast and coupled slow time models are investigated, respectively. The global stability of infectious-free and infectious equilibria for the isolated fast time model is established by means of the Lyapunov functions and LaSalle's invariance principle. The local asymptotic stability of disease-free and endemic equilibria for the coupled slow time model is obtained by using the linearization method. When there are two positive equilibria in the coupled slow time model, the existence of backward bifurcation also is established. Furthermore, the correctness of theoretical conclusions and the rationality of conjecture are verified by the numerical simulations.

Bifurcation and Sensitivity Analysis of a Malaria Model with Isolated Drug Resistant Population

A malaria model with isolated drug resistant population after the first line of treatment is presented using six systems of first order nonlinear differential equations. The disease free equilibrium point and the basic reproduction number are determined. Local stability of the disease free equilibrium is determined and the conditions for the existence of endemic equilibrium. Bifurcation analysis reveals the existence of backward bifurcation. Sensitivity analysis is used to determine the impact of the model parameter on the basic reproduction number. Early detection and using correct dosage will go a long way to prevent drug resistance.
 Keywords: Malaria, Bifurcation, Basic reproduction number, Stability, Equilibrium.

Model Persistency and Stability Analysis of HIV/AIDS Transmission Dynamics in the Presence of Oral Pre-Exposure Prophylaxis (PrEP) and Antiretroviral Therapy (ART)

Prevention by the use of both ART and PrEP are among the forefront strategies for tackling HIV/AIDS transmission. We propose a model by employing the Lipschitz continuity to check the basic properties. Existence of backward bifurcation is proved and it showed that the disease will continue to persist even if the basic reproduction number is less than unity. Thus, there is the need to lower R 0 to below a certain threshold, R_0∗ , for effective disease control. As a result, the model persistency is checked and the global stability of the disease free equilibrium for appropriate conditions is investigated. We consider the interplay of ART and PrEP rates on disease persistence with numerical simulations further confirming that simultaneously applying PrEP and ART reduces the viral loads of HIV/AIDS in the population.

Coupling the modeling of phage-bacteria interaction and cholera epidemiological model with and without optimal control

In this work, we assess the impact of the phage-bacteria infection and optimal control on the indirectly transmitted cholera disease. The phage-bacteria interactions are described by predator–prey system using the Smith functional response, which takes into account the number of bacteria binding sites. The study is done in two steps, namely the model without control and the model with control. For the first scenario, we explicitly compute the basic reproduction number R0 which serves as stability threshold and bifurcation parameter. The proposed model exhibits a bi-stability phenomenon via the existence of backward bifurcation, which implies that the classical requirement of bringing the reproduction number under unity, while necessary, is no longer sufficient for cholera elimination from the population. We intuitively introduce a new threshold number N0 needed for the global stability of the disease free equilibrium point which is achieved when R0⩽1 and N0⩽1. It is further shown that the phage absorption is a possible cause of bi-stability, since in its absence, the condition R0⩽1 is sufficient for cholera to die out. The existence of endemic equilibrium points depends on the range of both R0 and N0. Regarding the model extended to an optimal control problem, which involves the use of virulent vibriophages to reduce or eliminate the bacteria population, we use optimal control theory techniques. We establish the conditions under which the spread of cholera can be stopped, and examine the impact of control measures on the transmission dynamic of cholera. The Pontryagin’s maximum principle is used to characterize the optimal control. Numerical simulations suggest that, the release of lytic vibriophages can significantly reduce the spread of the disease. We discuss opportunities for phage therapy as treatment of some bacterial-borne diseases without side effects.

Optimal control analysis of vector-host model with saturated treatment.

Vector-host infectious diseases remain a challenging issue and cause millions of deaths each year globally. In such outbreaks, many countries especially developing or underdevelopment faces a situation where the number of infected individuals is getting larger and the medical facilities are limited. In this paper, we construct an epidemic model to explore the transmission dynamics of vector-borne diseases with nonlinear saturated incidence rate and saturated treatment function. This type of incidence rate, as well as the saturated treatment function, is also known as the Holling type II form and describes the effect of delayed treatment. Initially, we formulate a mathematical model and then present the basic analysis of the model including the positivity and boundedness of the solution. The threshold quantity {mathcal {R}}_{0} is presented and the stability analysis of the system is carried out for the model equilibria. The global stability results are shown using the Lyapunov function of Goh–Voltera type. The existence of backward bifurcation is discussed using the central manifold theory. Further, the global sensitivity analysis of the model is carried out using the Latin Hypercube sampling and the partial rank correlation coefficient techniques. Moreover, an optimal control problem is formulated and the necessary optimality conditions are investigated in order to eradicate the disease in a community. Four strategies are presented by choosing different set of controls combination for the disease minimization. Finally, the numerical simulations of each strategy are depicted to demonstrate the importance of suggesting control interventions on the disease dynamics and eradication.

Impact of Treatment on HIV-Malaria CoInfection Based on Mathematical Modeling

The distribution of HIV and malaria overlap globally. So there is always a chance of co-infection. In this paper the impact of medication on HIV-Malaria co-infection has been analyzed and we have developed a mathematical model using the idea of the models of Mukandavire, et al. [13] and Barley, et al. [3] where treatment classes are included. The disease-free equilibrium (DFE) of the HIV-only model is globally-asymptotically stable (GAS) when the reproduction number is less than one. But it is shown that in the malaria-only model, there is a coexistence of stable disease-free equilibrium and stable endemic equilibrium, for a certain interval of the reproduction number less than unity. This indicates the existence of backward bifurcation. Numerical simulations of the full model are performed to determine the impact of treatment strategies. It is shown that malaria-only treatment strategy reduces more new cases of the mixed infection than the HIV-only treatment strategy. Moreover, mixed treatment strategy reduces the least number of new cases compared to single treatment strategies.
 GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 45-62

Cholera‐schistosomiasis coinfection dynamics

SummaryThe present work investigates the coinfection dynamics of the cholera and schistosomiasis diseases. The steady states of the model are examined. We obtain results for the model in detail and present the stability results whenever the basic reproduction number is less than unity ( ). For each submodel, the existence of backward bifurcation is presented and for the coinfection model. Furthermore, we formulate an optimal control problem with an appropriate set of control variables. The optimal control problem and the associated results are derived and discussed. The optimal control problem and the suggested controls are utilized to obtain optimal control characterizations. Numerical results are presented by choosing various optimal control strategies for the early elimination of both infections from the population. It is suggested that appropriate uses and application to the population could significantly reduce the infection. Therefore, based on our findings, we suggest to the public health department that the only possible cost‐effective strategy for the elimination of schistosomiasis and cholera coinfection is the combination of both diseases' preventive measures and the treatment of schistosomiasis.

Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment

There is a wide range of works that have proposed mathematical models to describe the spread of infectious diseases within human populations. Based on such models, researchers can evaluate the effect of applying different strategies for the treatment of diseases. In this article, we generalize previous models by studying an SIR epidemic model with a nonlinear incidence rate, saturated Holling type II treatment rate, and logistic growth. We compute the basic reproduction number and determine conditions for the local stability of equilibria and the existence of backward bifurcation and Hopf bifurcation. We also show that, when the disease transmission rate and treatment parameter are varied, our model undergoes a Bogdanov‐Takens bifurcation of codimension 2 or 3. Simulations of the solutions and numerical continuation of equilibria are carried out to generate 2D and 3D bifurcation diagrams, as well as several related phase portraits that illustrate our results. Our work shows that incorporating these factors into epidemic models can lead to very complex dynamics.

A mathematical model of vertically transmitted vector diseases

A mathematical model of vector-borne infectious diseases is presented, which takes into account the local interactions between reservoirs and vectors, as well as the transmission from vectors to dilution hosts. In the model, vectors possess the ability to keep the virus within their own population through vertical transmission. The existence and the stability of disease free and endemic equilibria, together with the existence of backward bifurcation, are discussed.

Modeling treatment of cancer using virotherapy with generalized logistic growth of tumor cells

Virotherapy is an effective strategy in cancer treatment. It eliminates tumor cells without harming the healthy cells. In this article, a deterministic mathematical model to understand the dynamics of tumor cells in response to virotherapy is formulated and analyzed by incorporating cytotoxic T lymphocytes (CTLs). The basic reproduction number and the immune response reproduction number are computed and different equilibria of the proposed model are found. The local stability of different equilibria is discussed in detail. Further, the proposed model is extended to stochastic model. Numerical simulation is performed for both deterministic and stochastic models. It is observed that when both the reproduction numbers are greater than one, which corresponds to existence of unique nontrivial equilibrium point, dynamics of deterministic and stochastic models are almost same. The deterministic model shows a very complex dynamics when one or both the reproduction numbers are below one. The system exhibits both backward bifurcation and Hopf-bifurcation for suitable sets of parameters and in this situation it is not easy to predict the dynamics of cancer cells and virus particles. The existence of backward bifurcation demonstrates the fact that partial success of virotherapy can be achieved even if the immune response reproduction number is less than one.

Analysis of a tuberculosis model with undetected and lost-sight cases

A deterministic model of tuberculosis (TB) in sub-Saharan Africa including undetected and lost-sight cases is presented and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, when a stable disease-free equilibrium co-exists with one or more stable endemic equilibrium points when the associated basic reproduction number (R0) is less than unity. Analyzing the model obviously reveals that exogenous reinfection plays a key role on the existence of backward bifurcation. However, an analysis of the ranges of exogenous reinfection suggested that backward bifurcation occurs only for very high and unrealistic ranges of the exogenous reinfection rate. Random perturbation of reinfection rates was performed to gain insight into the role of this latter on the stability of the disease free equilibrium.

The effect of the incidence function on the existence of backward bifurcation

In modelling, the dynamics of infectious disease, the choice of the specific mathematical formulation of disease transmission (i.e. the incidence function) is one of the initial assumptions to be made. While inconsequential in many situations, we show that the incidence function can have an effect on the existence of backward bifurcation (the phenomenon where a disease can persist even when the basic reproductive number is less than 1). More specifically, we compare mass action (MA) and standard incidence (SI) (the most common incidence functions) versions of two hallmark models in the backward bifurcation literature and an original combination model. Our findings indicate that the SI formation of disease transmission is more conducive to backward bifurcation than MA, a trend seen in all the models analysed.

The Effect of the Incidence Function on the Existence of Backward Bifurcation

In modelling, the dynamics of infectious disease, the choice of the specific mathematical formulation of disease transmission (i.e. the incidence function) is one of the initial assumptions to be made. While inconsequential in many situations, we show that the incidence function can have an effect on the existence of backward bifurcation (the phenomenon where a disease can persist even when the basic reproductive number is less than 1). More specifically, we compare mass action (MA) and standard incidence (SI) (the most common incidence functions) versions of two hallmark models in the backward bifurcation literature and an original combination model. Our findings indicate that the SI formation of disease transmission is more conducive to backward bifurcation than MA, a trend seen in all the models analysed.

The impact of bed-net use on malaria prevalence

Malaria infection continues to be a major problem in many parts of the world including the Americas, Asia, and Africa. Insecticide-treated bed-nets have shown to reduce malaria cases by 50%; however, improper handling and human behavior can diminish their effectiveness. We formulate and analyze a mathematical model that considers the transmission dynamics of malaria infection in mosquito and human populations and investigate the impact of bed-nets on its control. The effective reproduction number is derived and existence of backward bifurcation is presented. The backward bifurcation implies that the reduction of R below unity alone is not enough to eradicate malaria, except when the initial cases of infection in both populations are small. Our analysis demonstrate that bed-net usage has a positive impact in reducing the reproduction number R. The results show that if 75% of the population were to use bed-nets, malaria could be eliminated. We conclude that more data on the impact of human and mosquito behavior on malaria spread is needed to develop more realistic models and better predictions.