Narrow Results

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 $S_h{I}_h{R}_h-S_v{I}_v$ 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 difficult 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 the forward Euler method.

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 ${\mathcal{ℛ}}_1^j$ and ${\mathcal{ℛ}}_2^j,j=1,2$, and show that the strain $j$ can not invade the system if ${\mathcal{ℛ}}_1^j+{\mathcal{ℛ}}_2^j<1$. Further, the disease-free equilibrium of the system is globally asymptotically stable if ${\mathrm{\text{max}}}_j\{{\mathcal{ℛ}}_1^j+{\mathcal{ℛ}}_2^j\}<1$. When ${\mathcal{ℛ}}_1^j>1$, strain $j$ dominance equilibrium ${\mathcal{ℰ}}_j$ exists, and is locally asymptotically stable when ${\mathcal{ℛ}}_1^j>1$, ${\mathcal{ℛ}}_1^i<\varsigma {\mathcal{ℛ}}_1^j,i,j=1,2,i\ne j$, $\varsigma \in (0,1)$. 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 ${\mathcal{ℛ}}_1^i/{\mathcal{ℛ}}_1^j<b_i/b_j<1,i\ne j$, where $b_j$ 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.

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, homo-clinic 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.

One of the promising strategies to reduce dengue transmission is to release Wolbachia-infected mosquitoes, which can reduce the reproductive success of wild female mosquitoes. We develop a dengue transmission model coupled with the Wolbachia infection to consider the impact of the increased mortality of Wolbachia-infected immature mosquitoes on Wolbachia invasion and dengue transmission. To begin with, we analyze the infection model of Wolbachia without dengue transmission dynamics. Next, we establish an optimal control model by introducing a control variable to simulate the continuous (daily) releases of Wolbachia-infected male mosquitoes and find the optimal control by using Pontryagin’s Maximum Principle. We determine the optimal release strategy by minimizing the total cost of releasing infected male mosquitoes. Then, the full dengue transmission model is analyzed. The basic reproduction number of dengue transmission is calculated using the next-generation matrix method. The stability of the dengue-free equilibrium is proved by using the method of monotone dynamical systems. Furthermore, we carry out sensitivity analysis to study the barrier effect of Wolbachia and the impact of the increased mortality of immature mosquitoes on dengue transmission. Our results suggest that the increased mortality of immature Wolbachia-infected mosquitoes is not conducive to Wolbachia establishment and dengue control, which also induces more Wolbachia-infected mosquitoes to be released. In particular, we estimate the threshold mortality rate of infected larvae by using bifurcation analysis, which provides a quantitative basis and theoretical support for rational selection of Wolbachia strains and scientific and effective practice of dengue control.

A mathematical model for a nonsterilizing vaccine is studied. The model considers a vaccination policy represented by the vaccine application rate, waning and an index of reduction of viral load. The model also incorporates the possibility of escape mutants that avoid vaccine action. The main result is that we can show the existence of an endemic equilibrium point when R₀ is less than one. The reason behind it is the existence of escape mutants that promote an increased rate of infection large enough to trigger an increase in the density of infected people even in the subthreshold case.

In this paper, a nonlinear deterministic model is proposed with a saturated treatment function. The expression of the basic reproduction number for the proposed model was obtained. The global dynamics of the proposed model was studied using the basic reproduction number and theory of dynamical systems. It is observed that proposed model exhibits backward bifurcation as multiple endemic equilibrium points exist when $R_0<1$. The existence of backward bifurcation implies that making $R_0<1$ is not enough for disease eradication. This, in turn, makes it difficult to control the spread of cholera in the community. We also obtain a unique endemic equilibria when $R_0>1$. The global stability of unique endemic equilibria is performed using the geometric approach. An extensive numerical study is performed to support our analytical results. Finally, we investigate two major cholera outbreaks, Zimbabwe (2008–09) and Haiti (2010), with the help of the present study.

A multiscale system for environmentally-driven infectious disease is proposed, in which control measures at three different scales are implemented when the number of infected hosts exceeds a certain threshold. Our coupled model successfully describes the feedback mechanisms of between-host dynamics on within-host dynamics by employing one-scale variable guided enhancement of interventions on other scales. The modeling approach provides a novel idea of how to link the large-scale dynamics to small-scale dynamics. The dynamic behaviors of the multiscale system on two time-scales, i.e. fast system and slow system, are investigated. The slow system is further simplified to a two-dimensional Filippov system. For the Filippov system, we study the dynamics of its two subsystems (i.e. free-system and control-system), the sliding mode dynamics, the boundary equilibrium bifurcations, as well as the global behaviors. We prove that both subsystems may undergo backward bifurcations and the sliding domain exists. Meanwhile, it is possible that the pseudo-equilibrium exists and is globally stable, or the pseudo-equilibrium, the disease-free equilibrium and the real equilibrium are tri-stable, or the pseudo-equilibrium and the real equilibrium are bi-stable, or the pseudo-equilibrium and disease-free equilibrium are bi-stable, which depends on the threshold value and other parameter values. The global stability of the pseudo-equilibrium reveals that we may maintain the number of infected hosts at a previously given value. Moreover, the bi-stability and tri-stability indicate that whether the number of infected individuals tends to zero or a previously given value or other positive values depends on the parameter values and the initial states of the system. These results highlight the challenges in the control of environmentally-driven infectious disease.

In this paper, we incorporate immune systems into an HIV model, which considers both logistic target-cell proliferation and viral cell-to-cell transmission. We study the dynamics of this model including the existence and stability of equilibria. Based on the existence of equilibria, we focus on the backward bifurcation and forward bifurcation. Considering the stability of equilibria, Hopf bifurcation is discussed by identifying the basic reproduction number $R_0$ as bifurcation parameter. The direction and stability of Hopf bifurcation are investigated by computing the first Lyapunov exponent. Specially, the effects of immune response on the basic reproduction number $R_0$ and viral dynamics are addressed by deriving the sensitivity analysis. As a result, we find that the removal rate of infected cells by cytotoxic T lymphocytes (CTLs), ${\gamma }_1$, is the predominant factor of $R_0$. However, we conclude from numerical results that it is unfeasible to decrease $R_0$ by increasing the value of ${\gamma }_1$ constantly. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions. These dynamics are investigated by the proposed model to point out the importance and complexity of immune responses in fighting HIV replication.

In this paper, we formulate a new-age structured heroin transmission model with respect to the age of vaccination which structures the vaccine wanes rate $\alpha (a)$ and infection ratio of vaccination individuals $\sigma (a)$. The well-posedness and the basic reproduction number ${{ℛ}}_0$ of our model are first presented. If ${{ℛ}}_0<1$, the drug-free steady state ${{ℰ}}^0$ is locally stable and there will be multiple positive steady states due to the imperfect vaccine. If ${{ℛ}}_0>1$, there is a unique drug spread steady state, and our model is uniformly persistent. To reveal the dynamics of our model in detail, we carry out a further analysis in some special cases, including the backward and forward bifurcation results of our model when $\alpha (a)=\alpha$ and $\sigma (a)=\sigma$, and the unique drug-spread steady state’s stability when ${{ℛ}}_0>1$. Finally, a brief conclusion and discussion are presented.

Recent studies have demonstrated that immune impairment is an essential factor in viral infection for disease development and treatment. In this paper, we formulate an age-structured viral infection model with a nonmonotonic immune response and perform dynamical analysis to explore the effects of both immune impairment and virus control. The basic infection reproduction number is derived for a general viral production rate, which determines the global stability of the infection-free equilibrium. For the immune intensity, we get two important thresholds, the post-treatment control threshold and the elite control threshold. The interval between the two thresholds is a bistable interval, where there are two immune-present infected equilibria. When the immune intensity is greater than the elite control threshold, only one immune-present infected equilibrium exists and it is stable. By assuming the death rate and virus production rate of infected cells to be constants, with the immune intensity as a bifurcation parameter, the system exhibits saddle-node bifurcation, transcritical bifurcation, and backward/forward bifurcation.

In this study, we have developed a novel SIR epidemic model by incorporating fractional-order differential equations and utilizing saturated-type functions to describe both disease incidence and treatment. The intricate dynamical characteristics of the proposed model, encompassing the determination of the conditions for the existence of all possible feasible equilibria with their local and global stability criteria, are investigated thoroughly. The model undergoes backward bifurcation with respect to the parameter representing the side effects due to treatment. This phenomenon emphasizes the critical role of treatment control parameters in shaping epidemic outcomes. In addition, to understand the optimal role of the treatment in mitigating the disease prevalence and minimizing the associated cost, we investigated a fractional-order optimal control problem. To further visualize the analytical results, we have conducted simulation works considering feasible parameter values for the model. Finally, we have employed local and global sensitivity analysis techniques to identify the factors that have the greatest potential to reduce the impact of the disease.

The aim of this paper is to show that, for some parameters, a two-dimensional cardiovascular system model can exhibit intrinsic equilibrium multiplicity generated by a backward bifurcation, regardless of the baroreflex effect. The model considers the dynamics of arterial and venous compartments and a feedback effect in the stroke volume induced by venous pressure changes. The results of the mathematical analysis indicate that multiple non-trivial equilibrium points exist when the stroke volume function is convex around the origin. Interestingly, this equilibrium point structure would imply that under certain stroke volume functions, the baroreflex system would have to stabilize and regulate an unstable operating condition produced by certain values of the stroke volume. The paper ends with the discussion of some implications for the reliability and robustness of the baroreflex-feedback mechanism.

Treatment of hepatitis C virus (HCV) is lengthy, expensive and fraught with side-effects, succeeding in only 50% of treated patients. In clinical settings, short-term treatment response (so-called sustained virological response (SVR)) is used to predict prolonged viral suppression. Although ordinary differential equation (ODE) models for within-host HCV infection have illuminated the mechanisms underlying treatment with interferon (IFN) and ribavirin (RBV), they have difficulty producing SVR without the introduction of an external extinction threshold. Here we show that bistability in an existing ODE model of HCV, which occurs when infected hepatocytes proliferate sufficiently faster than uninfected hepatocytes, can produce SVR without an external extinction threshold under biologically relevant conditions. The model can produce all clinically observed patient profiles for realistic parameter values; it can also be used to estimate the efficacy and/or duration of treatment that will ensure permanent cure for a particular patient.

We present a model describing the dynamics of an infectious disease for which the force of infection is diminished through a reaction of the susceptible to the number of infected individuals. We show that, even though the structure of the model is a simple one, different kinds of backward bifurcation can appear for values of the basic reproductive number bigger than one. Under some conditions on the parameters, multiple endemic equilibria may appear for values of the basic reproductive number less or greater than one.

A deterministic model for the effects on disease prevalence of the most advanced pre-erythrocytic vaccine against malaria is proposed and studied. The model includes two vaccinated classes that correspond to initially vaccinated and booster dose vaccinated individuals. These two classes are structured by time-since-initial-vaccination (vaccine-age). This structure is a novelty for vector–host models; it allows us to explore the effects of parameters that describe timed and delayed delivery of a booster dose, and immunity waning on disease prevalence. Incorporating two vaccinated classes can predict more accurately threshold vaccination coverages for disease eradication under multi-dose vaccination programs. We derive a vaccine-age-structured control reproduction number ${ℛ}$ and establish conditions for the existence and stability of equilibria to the system. The model is bistable when ${ℛ}<1$. In particular, it exhibits a backward (sub-critical) bifurcation, indicating that ${ℛ}=1$ is no longer the threshold value for disease eradication. Thus, to achieve eradication we must identify and implement control measures that will reduce ${ℛ}$ to a value smaller than unity. Therefore, it is crucial to be cautious when using ${ℛ}$ to guide public health policy, although it remains a key quantity for decision making. Our results show that if the booster vaccine dose is administered with delay, individuals may not acquire its full protective effect, and that incorporating waning efficacy into the system improves the accuracy of the model outcomes. This study suggests that it is critical to follow vaccination schedules closely, and anticipate the consequences of delays in those schedules.

In this paper, we develop an epidemic model with a nonlinear function that describes the saturated digital contact tracing. The model is theoretically and numerically analyzed based on its dynamics. The model equilibria points and the basic reproduction number are obtained. However, the proposed model reveals both backward and forward bifurcation, and backward bifurcation occurs when the digital contact tracing saturation parameter is larger than a specific threshold. Real data are used to fit the model and estimate the parameter values. Sensitivity analysis reveals that interventions such as reducing the infection rate, enhancing quarantine efforts, and accelerating vaccination can effectively reduce the basic reproduction number, $R_0$. Simulations show that improving tracing accuracy and encouraging greater participation in providing personal track information can effectively reduce the peak of undetected exposed individuals. Moreover, we analyze the expression for the specific threshold and determine that the increasing hospital resources and strengthening patient quarantine can reduce the potential occurrence of backward bifurcation. The analysis and simulations presented in this paper provide valuable suggestions for the prevention and control of the epidemic.

This study explores an epidemic model elucidating the dynamics of COVID-19 transmission amidst vaccination and saturated treatment interventions. The investigation encompasses both deterministic and stochastic frameworks, considering constant and fluctuating environments, utilizing COVID-19 data from India for empirical validation. Through rigorous mathematical and numerical analyses, we ascertain pivotal insights. Our deterministic model unveils a critical phenomenon: the occurrence of backward bifurcation at ${{ℛ}}_0=1$, underscoring that merely reducing the basic reproduction number below unity does not ensure disease eradication. Sensitivity analyses underscore the acceleration of epidemic spread with higher transmission rates, yet mitigation measures such as vaccination and comprehensive treatment can effectively reduce the basic reproduction number below unity. Within the stochastic framework, we establish the existence of a unique global positive solution. We delineate conditions for disease extinction or persistence and identify criteria for the emergence of stationary distribution, reflecting the sustained presence of infection within the community. Our findings elucidate that while smaller noise intensities sustain disease prevalence, heightened noise levels lead to complete eradication of the infection.

In this research work, we have developed and analyzed a deterministic epidemiological model with a system of nonlinear differential equations for controlling the spread of Ebola virus disease (EVD) in a population with vital dynamics (where birth and death rates are not equal). The model examines the disease transmission dynamics with isolation from exposed and infected human class and effect of vaccination in susceptible human population through stability analysis and bifurcation analysis. The model exhibits two steady state equilibria, namely, disease-free and endemic equilibrium. Next generation matrix method is used to find the expression for $R_0$ (the basic reproduction number). Local and global stability of diseases-free equilibrium are shown using nonsingular M-matrix technique and Lyapunov’s theorem, respectively. The existence and local stability of endemic equilibrium are explored under certain conditions. All numerical data entries are supported by various authentic sources. The simulation study is done using MATLAB code 45 which uses Runge–Kutta method of fourth order and we plot the time series and bifurcation diagrams which support our analytical findings. Stability analysis of the model shows that the disease-free equilibrium is locally as well as globally asymptotically stable if $R_0<1$ and endemic equilibrium is locally asymptotically stable in absence of vaccination if $R_0>1$. Using central manifold theorem, the presence of transcritical bifurcation for a threshold value of the transmission rate parameter $\beta$ when $R_0$ passes through unity and backward bifurcation (i.e. transcritical bifurcation in opposite direction) for some higher value of $R_0$ are established. Our simulation study shows that isolation of exposed and infected individuals can be used as a more effective tool to control the spreading of EVD than only vaccination.

An SIS epidemic model with the standard incidence rate and saturated treatment function 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 population 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 bifurcation. It is shown that the disease-free equilibrium is globally asymptotically stable under some conditions. Numerical simulations are consistent with our obtained results in theorems, which show that improving the efficiency and capacity of treatment is important for control of disease.

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 insecticide-treated 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 reproduction 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.