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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 R0 < 1. Further, we obtain two types of suffcient 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 R0 > 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 R0, but also with the growth rate of normal vascular endothelial cells promoted by the vascular endothelial growth factor. [ABSTRACT FROM AUTHOR]

In the face of the COVID-19 pandemic, understanding the dynamics of disease transmission is crucial for effective public health interventions. This study explores the concept of symmetry within compartmental models, employing compartmental analysis and numerical simulations to investigate the intricate interactions between compartments and their implications for disease spread. Our findings reveal the conditions under which the disease-free equilibrium is globally asymptotically stable while the endemic equilibrium exhibits local stability. Additionally, we investigate the phenomenon of backward bifurcation, shedding light on the critical role of quarantine measures in controlling outbreaks. By integrating the concept of symmetry into our model, we enhance our understanding of transmission dynamics and provide a robust framework for evaluating intervention strategies. The insights gained from this research are vital for policymakers and health authorities aiming to mitigate the impact of infectious diseases in the future. [ABSTRACT FROM AUTHOR]

Symptomatic and asymptomatic individuals play a significant role in the transmission dynamics of novel Coronaviruses. By considering the dynamical behaviour of symptomatic and asymptomatic individuals, this study examines the temporal dynamics and optimal control of Coronavirus disease propagation using an epidemiological model. Biologically and mathematically, the well-posed epidemic problem is examined, as well as the threshold quantity with parameter sensitivity. Model parameters are quantified and their relative impact on the disease is evaluated. Additionally, the steady states are investigated to determine the model's stability and bifurcation. Using the dynamics and parameters sensitivity, we then introduce optimal control strategies for the elimination of the disease. Using real disease data, numerical simulations and model validation are performed to support theoretical findings and show the effects of control strategies. [ABSTRACT FROM AUTHOR]

The HIV/AIDS model was dynamically analyzed in this study. The model has seven compartments: the uneducated, the educated, the HIV-positive who take antiretroviral therapy (ART), the HIV-positive who do not take ART, people receiving ART treatment, people with AIDS who do not receive any treatment (full-blown AIDS), and the recovered. This model takes into account the analysis of the multiple interactions between the uneducated and the educated subpopulations, the HIV-positive who take and who do not take ART. The free-disease and endemic equilibrium points, as well as the basic reproduction number ( R 0) as a limit condition for infection-free and endemic occurrence, were produced by a mathematical analysis. The center-manifold hypothesis was used to prove that a backward bifurcation exists. The free-disease and endemic equilibrium points coexist when R 0 < 1. This means that HIV/AIDS is still spreading. A basic reproduction number below one is insufficient to constitute a free-disease condition. In order to determine essential parameters that significantly contribute to HIV/AIDS transmission, we computed sensitivity index values using a sensitivity analysis. The HIV/AIDS model and bifurcation parameter both identified the rate of HIV transmission from uneducated individuals to HIV-positive individuals who do not receive ART as the most crucial parameter. A numerical simulation supports the dynamical analysis. [ABSTRACT FROM AUTHOR]

Based on the level of awareness of the population, an HIV/AIDS model is developed, which focused on the first 95% plan developed by UNAIDS. The threshold R0$$ {R}_0 $$ of model and the expressions of the disease‐free equilibrium and the endemic equilibrium are calculated, proving the existence of backward bifurcation. Backward bifurcation is caused by the imperfect protection rate of susceptible population due to education. Using China's actual data for parameter fitting, it is found that new HIV infections are on an upward trend. In response to this phenomenon, publicity and education, condoms, screening, and treatment of infected populations are considered as control measures. It is concluded that publicity and education is the primary strategy. This measure can not only effectively reduce the number of infected populations but also effectively increase the awareness rate of HIV‐infected populations. It is recommended to use condoms and have fewer sexual partners during sexual contact. Numerical simulation verifies that early stage publicity and education are much more important than post‐infection screening and treatment measures. [ABSTRACT FROM AUTHOR]

Let a population be composed of members of a criminal organization and judges of the judicial system, in which the judges can be co-opted by this organization. In this article, a model written as a set of four nonlinear differential equations is proposed to investigate this population dynamics. The impact of the rate constants related to judges' co-optation and ex-convicts' recidivism on the population composition is explicitly examined. This analysis reveals that the proposed model can experience backward and transcritical bifurcations. Also, if all ex-convicts relapse, organized crime cannot be eradicated even in the absence of corrupt judges. The results analytically derived here are illustrated by numerical simulations and discussed from a crime-control perspective. [ABSTRACT FROM AUTHOR]

The HIV/AIDS model was dynamically analyzed in this study. The model has seven compartments: the uneducated, the educated, the HIV-positive who take antiretroviral therapy (ART), the HIV-positive who do not take ART, people receiving ART treatment, people with AIDS who do not receive any treatment (full-blown AIDS), and the recovered. This model takes into account the analysis of the multiple interactions between the uneducated and the educated subpopulations, the HIV-positive who take and who do not take ART. The free-disease and endemic equilibrium points, as well as the basic reproduction number [Formula: see text] as a limit condition for infection-free and endemic occurrence, were produced by a mathematical analysis. The center-manifold hypothesis was used to prove that a backward bifurcation exists. The free-disease and endemic equilibrium points coexist when [Formula: see text] This means that HIV/AIDS is still spreading. A basic reproduction number below one is insufficient to constitute a free-disease condition. In order to determine essential parameters that significantly contribute to HIV/AIDS transmission, we computed sensitivity index values using a sensitivity analysis. The HIV/AIDS model and bifurcation parameter both identified the rate of HIV transmission from uneducated individuals to HIV-positive individuals who do not receive ART as the most crucial parameter. A numerical simulation supports the dynamical analysis.

The propagation of rumors indisputably inflicts profound negative impacts on society and individuals. This article introduces a new unaware ignorants-aware ignorants-spreaders-recovereds $ (2ISR) $ rumor spreading model that combines individual vigilance self-awareness with nonlinear spreading rate. Initially, the positivity of the system solutions and the existence of its positive invariant set are rigorously proved, and the rumor propagation threshold is solved using the next-generation matrix method. Next, a comprehensive analysis is conducted on the existence of equilibrium points of the system and the occurrence of backward bifurcation. Afterward, the stability of the system is validated at both the rumor-free equilibrium and the rumor equilibrium, employing the Jacobian matrix approach as well as the Lyapunov stability theory. To enhance the efficacy of rumor propagation management, a targeted optimal control strategy is formulated, drawing upon the Pontryagin's Maximum principle as a guiding framework. Finally, through sensitivity analyses, numerical simulations, and tests of real cases, we verify the reliability of the theoretical results and further consolidate the solid foundation of the above theoretical arguments.

The coinfection of HBV and TB diseases affects millions of individuals throughout the nations in the world. The main objective of this study is to formulate and analyze the HBV and TB coinfection dynamical system, investigate the impact of time‐dependent optimal control measures and cost‐effectiveness analysis, and tackle the coinfection transmission dynamics in the population. Each of the submodels and the coinfection model's qualitative analyses were carried out independently, whereas the TB‐only submodel and the coinfection model reveal the phenomenon of backward bifurcation. The coinfection model's optimal control problem with five time‐dependent optimal control measures is formulated and analyzed. Using the parameter values described in the paper, we carried out numerical simulations and verified the models' qualitative analysis results. To minimize invectives and the cost of the implementing effort toward the protection and the treatment, optimal control analysis is performed for the HBV and TB coinfection model using Pontryagin's minimum principle. Numerical simulations with different combinations of efforts are then carried out to explore the effect of protection in the presence of treatment for both diseases. Numerical simulations emphasize the fact that to reduce coinfection from the population, programs to accelerate the protection of both diseases are also required along with the treatment. The results reveal that the implementation of the combination of protective and treatment‐optimal control strategies suppresses the occurrence of HBV and TB coinfection, and protective control measures are more effective than treatment control measures for individuals who are coinfected with HBV and TB diseases. Implementing all the proposed protective and treatment control measures significantly minimized the transmission dynamics of HBV and TB confection in the community. From the cost‐effectiveness analysis of the proposed time‐dependent control strategies, treatment of HBV is the most cost‐effective control strategy required to tackle the spread of the HBV and TB coinfection in the community. [ABSTRACT FROM AUTHOR]

In this paper, we study a dynamic model of hepatitis C virus (HCV) infection with density‐dependent proliferation of uninfected and infected hepatocytes and two time delays, which is derived from a three‐dimensional model by the quasi‐steady‐state approximation. The model can exhibit forward bifurcation or backward bifurcation, and an explicit control threshold parameter Rc$R_c$ is obtained for the case of backward bifurcation. It is shown that if the proliferation rate of infected hepatocytes is greater than the proliferation rate of uninfected hepatocytes by a certain amount, it becomes more difficult for the virus to be removed. The model has rich dynamical properties: (i) In some parameter regions, bistability can occur; (ii) both time delays τ1$\tau _{1}$ (virus‐to‐cell delay) and τ2$\tau _{2}$ (cell‐to‐cell delay) can lead to Hopf bifurcations; (iii) same length of time delays τ1$\tau _{1}$ and τ2$\tau _{2}$ can lead to at most one stability switch, but different time delays can lead to multiple stability switches. Several sufficient conditions for the global stability of the disease‐free equilibrium and the endemic equilibrium are obtained for both forward and backward bifurcation scenarios. In particular, several sharp results on global stability are obtained. Theoretical and numerical results portray the complexity of viral evolutionary dynamics in chronic HCV‐infected patients. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and analyze a compartmental model of visceral leishmaniasis (VL). We validate our model by calibrating it to the yearly VL incidence data for India and Bangladesh. The proposed model's basic reproduction number ( R 0 ) has been derived and estimated. We have proved the existence of backward bifurcation in our system. The phenomenon of backward bifurcation has public health implications because the classical requirement of R 0 < 1 , while necessary, is no longer sufficient for effective disease control (or elimination). In such a (backward bifurcation) situation, the initial sizes of the model's subpopulations (state variables) would determine the effectiveness of disease control or elimination. As a result, it is the first attempt to represent and study VL disease dynamics using active case detection (ACD) as a control strategy, although a few experimental studies have been conducted to evaluate ACD in disease transmission. We use sensitivity analysis to investigate the effects of the model's controllable parameters for the basic reproduction number. We found that σ 1 , σ 2 , σ 3 (monitoring rate to infected individuals) have negative impacts on R 0 . The numerical result suggests that the ACD strategy can be useful for the VL elimination program. Due to a small increment in the monitoring rate, the dynamic behavior of infected individuals dramatically decreased. Successful employment of ACD strategy may reduce more than 60–80% symptomatic and post kala-azar dermal leishmaniasis (PKDL) individuals. We found that healthcare organizations should prioritize ACD in symptomatic and PKDL individuals over asymptomatic individuals. We also observed that the use of only culling effect to the reservoirs is not beneficial to society in the control of VL, but spraying of insecticides and the use of treated bednets can be effective control strategies to curtail the outbreak of the disease. [ABSTRACT FROM AUTHOR]

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

This article proposes and analyzes an SIRS infectious disease model with ratio-dependent incidence rate and saturated treatment rate functions. The incorporation of a ratio-dependent incidence rate provides a more intricate depiction of disease dynamics, encompassing not just the inhibitory effect of infected individuals but also the presence of susceptible individual for potential infection. Through the analysis of the model, it is discovered that when treatment capacity is limited, the infected population may survive even if the basic reproduction number is less than one. Consequently, the model displays endemicity through the coexistence of multiple steady states, and we observe a backward bifurcation. Further, a geometric approach is applied to derive the global stability of the endemic steady state. Bi-stability, Hopf bifurcation, and saddle-node bifurcation are some examples of nonlinear dynamics that are investigated. Additionally, this study also highlights the existence of Hopf bifurcation for the basic reproduction number less than one, showcasing the rare dynamics associated with ratio-dependent incidence rate function. We give numerical examples to demonstrate and validate the outcomes of our theoretical analysis. [ABSTRACT FROM AUTHOR]

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 ℛ0 = 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. [ABSTRACT FROM AUTHOR]

Pyrethroid-treated bed-nets (PTNs) protect individuals against malaria by blocking and repelling mosquitoes. We develop and analyze a PTNs malaria model that explicitly includes mosquito host choice (also known as feeding/biting preference) and Pyrethroid repellent effect. Our model reveals that mosquito biting/feeding preference on infectious hosts π and repellent effect r drive for the existence of both the endemic equilibrium points and the occurrence or elimination of backward bifurcation. The threshold parameters for the mosquito biting preference on infectious hosts π ∗ and repellent effect r ∗ for the occurrence and elimination of backward bifurcation are computed. Moreover, it is shown that, increasing the mosquito host choice rate or decreasing the repellent effect rate, annihilates backward bifurcation, thus facilitating the control of malaria. Furthermore, we prove that the threshold of mosquito biting preference is a monotone increasing function of the repellent effect r. We show that the model exhibits both trans-critical forward bifurcation and backward bifurcation when either the mosquito host choice π crosses a threshold value π 1 or the repellent effect r passes through a threshold repellent rate r 1 . Sufficient conditions for the global asymptotic stability of the equilibrium point are derived. On the other hand, it is established that, decreasing the mosquito biting preference or increasing the rate of the repellent effect (i.e personal protection) or the combining both actions, decreases the malaria control reproduction number R 0 . Finally, the interplay between the bed-nets treated repellent effect and mosquito host choice and its potential on the dynamics of malaria is investigated and illustrated numerically. [ABSTRACT FROM AUTHOR]

We propose a population dynamical model for SARS-CoV-2 that takes into account mask compliance and effectiveness, in the context of saturated treatment. This model also considers reinfection and relapse among individuals with comorbidities. Our findings indicate that global mask usage, in conjunction with other public health measures, effectively reduces the basic reproduction number ( R 0 ). We establish the local and conditional global stability of the disease-free equilibrium point. Notably, the model exhibits intriguing behavior due to saturated treatment and reinfection. Under specific parameter conditions, it demonstrates multiple endemic equilibria when R 0 < 1 resulting and backward and forward bifurcation. We conduct sensitivity analysis to pinpoint the key factors influencing disease spread. The existence of multiple equilibria contributes to intricate and diverse dynamics, showcasing a variety of bifurcations and oscillations through Hopf bifurcation. Under specific conditions, global asymptotic stability for the unique endemic equilibrium, when it exists, is established. Among further nonlinear dynamics exhibited by the proposed model, we establish backward Hopf bifurcation, Hopf–Hopf bifurcation and saddle-node bifurcation. Bistability of the equilibrium points is also observed through forward hysteresis. Additionally we provide the impact of parameters most effective in reducing in COVID-19 spread. Numerical simulations of the theoretical findings are offered to validate the results. [ABSTRACT FROM AUTHOR]

In this paper we study a two infection SIR-SIR compartmental model, considering biological features described in dengue fever epidemiology. Due to a progressive loss of protective antibodies there is waning immunity in the first infection stage and disease enhancement or protection effects by the second infection stage. Bifurcation analysis reveals two codim-2 bifurcations as organizing centers. The unfolding of a cusp bifurcation describes the transition of the disease-free equilibrium into an endemic equilibrium by varying a parameter. These equilibria allow an analytical solution with explicit expressions which allow for a full geometrical interpretation of the occurring bifurcations related to stationary dynamics. A Bogdanov-Takens point is the starting point in the parameter space where oscillatory endemic dynamics occurs including a homoclinic connection. These findings bring additional insights on biological mechanisms able to generate rich and complicated dynamical behavior in simple epidemic models that are, so far, largely unexplored. [ABSTRACT FROM AUTHOR]

Symptomatic and asymptomatic individuals play a significant role in the transmission dynamics of novel Coronaviruses. By considering the dynamical behaviour of symptomatic and asymptomatic individuals, this study examines the temporal dynamics and optimal control of Coronavirus disease propagation using an epidemiological model. Biologically and mathematically, the well-posed epidemic problem is examined, as well as the threshold quantity with parameter sensitivity. Model parameters are quantified and their relative impact on the disease is evaluated. Additionally, the steady states are investigated to determine the model's stability and bifurcation. Using the dynamics and parameters sensitivity, we then introduce optimal control strategies for the elimination of the disease. Using real disease data, numerical simulations and model validation are performed to support theoretical findings and show the effects of control strategies.

In this article, we have constructed a stochastic SIR model with healthcare resources and logistic growth, aiming to explore the effect of random environment and healthcare resources on disease transmission dynamics. We have showed that under mild extra conditions, there exists a critical parameter, i.e., the basic reproduction number $ R_0^s $, which completely determines the dynamics of disease: when $ R_0^s < 1 $, the disease is eradicated; while when $ R_0^s > 1 $, the disease is persistent. To validate our theoretical findings, we conducted some numerical simulations using actual parameter values of COVID-19. Both our theoretical and simulation results indicated that (1) the white noise can significantly affect the dynamics of a disease, and importantly, it can shift the stability of the disease-free equilibrium; (2) infectious disease resurgence may be caused by random switching of the environment; and (3) it is vital to maintain adequate healthcare resources to control the spread of disease.

In this work, we developed, validated, and analysed the behaviour of a compartmental model of COVID-19 transmission in Saudi Arabia. The population was structured into four classes: susceptible (S), exposed (E), infectious (I), and removed (R) individuals. This SEIR model assumes a bilinear incidence rate and a nonlinear recovery rate that depends on the quality of health services. The model also considers a treatment function and incorporates the effect of fear due to the disease. We derived the expression of the basic reproduction number and the equilibrium points of the model and demonstrated that when the reproduction number is less than one, the disease-free equilibrium is stable, and the model predicts a backward bifurcation. We further found that when the reproduction number is larger than one, the model predicts stable periodic behaviour. Finally, we used numerical simulations with parameter values fitted to Saudi Arabia to analyse the effects of the model parameters on the model-predicted dynamic behaviours. [ABSTRACT FROM AUTHOR]

In this work, we formulate a random Wolbachia invasion model incorporating the effects of imperfect maternal transmission and incomplete cytoplasmic incompatibility (CI). Under constant environments, we obtain the following results: Firstly, the complete invasion equilibrium of Wolbachia does not exist, and thus the population replacement is not achievable in the case of imperfect maternal transmission; Secondly, imperfect maternal transmission or incomplete CI may obliterate bistability and backward bifurcation, which leads to the failure of Wolbachia invasion, no matter how many infected mosquitoes would be released; Thirdly, the threshold number of the infected mosquitoes to be released would increase with the decrease of the maternal transmission rate or the intensity of CI effect. In random environments, we investigate in detail the Wolbachia invasion dynamics of the random mosquito population model and establish the initial release threshold of infected mosquitoes for successful invasion of Wolbachia into the wild mosquito population. In particular, the existence and stability of invariant probability measures for the establishment and extinction of Wolbachia are determined. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this investigation, we explore a predator-prey system characterized by generalist predators exhibiting cooperative behavior during hunting, subject to nonlinear harvesting rate. Employing a Beverton-Holt type functional response to capture the impact of additional food sources on predator growth, our numerical findings affirm the destabilizing influence of hunting cooperation, excessive predation, and supplementary food sources. Conversely, the system demonstrates stabilization in response to increased prey species growth. We unveil the occurrence of both forward and backward bifurcations in the system due to the predator growth attributed to additional food sources, contingent on the degree of hunting cooperation. To broaden the scope, we extend our proposed model to its stochastic counterpart, introducing environmental white noises. Our numerical results anticipate that higher intensities of white noises lead to fluctuations of greater amplitude, while smaller intensities exhibit a more modest impact. Additionally, we elucidate the dynamics of prey and predator populations through histogram plots. The theoretical and numerical insights derived from this study provide a deeper understanding of the intricate dynamics within predator-prey systems of ecological communities, emphasizing the significance of additional food sources for cooperative predators subject to harvesting in both constant and fluctuating environments. [ABSTRACT FROM AUTHOR]

HPV is the most common sexually transmitted infection and can lead to cancer. Vaccination has been considered as a very effective measure against HPV and along with regular screenings is recommended for the prevention of cervical cancer. In this study we propose and analyze a model for the transmission dynamics of Human Papilloma Virus (HPV). We discuss the optimal vaccination strategy in the case when multiple vaccines are available. We have considered an ODE based compartmental model, incorporating sex structure, we also consider HPV leading to cervical cancer by including pre-cancerous and cancerous compartments in the model. Adding the pre-cancerous and cancerous compartments will help us better understand the role of vaccination in prevention of cancers due to HPV. Using standard techniques from dynamical systems theory, we determine the disease free (DFE) and endemic steady states (EE). We determine a threshold quantity, the basic reproductive number R0 in terms of the model parameters, such that, the DFE is stable when R01. The goal of control measures is to try an reduce R0 to be below the threshold value of 1, the regime in which the disease will ultimately be driven to extinction. We also prove the existence of a backward bifurcation in the model, epidemiologically this means that the long time behavior of the model depends on the initial infected population and that the disease may be harder to control, and the condition R0

The novel coronavirus disease (COVID-19) pandemic has profoundly impacted the global economy and human health. The paper mainly proposed an improved susceptible-exposed-infected-recovered (SEIR) epidemic model with media coverage and limited medical resources to investigate the spread of COVID-19. We proved the positivity and boundedness of the solution. The existence and local asymptotically stability of equilibria were studied and a sufficient criterion was established for backward bifurcation. Further, we applied the proposed model to study the trend of COVID-19 in Shanghai, China, from March to April 2022. The results showed sensitivity analysis, bifurcation, and the effects of critical parameters in the COVID-19 model.

A new network-based SIR epidemic model with saturated incidence rate and nonlinear recovery rate is proposed. We adopt an edge-compartmental approach to rewrite the system as a degree-edge-mixed model. The explicit formula of the basic reproduction number $ \mathit{\boldsymbol{R_{0}}} $ is obtained by renewal equation and Laplace transformation. We find that $ \mathit{\boldsymbol{R_{0}}} < 1 $ is not enough to ensure global asymptotic stability of the disease-free equilibrium, and when $ \mathit{\boldsymbol{R_{0}}} > 1 $, the system can exist multiple endemic equilibria. When the number of hospital beds is small enough, the system will undergo backward bifurcation at $ \mathit{\boldsymbol{R_{0}}} = 1 $. Moreover, it is proved that the stability of feasible endemic equilibrium is determined by signs of tangent slopes of the epidemic curve. Finally, the theoretical results are verified by numerical simulations. This study suggests that maintaining sufficient hospital beds is crucial for the control of infectious diseases.

In this article, we present a susceptible, exposed, infected, hospitalized and recovered compartmental model for COVID-19 with vaccination strategies and mask efficiency. Initially, we established the positivity and boundedness of the solutions to ensure realistic predictions. To assess the epidemiological relevance of the system, an examination is conducted to ascertain the local stability of the endemic equilibrium and the global stability across two equilibrium points are carried out. The global stability of the system is demonstrated using Lyapunov’s direct method. The disease-free equilibrium is globally asymptotically stable when the basic reproduction number (BRN) is less than one, whereas the endemic equilibrium is globally asymptotically stable when BRN is greater than one. A sensitivity analysis is performed to identify the influential factors in the BRN. The impact of various time-dependent strategies for managing and regulating the dynamic transmission of COVID-19 is investigated. In this study, Pontryagin’s maximum principle for optimal control analysis is used to identify the most effective strategy for controlling the disease, including single, coupled, and threefold interventions. Single-control interventions reveal physical distancing as the most effective strategy, coupled measures reduce exposed populations, and implementing all controls reduces susceptibility and infections.

Roguing and elimination of vectors are the most commonly seen biological control strategies regarding the spread of plant viruses. It is practically significant to establish the mathematical models of plant virus transmission and regard the effect of removing infected plants as well as eliminating vector strategies on plant virus eradication. We proposed the mathematical models of plant virus transmission with nonlinear continuous and pulse removal of infected plants and vectors. In terms of the nonlinear continuous control strategy, the threshold values of the existence and stability of multiple equilibria have been provided. Moreover, the conditions for the occurrence of backward bifurcation were also provided. Regarding the nonlinear impulsive control strategy, the stability of the disease-free periodic solution and the threshold of the persistence of the disease were given. With the application of the fixed point theory, the conditions for the existence of forward and backward bifurcations of the model were presented. Our results demonstrated that there was a backward bifurcation phenomenon in continuous systems, and there was also a backward bifurcation phenomenon in impulsive control systems. Moreover, we found that removing healthy plants increased the threshold $ R_{1}. $ Finally, numerical simulation was employed to verify our conclusions.

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 ShIhRh − SvIv 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, ℛ0, 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. [ABSTRACT FROM AUTHOR]

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 ℛ1j and ℛ2j,j = 1, 2, and show that the strain j can not invade the system if ℛ1j +ℛ 2j < 1. Further, the disease-free equilibrium of the system is globally asymptotically stable if maxj{ℛ1j +ℛ 2j} < 1. When ℛ1j > 1, strain j dominance equilibrium ℰj exists, and is locally asymptotically stable when ℛ1j > 1, ℛ1i < ςℛ 1j,i,j = 1, 2,i≠j, ς ∈ (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 ℛ1i/ℛ 1j < b i/bj < 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. [ABSTRACT FROM AUTHOR]

The immune response is a crucial factor in controlling HIV infection. However, oxidative stress poses a significant challenge to the HIV-specific immune response, compromising the body's ability to control viral replication. In this paper, we develop an HIV infection model to investigate the impact of immune impairment on virus dynamics. We derive the basic reproduction number (R0) and threshold (Rc). Utilizing the antioxidant parameter as a bifurcation parameter, we establish that the system exhibits saddle-node bifurcation backward and forward bifurcations. Specifically, when R0 > Rc, the virus will rebound if the antioxidant parameter falls below the post-treatment control threshold. Conversely, when the antioxidant parameter exceeds the elite control threshold, the virus remains under elite control. The region between the two thresholds represents a bistable interval. These results can explain why some HIV-infected patients experience rapid viral rebound after treatment cessation while others achieve post-treatment control for a longer time. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The model is governed by a system of five ordinary differential equations namely: five compartments namely: the susceptible (S), Exposed (E), Infected (I), Hospitalized (H), and Recovery (R). The model was analyzed to find the global stability, reproduction number, bifurcation, endemic equilibrium, and disease-free equilibrium. The analysis's findings show that the bifurcation displays reverse bifurcation. There is now global asymptotic stability in the system. Based on numerical simulations of the model, it was found that the first week of the outbreak was when Ebola was most common. But as safety precautions were put in place, it gradually decreased. It is significant to remember that in order to avoid a relapse into the original high levels, treatment is required. The contaminated compartment remained high throughout the first five weeks of the outbreak, but with the right therapy, it began to diminish. Due to drug experimenting, the hospitalized compartment saw a large number of patients during the first week. Nonetheless, it kept declining with the right medication administration. The recovery section illustrates that prompt identification and appropriate care are essential for each person to recover from Ebola. [ABSTRACT FROM AUTHOR]

Ontogenic resistance has been described for many plant-pathogen systems. Conversely, coffee leaf rust, a major fungal disease that drastically reduces coffee production, exhibits a form of ontogenic susceptibility, with a higher infection risk for mature leaves. To take into account stage-dependent crop response to phytopathogenic fungi, we developed an SEIR-U epidemiological model, where U stands for spores, which differentiates between young and mature leaves. Based on this model, we also explored the impact of ontogenic resistance on the sporulation rate. We computed the basic reproduction number R 0 , which classically determines the stability of the disease-free equilibrium. We identified forward and backward bifurcation cases. The backward bifurcation is generated by the high sporulation of young leaves compared to mature ones. In this case, when the basic reproduction number is less than one, the disease can persist. These results provide useful insights on the disease dynamics and its control. In particular, ontogenic resistance may require higher control efforts to eradicate the disease. [ABSTRACT FROM AUTHOR]

Tuberculosis is a pressing global health concern, particularly pervasive in many developing nations. This study investigates the influence of treatment failure on tuberculosis control strategies, incorporating vaccination interventions using a deterministic compartmental epidemiological model. Mathematical analysis unveils disease-free and endemic equilibrium points, with the control reproduction number determined using next-generation methods. Identifying endemic equilibrium points and determining the control reproduction number provide essential metrics for assessing the effectiveness of control strategies and guiding policy decisions. The model exhibits a backward bifurcation phenomenon, leading to multiple endemic equilibria despite a reproduction number below one due to reinfection. Sensitivity analysis using Latin Hypercube Sampling/Partial Rank Correlation Coefficient elucidates parameter impacts on the control reproduction number. Vaccination efficacy is crucial for quality and validity, with superior quality and longer validity yielding more significant effects. While reinfection may not directly affect the reproduction number, its influence is pivotal in determining tuberculosis persistence or extinction. This study underscores the intricate interplay of factors in tuberculosis control strategies, providing insights vital for effective interventions and policy formulation.

This paper introduces a mathematical model that simulates the transmission of the dengue virus in a population over time. The model takes into account aspects such as delays in transmission, the impact of inhibitory effects, the loss of immunity, and the presence of partial immunity. The model has been verified to ensure the positivity and boundedness. The basic reproduction number R0 of the model is derived using the advanced next-generation matrix approach. An analysis is conducted on the stability criteria of the model, and equilibrium points are investigated. Under appropriate circumstances, it was shown that there is local stability in both the virus-free equilibrium and the endemic equilibrium points when there is a delay. Analyzing the global asymptotic stability of equilibrium points is done by using the appropriate Lyapunov function. In addition, the model exhibits a backward bifurcation, in which the virus-free equilibrium coexists with a stable endemic equilibrium. By using a sensitivity analysis technique, it has been shown that some factors have a substantial influence on the behavior of the model. The research adeptly elucidates the ramifications of its results by effortlessly validating theoretical concepts with numerical examples and simulations. Furthermore, our research revealed that augmenting the rate of inhibition on infected vectors and people leads to a reduction in the equilibrium point, suggesting the presence of an endemic state.

Chagas disease is a zoonosis caused by the protozoan parasite Trypanosoma cruzi and transmitted by a broad range of blood-sucking triatomine species. Recently, it is recognized that the parasite can also be transmitted by host ingestion. In this paper, we propose a Chagas disease model incorporating two transmission routes of biting-defecation and host predation between vectors and hosts with Holling II functional response. The basic reproduction number R_v of triatomine population and basic reproduction numbers R_0 of disease population are derived analytically, and it is shown that they are insufficient to serve as threshold quantities to determine dynamics of the model. Our results have revealed the phenomenon of bistability, with backward and forward bifurcations. Specifically, if R_v>1, the dynamic is rather simple, namely, the disease-free equilibrium is globally asymptotically stable as R_01. However, if R_v1. In conclusion, predation transmission in general reduces the risk of Chagas disease, whilst it makes the complexity of Chagas disease transmission, requiring an integrated strategy for the prevention and control of Chagas disease.

Let a population be composed of members of a criminal organization and judges of the judicial system, in which the judges can be co-opted by this organization. In this article, a model written as a set of four nonlinear differential equations is proposed to investigate this population dynamics. The impact of the rate constants related to judges’ co-optation and ex-convicts’ recidivism on the population composition is explicitly examined. This analysis reveals that the proposed model can experience backward and transcritical bifurcations. Also, if all ex-convicts relapse, organized crime cannot be eradicated even in the absence of corrupt judges. The results analytically derived here are illustrated by numerical simulations and discussed from a crime-control perspective.

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 influences these conditions. Numerical examples highlight the effect of temporarily recovered individuals and initial conditions on infected population persistence.

Oral pre-exposure prophylaxis (PrEP) is a highly effective method of HIV prevention. However, despite its effectiveness, the use of PrEP is not without challenges. These challenges include issues such as adherence, societal stigma, as well as accessibility and cost. PrEP effectiveness measures the drug’s performance in the real world while efficacy measures the degree to which PrEP inhibits HIV infection and its transmission. In this paper, we address the problem of reduced effectiveness of PrEP, primarily due to non-adherence, and the impact on the dynamics of HIV infection. The study aims to determine the critical threshold of PrEP effectiveness required to avoid the occurrence of a backward bifurcation. The conditions for a stable infection-free equilibrium are stated. Through a rigorous analysis of the proposed mathematical model using Castillo-Chavez and Song’s bifurcation theorem, it was found that backward bifurcation occurs when PrEP effectiveness falls below 100%. In contrast, if the effectiveness is 100%, which implies full adherence to PrEP, the model undergoes forward bifurcation. The existence of a backward bifurcation bears significant consequences, most notably the co-existence of the infection-free equilibrium and the endemic equilibrium. Under these circumstances, the eradication of HIV within a particular community becomes very difficult. The results of the numerical analysis demonstrate the important role of proper adherence in augmenting the effectiveness of PrEP and, consequently, curbing HIV transmission within communities. Nevertheless, intensive efforts are required to boost adherence to PrEP. Therefore, other HIV control measures need to be promoted to further reduce transmission.

The main goal of this paper is to build a nonlinear compartmental pneumonia infection model and examine the impact of treatment, immunization, and preventative measures (such as maintaining excellent cleanliness, avoiding close contact with patients, and restricting smoking). The developed model exhibits two different types of equilibrium points: endemic and disease-free pneumonia equilibrium points. In order to confirm that the pneumonia model displays the phenomena of backward bifurcations whenever its effective reproduction number is less than unity, we have employed the center manifold criteria. This finding leads us to the conclusion that lowering the basic reproduction number of pneumonia infections does not ensure that the infection will be completely eradicated in the community. But as this paper's conclusion shows, reducing the rate at which pneumonia spreads has a significant impact on community-wide pneumonia infection control. Sensitivity analysis research reveals that altering the qualitative dynamics of pneumonia infection is mostly dependent on the transmission rate of the illness. Ultimately, based on the outcomes of numerical simulations, we deduce that the most effective ways to reduce the spread of pneumonia in the community are through immunization, treatment, and prevention. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive HIV transmission model that incorporates the influence of media campaigns. Our model aims to explore the relationship between media campaigns and disease spread by deriving the basic reproduction number, a critical indicator of HIV transmission. To gain deeper insights into the dynamics of the model, we conduct a thorough bifurcation analysis using center-manifold theory, establishing the necessary conditions for the occurrence of a backward bifurcation. The obtained results are then numerically validated and extensively discussed, providing valuable insight into both the mathematical and epidemiological implications of the model. In addition, we perform numerical simulations to offer a more detailed understanding of the intricate dynamics involved in HIV transmission. Moreover, we conduct sensitivity analysis using Latin hypercube sampling (LHS), which underscores the significant role of the effective contact rate in driving the proliferation of the HIV epidemic. These findings not only contribute to a better understanding of the complex dynamics underlying HIV transmission but also emphasize the crucial impact of media campaigns in effectively controlling and managing the epidemic. By highlighting the interplay between media campaigns and disease spread, our study provides valuable insights that can inform public health strategies and interventions for combating HIV transmission. [ABSTRACT FROM AUTHOR]