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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 quarantine strategy plays a crucial role in the prevention and control of infectious disease. In this paper, a two-layer network model coupling the transmission of infectious diseases and the dynamics of human behavior based on game theory is proposed. The basic reproduction number of the infectious disease in our proposed model is obtained by the next-generation matrix method and the stability of the disease-free equilibrium is analyzed. Theoretical results show that the spread of infectious diseases can be controlled when the voluntary quarantined individuals reach a certain proportion. The sensitivities of the parameters are analyzed by simulations, and the results show that increasing propaganda can directly accelerate quarantine, and reducing the relative cost of quarantine has a significant effect on preventing the infectious diseases. Increasing the detection rate will lead to overestimating the proportion of undiagnosed infected individuals, and can also promote individuals to quarantine. [ABSTRACT FROM AUTHOR]

The decline in amphibian populations in recent decades may be linked to the occurrence of infectious diseases such as chytridiomycosis, which is caused by the chytrid fungus Batrachochytrium dendrobatidis (Bd). It is known that symbiotic bacteria protect the host due to their inhibitory nature. However, how the population dynamics of amphibians is affected by additional effects provided by symbiotic bacteria has not been analyzed in depth. In this paper, a model is proposed to describe the interaction among susceptible amphibians, susceptible amphibians with symbiotic bacteria and amphibians with chytrid fungus. When the modeling takes into account the additional reproductive benefits that the symbiont Pseudomonas sp. grants to the host, multiple endemic equilibrium points can exist if R0 < 1 (R0 is the basic reproduction number for Bd). In this scenario, the existence of a subcritical bifurcation at R0 = 1, which can occur in two different disease-free equilibrium points, gives rise to complex dynamics and stability scenarios. Particularly, the analysis of the model shows that a sudden increase of fungus-infected amphibians can occur even when R0 < 1 due to bistability phenomena. In this scenario, the existence of a subcritical bifurcation, which translates for the fungus into colonization even for values of R0 less than one, represents an advantage for the chytrid fungus Batrachochytrium dendrobatidis since the pathogen should benefit from remaining as close as possible to an endemic equilibrium. To control the fungal infection, R0 must be reduced to a value below one until the endemic equilibrium points disappear. Finally, we show that the amphibian population can reach a critical population level close to an extinction scenario when R0 increases. [ABSTRACT FROM AUTHOR]

Infectious disease outbreaks have caused significant losses to populations worldwide in terms of morbidity, social and economic costs. This has prompted governments to seek suitable intervention policies to suppress the outbreaks. This paper aims to explore the intrinsic impact of intervention strategies and the psychological behavior of people on the spread of infectious diseases. Employing the finite state Markov chain, we formulate a new stochastic SIS epidemic model under regime switching by using the Beddington–DeAngelis transmission function, in the presence of some governmental control measures. The study of the model is centered on three different scenarios in terms of varying two control parameters. We analyze the deterministic system for the global dynamics in terms of basic reproductive number. Some new threshold parameters are defined to discuss the extinction and the persistence of the disease in the stochastic setting. Numerical assessments were undertaken to validate theoretical findings for the deterministic and stochastic systems. Our findings strongly suggest that heightened awareness among susceptible individuals contributes significantly to the eradication of infection and fosters a stable epidemiological situation. As levels of public awareness increase, particularly within the susceptible population, a discernible trend toward the eradication of the infection emerges, ultimately ensuring sustained stability within the epidemiological landscape. [ABSTRACT FROM AUTHOR]

The summary measurement of an epidemic such as the final size is an important quantity that allows us to approximate the impact of the disease on the affected region. In recent years, final size measurements for vector-transmitted diseases have acquired importance because of the increased concern for mosquito-borne diseases like dengue, Zika, Chikungunya, etc. However, analytical expressions for this estimate in the stage-structured models applicable to vector-borne diseases are less, mostly focused on the classical Kermack–McKendrick model. In this paper, we first calculate the final size expressions for an SIR model with carrier state and a SEIR–SI host–vector model. Then we extend this for the SEIR–SI host–vector model with treatment class in the host, as well as for the model with vertical transmission in the vector population. Finally, we verify our final size expression with some real scenarios. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to conduct an in-depth analysis of the intricacies inherent in a gender-structured model depicting the dynamics of HIV–TB coinfection. The model is articulated through a set of nonlinear fractional-order differential equations. Our focus is on elucidating the stability characteristics around equilibrium points, employing advanced fractional stability techniques. We leverage sensitivity analysis as a powerful tool to discern the specific model parameters exerting significant influence on the transmission and control of the coinfection, as gauged by the basic reproduction number. Through this, we aim to identify key factors that drive the spread or containment of the dual affliction. Furthermore, we delve into the realm of optimization by formulating a fractional optimal control problem. The objective is to design and implement two control strategies that effectively mitigate the prevalence of coinfection, demonstrating a state of perfect protection. Notably, our findings reveal that the implementation of control measures at a rate exceeding 80% suffices to eliminate the infection entirely. We anticipate that the insights garnered from this research will prove instrumental for policymakers in crafting strategic interventions to curtail the tragic impact of HIV–TB coinfection. Our scientifically grounded study serves as a foundation upon which comprehensive plans can be devised to stem the devastating consequences of this dual health challenge. [ABSTRACT FROM AUTHOR]

A mathematical model considering two strains of hepatitis B virus (HBV) chronic carriers, to assess the impact of dose-structured imperfect vaccine, in a population, is designed and analyzed. The model is shown to have a locally and globally asymptotically stable disease-free equilibrium (DFE) whenever its associated reproduction number is numerically less than unity. Numerical analysis of the model shows that with the expected 50% minimum efficacy of the first vaccine dose, vaccinating 55% of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of hepatitis B infection. Such effective control can also be achieved if 50% of the first vaccine dose recipients take the second dose. Threshold analysis reveals that an imperfect HBV vaccine should have positive or negative population-level effect. Latin hypercube sampling–PRCC analysis illustrates that disease transmission rate, birth rate, natural death rate and proportion of children born with maternal immunity are most influential parameters in the disease dynamics. In this paper, the sensitivity analysis based on mathematical and in addition statistical techniques have been performed to determine the significance of the model parameters. It is observed that a number of the parameters play an important role to determine the magnitude of the basic reproduction number. Sensitivity analysis is achieved to determine model parameters' importance in disease dynamics. It is observed that the reproduction number is the most responsive quantity to the potent transmission rate of HBV and in addition also vital to control the spread of the disease. [ABSTRACT FROM AUTHOR]

An age-structured vector-borne disease model with horizontal transmission is proposed and studied in this paper, where the incubation ages of both host and vector and the immunity age of host are also introduced to consider the effects of multi-class-age structure. The reproductive number ℛ 0 is derived as a threshold value to determine the existence and stability of the disease-free and endemic steady states. Furthermore, by constructing suitable Lyapunov functionals, the global threshold dynamics of this model is established by ℛ 0 , that is, the disease-free equilibrium is globally asymptotically stable when ℛ 0 < 1 , while if ℛ 0 > 1 the endemic equilibrium is globally asymptotically stable. In addition, considering the limited budget of the centers for disease control and prevention (CDC) in the process of disease control, we present an optimal control problem with a fixed total expenditure, and discuss the existence of the most control strategy for this disease. Finally, some numerical simulations are performed to support the theoretical results. [ABSTRACT FROM AUTHOR]

Age structure and delay play a significant role in determining the dynamics of the diseases. In this paper, an SEIRS epidemic model with age structure and time delay is investigated, where the loss of the acquired immunity and delay are incorporated. Through some rigorous analyses, an explicit formula for the basic reproduction number of the model is calculated, and some results about stability and instability of equilibria for the model are established. The findings show that the age structure and delay can produce Hopf bifurcation for an SEIRS model. The numerical examples are executed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Mathematical models for the transmission dynamics of infectious diseases have aided our understanding of the important factors that drive epidemic patterns. In this paper, we formulate and analyze a stochastic epidemic model, a continuous-time Markov chain, in order to understand rotavirus dynamics with a contaminated environment. The assumptions of the deterministic model are utilized in the formulation of the corresponding stochastic model. We perform both local and global stability analyses of the equilibria of the deterministic model with respect to the basic reproduction number. The extinction threshold for the stochastic model and conditions for either disease extinction or persistence are derived by employing the branching process to the infectious classes only. It is shown that the probability of rotavirus extinction obtained from the branching process is in excellent agreement with the numerically approximated probability. Numerical results indicate that the probability of rotavirus extinction is the highest if the contaminated environment introduces the virus into a totally susceptible population at the beginning of the epidemic process. Thus, a major rotavirus outbreak is likely if the virus emanates from infectious children at the onset of the epidemic. Results of sensitivity analysis showed that shedding of the virus into the environment by infectious children is the most sensitive parameter of the model. Further, it is shown that decreasing the shedding rate leads to an increase in the probability of disease extinction and vice versa. This, therefore, implies that disposal of stool of infectious children should be well managed if efforts to curb further spread of the disease or even eliminating it are to bear desirable fruits. [ABSTRACT FROM AUTHOR]

This paper deals with the dynamics of tuberculosis transmission model with different genders and age structures. The basic reproduction numbers ℛ 0 = ℛ F + ℛ M are defined, where ℛ F and ℛ M are the basic reproduction number of tuberculosis transmission in female and male populations, respectively. The global stability of the disease-free equilibrium was discussed when ℛ 0 < 1 ; The existence, uniqueness and global stability of the endemic equilibrium, and the persistence of the system both were proved when ℛ 0 > 1. The findings indicate that the tuberculosis dynamical model with different genders and age structures will experience the classical threshold dynamic behaviors, that is, if ℛ 0 < 1 , the disease will disappear; while if ℛ 0 > 1 , the disease will spread within the population, and eventually lead to a endemic disease. Finally, the theoretical analysis of the model is verified by numerical simulation and we proposed that male infected individuals should be focused on in the prevention and control of tuberculosis in the future. [ABSTRACT FROM AUTHOR]

Influenced by seasonal changes, the infection rate of many infectious diseases fluctuates in cycles. In this paper, we propose and investigate an SIRS model on a scale-free network. To model seasonality, we assume that the infection rate is periodic. The existence and positivity of solutions of the proposed model are proved and the basic reproduction number ℜ 0 is defined. The global stability of steady states is determined by rigorous mathematical analysis. When ℜ 0 < 1 , the disease-free equilibrium E 0 is globally asymptotically stable. When ℜ 0 > 1 , the system has a unique positive periodic solution E * , and E * is globally asymptotically stable. Numerical simulations are performed to support our theoretic results, and the effects of various parameters on the amplitude and mean of infected individuals are studied. The sensitivity of parameters of the basic reproduction number ℜ 0 is solved by the Sobol global sensitivity analysis method, and the results show that the effects of the parameters β 0 and α on ℜ 0 are remarkable. [ABSTRACT FROM AUTHOR]

This paper studies an SIS-type Lyme transmission model incorporating insecticides spraying and vaccination as interventions. We obtain the positivity and boundedness of solutions, calculate the basic reproduction number, and discuss the global stability of disease-free and endemic equilibria when the basic reproduction number ℛ 0 < 1 and ℛ 0 > 1 , respectively. We apply Pontryagin's maximum principle to explore an optimal control strategy to minimize the number of infected ticks and hosts and the cost of using insecticides and vaccination. We design numerical simulations to illustrate the effectiveness of theoretical analysis. [ABSTRACT FROM AUTHOR]

Anthrax is a disease caused by Bacillus anthracis, commonly affects animals as well as humans health. In this paper, a nonlinear deterministic anthrax model involving human and animal is proposed and analyzed. The reproduction number ℛ 0 and equilibrium points are explored to study the dynamic behavior of the disease. The existence and stability of equilibrium points are discussed. For ℛ 0 < 1 , the disease-free equilibrium E 0 is globally stable. However, it is unstable when ℛ 0 > 1 and a locally stable endemic equilibrium point E * exists. The model is then extended to optimal control model considering human vaccination, animal vaccination and proper removal of carcass. The vaccination class of human and animal population appears separately in a model. The existence and characterization of optimal control are discussed. The numerical simulations are carried out for the choice of parametric values and initial conditions. These illustrate scavengers in the suspected area which eat infected dead body of animals contributing to the effort of reducing the expansion of disease. In addition, numerical comparison analysis with four distinct control strategies is carried out. Our findings show that each control technique has its own influence on reducing the total number of infections in the human and animal populations. The cumulative impact of all control measures is found to be extremely effective in lowering the prevalence of the disease. [ABSTRACT FROM AUTHOR]

Media impact has significant effect on reducing the disease prevalence, meanwhile sanitation and awareness can control the epidemic by reducing the growth rate of bacteria and direct contacts with infected individuals. In this paper, we investigate the impacts of media and sanitation coverage on the dynamics of epidemic outbreak. We observe that the growth rate of social media advertisements carries out a destabilizing role, while the system regains stability if the baseline number of social media advertisements exceeds a certain threshold. The dissemination of awareness among susceptibles first destabilizes and then stabilizes the system. The disease can be wiped out if the baseline level of awareness or the rate of spreading global information about the disease and its preventive measures is too high. We obtain an explicit expression for the basic reproduction number ℛ 0 and show that ℛ 0 < 1 leads to the total eradication of infection from the region. To capture a more realistic scenario, we construct the forced delay model by seasonally varying the growth rate of social media advertisements and incorporating the time lag involved in reporting of total infective cases to the policy makers. Seasonal pattern in the growth rate of social media advertisements adds complexity to the system by inducing chaotic oscillations. For gradual increase in the delay in reported cases of infected individuals, the nonautonomous system switches finitely many times between periodic and chaotic states. [ABSTRACT FROM AUTHOR]

In this paper, we proposed and analyzed a nonlinear deterministic model for the impact of temperature variability on the epidemics of the malaria. The model analysis showed that all solutions of the systems are positive and bounded with initial conditions in a certain set. Thus, the model is proved to be both epidemiologically meaningful and mathematically well-posed. Using the next-generation matrix approach, the basic reproduction number with respect to the disease-free equilibrium (DFE) point is obtained. The local stability of the equilibria points is shown using the Routh–Hurwitz criterion. The global stability of the equilibria points is performed using the Lyapunov function. Also, we proved that if the basic reproduction number is less than one, the DFE is locally and globally asymptotically stable. But, if the basic reproduction number is greater than one, the unique endemic equilibrium exists, locally and globally asymptotically stable. The sensitivity analysis of the parameters is also described. Moreover, we used the method implemented by the center manifold theorem to identify that the model exhibits forward and backward bifurcations. From our analytical results, we confirmed that the variation of temperature plays a significant role on the transmission of malaria. Lastly, numerical simulations are demonstrated to enhance the analytical results of the model. [ABSTRACT FROM AUTHOR]

In this paper, we study a within-host malaria infection model recently proposed by Schneider et al. in 2018. The stability and Hopf bifurcation analysis at the interior equilibrium are carried out, finding that the basic reproduction number plays a key role in the dynamics of the model, and incrementing the time delay will induce Hopf bifurcation at this equilibrium. The global extension of the local Hopf branch is further tracked numerically by the MatLab package DDE-BIFTOOL. Neimark-Sacker bifurcation of Poincaré map and period-doubling bifurcation of the bifurcated periodic solution are also detected, resulting in the existence of quasi-periodic and multiple periodic solutions, respectively. These results reveal that Hopf bifurcation will indeed bring about the rich dynamics of the model. [ABSTRACT FROM AUTHOR]

An epidemic model is proposed to comprehend the disease dynamics between humans and animals and back to humans with a culling intervention strategy. The proposed model is separated into two cases with two different culling rates: (1) at a per-capita constant rate and (2) constant population being culled. The global asymptotic stability of equilibria is determined in terms of the basic reproduction numbers. Further, we find that the culling rate (2) considered in the model could change the dynamics by having multiple positive equilibria. Sensitivity analysis recommends developing a strategy that promotes animals' natural and disease-related death rates. By ranking the efficacies of various intervention strategies, we obtain that vaccination in the human population, isolation and public awareness are the largely effective control interventions. Our general theory raises concerns about both human and animal populations becoming reservoirs of the disease and affecting each other dynamically. [ABSTRACT FROM AUTHOR]

We propose and analyze a TB transmission model with nonlinear incidence rate, immunization and medical treatment. First, the existences and stabilities of the equilibrium are studied. The results indicate the basic reproduction number ℜ 0 is the threshold of disease extinction and persistence. The disease-free equilibrium is globally asymptotically stable when ℜ 0 < 1 , and the disease will gradually disappear. The unique positive equilibrium is local stability and the disease is uniformly persistent when ℜ 0 > 1. Second, optimal control is added to the original model because of limited resources. Finally, the stability of the equilibrium and the theoretical results of optimal control are verified by numerical simulations, and the sensitivity of the parameters is analyzed by the PRCC method. [ABSTRACT FROM AUTHOR]

We propose and study the transmission dynamics of susceptible-exposed-infected-recovered (X Y Z W) epidemic model with saturated treatment function. We consider saturated treatment function in the epidemic system to understand the effect of delayed treatment on the disease transmission. The indiscriminately perturbation which is considered as a type of white noise is proportional to the distance of state variables from the values of endemic equilibria. Choosing the suitable Lyapunov function and using the It ô 's formula, the existence and the uniqueness of the positive solution of the system are examined. Stochastic boundedness, permanence and extinction of the epidemic model are investigated with proper conditions. Numerical simulations are performed to illustrate our results. The sensitivity analysis of the basic reproduction number is performed. The effect of control parameter is determined on the model dynamics. It is our main finding that the different intensities of white noises can fluctuate the susceptible, exposed, infected, recovered individuals around its equilibrium points. [ABSTRACT FROM AUTHOR]

A model for assessing the impact of key population at risk in spreading HIV/ AIDS is designed. The model is shown to have a globally asymptotically stable (GAS) disease-free equilibrium whenever the associated reproduction number is less than unity. It has a unique GAS endemic equilibrium whenever reproduction number exceeds unity, if there is no back and forth movement in susceptible classes. It is shown that if infected individuals with high risk abandon their risky behavioral practices, then the disease burden will reduce. Furthermore, if infected individuals with low risk do not adopt risky behaviors then the transmission of the disease will decrease. Partial Rank Correlation Coefficient (PRCC) indices are assessed using all the infection-related classes and basic reproduction number of the model. Uncertainty and sensitivity analysis results show that controlling the effective contact rate, disease-induced death rate in AIDS stage with low and high risk classes will help controlling the transmission of the disease. [ABSTRACT FROM AUTHOR]

An age-structured deterministic model with multiple infections that accounts for decaying maternal antibody, waning infection-acquired and vaccine-induced immunity is formulated to study the transmission dynamics of pertussis and the effect of childhood DTaP and adolescent Tdap vaccines. The expression of the reproduction number ℛ is derived for the ODE model in the case of proportionate mixing. Estimated age-dependent transmission probability and empirical contact data are used in the simulation of the ODE model from which the basic reproduction number ℛ 0 is estimated to be around 15. The combination of DTaP and Tdap vaccines fails to bring ℛ 0 under one and thus pertussis remains endemic despite sustained high coverage of vaccination. While both DTaP and Tdap vaccines have remarkable effect on reducing the incidences of the age groups being directly vaccinated, the adolescent booster dose Tdap is also found to provide some indirect protection for infants though very limited (< 5 % incidence reduction). [ABSTRACT FROM AUTHOR]

In this study, we propose a deterministic model describing the dynamics of anthrax transmission in animals. It is assumed in the model that anthrax is contracted when susceptible animal comes into contact with infected animals, infected carcasses or the spores of Bacillus anthracis. Conditions for disease-free and endemic equilibria are derived and the basic reproduction number ( R 0 ) is computed. The disease-free equilibrium is stable when R 0 < 1 and the disease will be eliminated. If R 0 > 1 , the disease persists, therefore, the endemic equilibrium is globally stable. Further, we extend our model by applying optimal control problem in which vaccination for susceptible animals and sanitation are control variables to minimize the anthrax transmission. We study three strategies to investigate the impact of the controls. Our numerical results demonstrate that both vaccination and sanitation help minimizing the transmission although vaccination alone gives more significant impact than sanitation alone. However, the combination of both controls gives the best result in wiping out the anthrax transmission overall. Further, cost-effectiveness analysis shows that the most effective strategy to control anthrax disease is a combination of vaccination and sanitation of the infected areas, therefore these two interventions should be encouraged. [ABSTRACT FROM AUTHOR]

In epidemiological modeling, the basic reproduction number is usually defined as being the spectral radius of the next-generation matrix evaluated at the trivial equilibrium. The global stability of the trivial equilibrium point was determined by the left eigenvector associated with that next-generation matrix. More recently, the fraction of susceptible individuals was also obtained from the next generation matrix. The gross reproduction number and the fraction of susceptible individuals were calculated by revisiting the drug-sensitive and resistant tuberculosis model. Hence, the next-generation matrices shed light on the evolution of the dynamics: the beginning of the epidemic via the basic reproduction number and approaching the epidemic's steady-state via the susceptible individuals' asymptotic fraction. [ABSTRACT FROM AUTHOR]