Saturated Incidence Rate Research Articles

A novel nonlinear SAZIQHR epidemic transmission model: mathematical modeling, simulation, and optimal control

Abstract This study presents a seven-compartmental nonlinear epidemic model to explore the dynamics and control of the epidemic. The model extends the traditional susceptible-infected-recovered (SIR) framework by including additional compartments for aware, infodemic, hospitalized, and quarantined populations. It also incorporates three Holling type II saturated nonlinear incidence rates to better represent the disease transmission process. The ‘infodemic population’ refers to those spreading misinformation about the disease, its spread, control, and treatment. The study examines disease dynamics in the absence of a vaccine and identifies two key equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). It is found that the DFE is locally asymptotically stable when the basic reproduction number ( R 0 ) is below one and becomes unstable when R 0 exceeds one. The model’s behavior at R 0 = 1 is analyzed using center manifold theory, revealing a forward bifurcation. Additionally, the existence of a positive endemic equilibrium is confirmed for R 0 > 1 , with no backward bifurcation observed when R 0 ≤ 1 . The EE is also shown to be locally asymptotically stable when R 0 is greater than one. Furthermore, the study formulates and mathematically analyzes an optimal control problem. Finally, numerical simulations are presented to support the analytical results.

Fractional-Calculus Analysis of the Dynamics of a Vector-Borne Infection with Preventive Measures

Vector-borne infections pose serious public health challenges due to the complex interplay of biological, environmental, and social factors. Therefore, comprehensive approaches are essential to mitigate the burden of vector-borne infections and minimize their impact on public health. In this research, an epidemic model for the vector-borne disease malaria is structured with a saturated incidence rate via fractional calculus and preventive measures. The essential results and concepts are introduced to examine the proposed model. The solution of the system is examined for some necessary results, and the threshold parameter of the model, indicated by R0, is calculated. In this paper, the proposed malaria model is analyzed both quantitatively and qualitatively. The fixed-point theorems of Banach and Schaefer are utilized to examine the uniqueness and existence of the solution dynamics. Furthermore, the necessary conditions for the stability of the model have been determined. A numerical approach is offered to visualize the solution pathways of the system and identify its key factors. Through the results, the most influential factors for the control and management of the disease are highlighted.

Threshold dynamics of a degenerated diffusive incubation period host–pathogen model with saturation incidence rate

In this paper, we consider an incubation period host–pathogen system with degenerated diffusion. The global compact attractor of the solution of the model is investigated using the κ-contraction method. Furthermore, the basic reproduction number is defined, and we discuss the dynamic analysis of a degenerated diffusion model. The obtained theoretical results are nontrivial and can be considered a continuation of the work by Wang et al. in 2022.

Impact of awareness and time-delayed saturated treatment on the transmission of infectious diseases

We have proposed a mathematical model with saturated incidence and treatment rates along with awareness and delay in treatment. We analyze the model and find the equilibrium points and their stability. We also find the basic reproduction number R0 to understand the disease dynamics. For R0=1, we get transcritical backward bifurcation which means that the disease persists in the population even if R0<1. We performed the sensitivity analysis and found that treatment along with awareness plays a significant role in controlling infectious disease. We deduce that awareness about the disease affects the transmission rate of infection. As people become aware of aspects of the infectious disease, they amend their behavior so that they fend themselves from catching the disease. We have introduced a time lag in treatment and found the threshold value of the time delay. It is observed that when the value of the time delay crosses the threshold value, we get a Hopf bifurcation i.e. endemic steady state becomes unstable above the threshold value and it may become difficult to control the disease beyond the threshold value of delay in treatment.

Stability analysis of a [formula omitted] epidemic compartmental model with saturated incidence rate, vaccination and elimination strategies

In this study, we developed and analyzed a compartmental epidemic model and a system model that incorporate vaccination, elimination, and quarantine techniques, with a saturated incidence rate, by theoretical and numerical means. The model is described by a system of four nonlinear differential equations. Our analysis included determining the reproduction number Rq and examining equilibrium solutions. The outcomes of the disease are identified through the threshold Rq. When Rq<1, the disease-free equilibrium is globally asymptotically stable, as proved by the LaSalle invariance principle and disease extinction analysis. However, using the Routh–Hurwitz criterion, we have proved that the disease-free equilibrium is not stable, and when Rq>1, the unique endemic equilibrium is asymptotically stable locally. The stability of the endemic and disease-free equilibria globally has been examined using the Routh–Hurwitz and Dulac criteria. Subsequently, numerical simulations were utilized to illustrate the theoretical findings effectively.

Developing an SEIHR epidemiological model featuring saturated incidence rate and investigating its stability

Each and every person will inevitably come into contact with infectious diseases over their lifetime. This unavoidable phenomena is a characteristic of all people. Ideally, the effects of such diseases are still controllable, but under unfavourable circumstances, they may worsen and become full-fledged outbreaks. As a result, this article tries to offer an SEIHR model whose main goal is to reduce and stop the transmission of infectious diseases. Notably, infectious diseases continue to plague many countries, represented by the global threat of Covid-19, underlining the urgent need for efficient control methods. This model stands out because it carefully takes into account the saturated incidence rate of infectivity. The concept also divides the infected population into two different groups, enabling more focused and customized medical interventions. The article begins by closely examining the positivity and boundedness of the model, then conducts a thorough analysis of the fundamental reproductive number. The model's stability is then categorically proven at both equilibrium points. In summary, the article concludes with convincing numerical simulations that validate the suggested model, providing convincing evidence of its effectiveness as a whole. In conclusion, the suggested SEIHR model, which is supported by a wealth of empirical evidence confirming its efficacy and optimality, represents a significant advancement in the management and containment of infectious diseases. It offers a promising route for addressing the enduring threat of infectious diseases on a global scale, and its unique form and extensive analysis highlight its potential to alter our approach to infectious disease control.

Stability analysis and Hopf bifurcation of a fractional order HIV model with saturated incidence rate and time delay

In this paper, a fractional order HIV model with saturated incidence rate and time delay is proposed and analyzed. Firstly, the existence and uniqueness of positive solutions are proved. Secondly, the basic reproduction number and the sufficient conditions for the stability of two equilibriums are obtained. Thirdly, by using time delay as the bifurcation parameter, it is found that Hopf bifurcation may occur when the time delay passes through a sequence of critical values. After that, some numerical simulations are performed to verify the theoretical results. Finally, some discussions and conclusions are listed.

Stochastic two-strain epidemic model with saturated incidence rates driven by Lévy noise

In this paper, we introduce a stochastic two-strain epidemic model driven by Lévy noise describing the interaction between four compartments; susceptible, infected individuals by the first strain, infected ones by the second strain and the recovered individuals. The forces of infection, for both strains, are represented by saturated incidence rates. Our study begins with the investigation of unique global solution of the suggested mathematical model. Then, it moves to the determination of sufficient conditions of extinction and persistence in mean of the two-strain disease. In order to illustrate the theoretical findings, we give some numerical simulations.

A note on the dynamics of a COVID-19 epidemic model with saturated incidence rate

A note on the dynamics of a COVID-19 epidemic model with saturated incidence rate

Ergodicity and stationary distribution of a stochastic SIRI epidemic model with logistic birth and saturated incidence rate

Ergodicity and stationary distribution of a stochastic SIRI epidemic model with logistic birth and saturated incidence rate

Based on epidemiological parameter data, probe into a stochastically perturbed dominant variant of the COVID-19 pandemic model

During the COVID-19 pandemic, the SARS-CoV-2 gene mutation has been rapidly emerging and spreading all over the world. Experts worldwide regularly monitor genetic mutations and variants through genome-sequence-based surveillance, laboratory testing, outbreak investigation, and epidemiological probing. Clinical pathologists and medical laboratory scientists prefer developing or endorsing COVID-19 vaccines with a broader immune response involving various antibodies and cells to protect against mutations or new variants. Randomness plays an enormous role in pathology and epidemiology. Hence, based on epidemiological parameter data, we construct and probe a stochastically perturbed dominant variant of the coronavirus epidemic model with three nonlinear saturated incidence rates. We reveal the existence of a unique global positive solution to the constructed stochastic COVID-19 model. The Lyapunov function method is used to determine the presence of a stationary distribution of positive solutions. We derive sufficient conditions for the coronavirus to be eradicated. Eventually, numerical simulations validate the effectiveness of our theoretical outcomes.

Analysis of a fractional-order model for dengue transmission dynamics with quarantine and vaccination measures

A comprehensive mathematical model is proposed to study two strains of dengue virus with saturated incidence rates and quarantine measures. Imperfect dengue vaccination is also assumed in the model. Existence, uniqueness and stability of the proposed model are proved using the results from fixed point and degree theory. Additionally, well constructed Lyapunov function candidates are also applied to prove the global stability of infection-free equilibria. It is also demonstrated that the model is generalized Ulam-Hyers stable under some appropriate conditions. The model is fitted to the real data of dengue epidemic taken from the city of Espirito Santo in Brazil. For the approximate solution of the model, a non-standard finite difference(NSFD) approach is applied. Sensitivity analysis is also carried out to show the influence of different parameters involved in the model. The behaviour of the NSFD is also assessed under different denominator functions and it is observed that the choice of the denominator function could influence the solution trajectories. Different scenario analysis are also assessed when the reproduction number is below or above one. Furthermore, simulations are also presented to assess the epidemiological impact of dengue vaccination and quarantine measures for infected individuals.

Delayed dynamics analysis of SEI2RS malware propagation models in cyber–Physical systems

Cyber-physical systems facilitate seamless interaction between the physical and digital elements for improved efficiency, automation, and real-time monitoring across domains. This study analyzes a novel virus-spreading model called the delayed SEI2RS model, which is specifically designed for cyber-physical systems. This model incorporates a saturated incidence rate and treatment. An emphasis of this research is to explore the impact of time delay on the transient immunity interval of restored nodes. By using the time delay associated with the transitory immunity interval of recovered nodes as the bifurcation parameter, we derive a comprehensive set of appropriate conditions to assess the local stability of the malware-existence equilibrium and determine Hopf bifurcation. The center manifold theorem and normal form theory are employed to investigate the path and stability of Hopf bifurcation. Numerical calculations were used to validate the results, providing empirical evidence for the proposed model and its implications.

Fractional-order brucellosis transmission model between interspecies with a saturated incidence rate

In this study, brucellosis dynamics between interspecies are discussed with the Atangana-Baleanu fractional derivative to examine the transmission of brucellosis by its behavior. The recovered compartment, recruitment, and natural death rate for humans are considered for the fractional order model to analyze the transmission dynamics in more detail from an epidemiological point of view. Additionally, the saturated incidence rate is suggested for brucellosis as indirectly transmitted to individuals from the environment. By fixed point theory, it is verified that developed fractional transmission dynamics have a unique solution. The model under consideration employs the Adams-type predictor-corrector method for numerical solution. All comparative results are plotted by MATLAB.

Co-infection dynamics between HIV-HTLV-I disease with the effects of Cytotoxic T-lymphocytes, saturated incidence rate and study of optimal control

The spreading of HIV or HTLV-I among the cells has received the great attention in recent modelling study to explore the virus infection dynamics. The co-infection of HIV and HTLV-I with the effect of Cytotoxic T-lymphocytes (CTLs) immune response is also important from epidemiological point of view. To identify the co-infection scenario of HIV and HTLV-I with the CTLs effect we proposed in this paper a six compartmental ODE-model with uninfected, HIV-infected, HTLV-I infected CD4+T cells and free HIV virus particles with HIV specific CTLs and HTLV-I specific CTLs. The rates of infection of the cases are considered here saturated type and proliferation rate of uninfected and HIV infected CD4+ T-cells are of logistic terms. To establish the well-posedness of the model we have shown that the solution of the proposed model is non-negative and bounded. We obtain the basic reproduction number which is the maximum of the HIV-related reproduction and the HTLV-I related reproduction number. Along with the disease free equilibrium point the system contains other seven endemic equilibrium points containing infection by single disease or both. Analytically, we establish the local and global stability conditions of the equilibrium points and also we establish that the system experiences transcritical bifurcation by the generation of only HIV or HTLV-I infected endemic equilibrium point. Using numerical simulations, we validate the theoretical results and found two infection paths, one initiating with HIV and other with HTLV-I, both cases ultimately become co-infected. Finally, using the optimal control analysis we found the optimal policy for treatment using AVR, RTI & PI for HIV or AZT for HTLV-I control and lastly concluded by some recommendations.

Stationary distribution of a double epidemic stochastic model driven by saturated incidence rates

The double epidemic study is one of the critical studies in recent times as humankind experiences various simultaneous spreads of diseases. Reproduction numbers are important, which helps to derive sufficient conditions for extinction, persistence and co-persistence of diseases. This work aims to establish the stationary distribution of a stochastic double epidemic model comprised of SIR and SIRS transmission hypothesis through saturated incidence rate, with the aid of Markov semigroup theory. By constructing a suitable Lyapunov functional, the required result is sufficient by conditions on reproduction numbers.

Modelling the Transmission Dynamics of Anthrax Disease with Simultaneous Vaccination for both Human and Animal Population Using the Saturated Incidence Rate

In this study, a mathematical model based on saturated incidence rate that explores the transmission dynamics of anthrax as a zoonotic disease with vaccination compartment introduced into the human and animal population was formulated using ordinary differential equations. The model’s flow diagram was presented with qualitative and quantitative analysis carried out on the model. The local and global stability analysis of the equilibrium points investigated using the Ruth-Hurwitz criterion and Castillo-Chaves method respectively were found to be locally asymptotically stable if the basic reproductive number is less than one and unstable if the basic reproductive number is greater than one. Test for backward bifurcation was done with the result showing no backward bifurcation in the system. The sensitivity analysis of the model's parameters was performed to determine the contribution of each parameter to the basic reproduction number and a plot of partial ranking correlation (PRCC) obtained. The analysis revealed that, by decreasing human and animal contact rate, it would cause a decrease in the basic reproduction number. Also, numerical simulation further showed that increase in vaccination for both human and animal population reduces the spread of anthrax disease in the population. Anthrax disease, Bifurcation, Mathematical modeling, Sensitivity index, Stability

BIFURCATION ANALYSIS OF AN SIR MODEL WITH SATURATED INCIDENCE RATE AND STRONG ALLEE EFFECT

In this paper, rich dynamics and complex bifurcations of an SIR epidemic model with saturated incidence rate and strong Allee effect are investigated. First, the existence of disease-free and endemic equilibria is explored, and we prove that the system has at most three positive equilibria, which exhibit different types such as hyperbolic saddle and node, degenerate unstable saddle (node) of codimension 2, degenerate saddle-node of codimension 3 at disease-free equilibria, and cusp, focus, and elliptic types Bogdanov–Takens singularities of codimension 3 at endemic equilibria. Second, bifurcation analysis at these equilibria are investigated, and it is found that the system undergoes a series of bifurcations, including transcritical, saddle node, Hopf, degenerate Hopf, homoclinic, cusp type Bogdanov–Takens of codimensional 2, and focus and elliptic type Bogdanov–Takens bifurcation of codimension 3 which are composed of some bifurcations with lower codimension. The system shows very rich dynamics such as the coexistence of multiple periodic orbits and homoclinic loops. Finally, numerical simulations are conducted on the theoretical results.

Minimal Wave Speed for a Nonlocal Viral Infection Dynamical Model

To provide insights into the spreading speed and propagation dynamics of viruses within a host, in this paper, we investigate the traveling wave solutions and minimal wave speed for a degenerate viral infection dynamical model with a nonlocal dispersal operator and saturated incidence rate. It is found that the minimal wave speed c∗ is the threshold that determines the existence of traveling wave solutions. The existence of traveling fronts connecting a virus-free steady state and a positive steady state with wave speed c≥c∗ is established by using Schauder’s fixed-point theorem, limiting arguments, and the Lyapunov functional. The nonexistence of traveling fronts for c&lt;c∗ is proven by the Laplace transform. In particular, the lower-bound estimation of the traveling wave solutions is provided by adopting a rescaling method and the comparison principle, which is a crucial prerequisite for demonstrating that the traveling semifronts connect to the positive steady state at positive infinity by using the Lyapunov method and is a challenge for some nonlocal models. Moreover, simulations show that the asymptotic spreading speed may be larger than the minimal wave speed and the spread of the virus may be postponed if the diffusion ability or diffusion radius decreases. The spreading speed may be underestimated or overestimated if local dispersal is adopted.

An HIV / AIDS Epidemic Model with Non-Linear Saturated Incidence Rate and Treatment

An HIV / AIDS Epidemic Model with Non-Linear Saturated Incidence Rate and Treatment