Disease-free Equilibrium Research Articles

Modeling the transmission dynamics of COVID-19 with genetically resistant humans

The emergence of a coronavirus disease (COVID-19) pandemic caused by severe acute respiratory syndrome coronavirus (SARS-COV 2) has caused an unprecedented global societal and public health crisis. A mathematical model that incorporates an individual’s ability to resist COVID-19 infection is considered to study the dynamics of COVID-19. Detection and testing for COVID-19 in individuals who has ability to resist the infection, proper preventive measures, and timely medical care are essential in managing and controlling COVID-19 outbreak. These allows us to make quantifiable predictions about the effects of testing utilization on the COVID-19 prevalence. First, we analyze the disease-free state of the COVID-19 model, calculate the baseline control reproduction number, and obtain the local stability of disease-free equilibrium. The model is calibrated using data obtained from Nigeria Centre for Disease Control and key parameters of the model are estimated. Sensitivity analysis is carried out to investigate the influence of the parameters in curtailing the disease. Numerical simulations are then used to explore the behavior of the model solutions involving the novel variable (genetically resistant humans) and control measures. Our results suggest that strict intervention effort (early detection of genetically resistant individuals, testing and isolation of the infected individuals) is required for quick suppression of the disease.

Stability analysis of a delayed SEIRQ epidemic model with diffusion

In this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SEIRQ epidemic model. The introduction of the delay in this model makes it more realistic and modelizes the latency period. In addition, the consideration of an SEIRQ model with diffusion aims to better understand the impact of the spatial heterogeneity of the environment and the movement of individuals on the persistence and extinction of disease. First, we determined a threshold value $R_0$ of the delayed SEIRQ model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that the unique disease-free equilibrium is asymptotically stable , what is proven by the numericals scchema. Moreover,we search under their condition the endemic equilibrium is asymptotically stable .

Analysis of mathematical model for asthma caused by the effects of environmental pollution

Asthma is a condition in which your airways narrow and swell and may produce extra mucus. In this study, an improved mathematical model of asthma which includes infected class not detected, infected detected class and recovered (relieved) class was formulated. The model exhibits two equilibrium, namely disease free equilibrium and endemic equilibrium. The analysis of positivity and boundedness of solutions showed that the model is epidemiologically well pose d. Simulation results showed that when pollutants are discharged into the environment at a constant rate, the asthma disease cases also increase in the population due to the fact that the interaction rate of both susceptible and infected undetected with pollutants in creases.Finally, asthma can be control by restricting the smokers from the population and the rate of release of pollutants into the environment.

Numerical analysis of a nonlinear age-structured HBV model with saturated incidence and spatial diffusion

In this paper, the numerical properties of a nonlinear age-structured hepatitis B virus model with saturated incidence and spatial diffusion are studied. Applying linearly implicit Euler method in time integration, a numerical scheme which can preserve the biological meanings is constructed. The convergence of the numerical solution in finite time is explored. In stability analysis, a threshold is proposed, which is called numerical basic reproduction number and denoted by R0h. It is proved that the numerical solution is locally asymptotically stable at the disease-free equilibrium when R0h<1. Moreover, it is proved the numerical basic reproduction number converges to the exact basic reproduction number of the model with first order accuracy. Furthermore, it is shown that a numerical space independent equilibrium exists and is asymptotically stable if R0h>1, which implies the threshold stability of the model can be preserved by numerical solution proposed. Eventually, our conclusions are tested through numerical experiments.

Dynamical analysis of HIV-TB co-infection transmission model in the presence of treatment for TB

The immense disease burden of tuberculosis (TB) infection is well-documented, particularly among those co-infected with HIV and TB. To better understand the transmission dynamics of HIV-TB co-infection in the absence of readily available HIV treatment, we develop a deterministic compartmental co-infection model. Our model helps to identify the effects of TB infection on the co-infection dynamics of the two diseases, especially when treatment for TB is readily available. We find that susceptibility to TB reinfection after a previous infection leads to backward bifurcation in the TB-only model when the associated reproduction number (R_0) is less than unity. However, when we make the susceptibility to TB re-infection insignificant in the model, the disease-free equilibrium of the TB-only model is locally asymptotically stable when the associated R_0 is less than unity. We conduct sensitivity and uncertainty analyses to identify the key parameters driving TB infection dynamics, using the R_0 as the response function. We discover that the transmission rate for TB, the modification parameters accounting for the infectiousness of infected individuals with TB-only, and the treatment rates for singly infected individuals with latently infected TB are the top drivers of TB infection in the given population. Our numerical simulations suggest that concentrating treatment on TB-infected individuals in the diagnosed latently infected stage (singly or dually infected with HIV) could effectively reduce the co-infection disease burden and HIV incidence in the population under study.

Effects of Poor Sanitation and Public Awareness in Modeling Bacterial Infection amongst the Students of a Tertiary Institution in Kaura Namoda, Zamfara State, Nigeria

Acute bacterial infection of the intestine is caused by ingestion of food or water containing vibrio cholera. The symptoms include acute water diarrhea and vomiting which can result in severe dehydration or water loss. Sanitary conditions in the environment play an important role. Hence, the objective of this paper as to evaluate the effects of poor sanitation and public awareness in modeling bacterial infection amongst the students of a tertiary institution in Kaura Namoda, Zamfara State, Nigeria. We incorporated effectiveness of drug and awareness for proper hygiene and sanitation into our model. The disease free and endemic equilibrium were determined. The effective reproduction number was showed. Numerical results of the dynamics system of the transmission of bacterial infection were presented and we found that as the effective contact rate increases, the effective reproduction number increases. Also as the effectiveness of compliance of good hygiene increases, the effective reproduction number decreases by varying the contact rate. More so, as production rate of acute diarrhea bacteria increases, it increases the secondary cases of the infected individuals.

A novel study of the impact of vaccination on pneumonia via fractional approach

Substantial advancements have been made in applying fractional calculus to understanding the epidemiology of viral diseases. However, it remains troubling that over 2,000,000 children, particularly those under the age of 5 and elderly individuals over 65, succumb to pneumonia annually in developing nations. This paper uses a deterministic SVEIR (susceptible, vaccinated, exposed, infected, and recovered) model to investigate pneumonia disease dynamics from a mathematical perspective. The governing model has been generalized from Caputo fractional derivatives, and the generalized Euler’s method has been used to calculate the approximate solution of the governing model. The dynamical systems theory examines the model’s stability analysis, basic reproduction number, and equilibrium points. The linearization method for spatial equilibrium and the Lyapunov functional method demonstrate that the model is locally asymptotic stable for disease-free equilibrium. Graphical results are showcased through MATLAB21 to illustrate the resolution of the problem. The research findings suggest that the disease will eventually disappear from the population if vaccination coverage rises over the necessary vaccination threshold. According to sensitivity analysis, the most sensitive factors are the transmission rate and the pace at which exposed individuals become contagious. According to the study, we must strengthen treatment effectiveness to prevent disease spreading.

Epidemicity indices and reproduction numbers from infectious disease data in connected human populations

We focus on distinctive data-driven measures of the fate of ongoing epidemics. The relevance of our pursuit is suggested by recent results proving that the short-term temporal evolution of infection spread is described by an epidemicity index related to the maximum instantaneous growth rate of new infections, echoing concepts and tools developed to study the reactivity of ecosystems. Suitable epidemicity indices can showcase the dynamics of infections, together with commonly employed effective reproduction numbers, especially when the latter assume values less than 1. In particular, epidemicity evaluates the short-term reactivity to perturbations of a disease-free equilibrium. Here, we show that sufficient epidemicity thresholds to prevent transient epidemic outbreaks in a spatially connected setting can be estimated by generalizing existing analogues derived when spatial effects are neglected. We specifically account for the discrete nature, in both space and time, of surveillance data of the type typically employed to estimate effective reproduction numbers that formed the bulk of the communication of the state of the COVID-19 pandemic and its controls. After analyzing the effects of spatial heterogeneity on the considered prognostic indicators, we perform a short- and long-term analysis on the COVID-19 pandemic in Italy, showing that endemic conditions were maintained throughout the duration of our simulation despite stringent control measures. Our method provides a portfolio of prognostic indices that are essential to pinpoint the ongoing pandemic in both a qualitative and quantitative manner, as our results demonstrate. We base our conclusions on extended investigations of the effects of spatial fragmentation of communities of different sizes owing to connectivity by human mobility and contact scenarios, within real geographic contexts and synthetic setups designed to test our framework.

Stability and codimension 2 bifurcations in an [formula omitted] model with incubation delay

The aim of this paper is to investigate the complex nonlinear dynamics, such as the existence and effect of various types of bifurcations, in an infectious disease model system. An SIR delay model that accounts for saturated incidence, saturated treatment, and self-protection in the susceptible population is considered. The model system has a disease free equilibrium, and its local stability is determined by the basic reproduction number for any delay value. The existence of multiple endemic equilibria is obtained, and these equilibria may switch stability due to variation of bifurcation parameters. When incubation delay is zero, Bogdanov–Takens bifurcation occurs, which implies the existence of Hopf bifurcation, saddle–node, and homoclinic bifurcations. Thus, the disease will persist within or on the homoclinic loop and otherwise dies out. We also find the existence of backward bifurcation and forward (transcritical) bifurcation. The delay-induced stability switches are observed due to Hopf bifurcations, leading to endemic bubbles. Incubation delay affects the stability of equilibria when the model system has multiple endemic equilibria, so multi-stability is observed. Due to incubation delay, the two frequency Hopf–Hopf bifurcation and Bogdanov–Takens bifurcation are found. Thus, both codimension 1 and codimension 2 bifurcations in the effect of incubation delay are exhibited in this study. Numerically, we notice that with increasing incubation delay, the endemic equilibrium either changes stability (from stable to unstable or unstable to stable) or remains unstable with different frequencies of oscillations.

Mathematical assessment of the roles of age heterogeneity and vaccination on the dynamics and control of SARS-CoV-2

The COVID-19 pandemic, caused by SARS-CoV-2, disproportionately affected certain segments of society, particularly the elderly population (which suffered the brunt of the burden of the pandemic in terms of severity of the disease, hospitalization, and death). This study presents a generalized multigroup model, with m heterogeneous sub-populations, to assess the population-level impact of age heterogeneity and vaccination on the transmission dynamics and control of the SARS-CoV-2 pandemic in the United States. Rigorous analysis of the model for the homogeneous case (i.e., the model with m = 1) reveal that its disease-free equilibrium is globally-asymptotically stable for two special cases (with perfect vaccine efficacy or negligible disease-induced mortality) whenever the associated reproduction number is less than one. The model has a unique and globally-asymptotically stable endemic equilibrium, for special a case, when the associated reproduction threshold exceeds one. The homogeneous model was fitted using the observed cumulative mortality data for the United States during three distinct waves (Waves A (October 17, 2020 to April 5, 2021), B (July 9, 2021 to November 7, 2021) and C (January 1, 2022 to May 7, 2022)) chosen to align with time periods when the Alpha, Delta and Omicron were, respectively, the predominant variants in the United States. The calibrated model was used to derive a theoretical expression for achieving vaccine-derived herd immunity (needed to eliminate the disease in the United States). It was shown that, using the one-group homogeneous model, vaccine-derived herd immunity is not attainable during Wave C of the pandemic in the United States, regardless of the coverage level of the fully-vaccinated individuals. Global sensitivity analysis was carried out to determine the parameters of the model that have the most influence on the disease dynamics and burden. These analyses reveal that control and mitigation strategies that may be very effective during one wave may not be so very effective during the other wave or waves. However, strategies that target asymptomatic and pre-symptomatic infectious individuals are shown to be consistently effective across all waves. To study the impact of the disproportionate effect of COVID-19 on the elderly population, we considered the heterogeneous model for the case where the total population is subdivided into the sub-populations of individuals under 65 years of age and those that are 65 and older. The resulting two-group heterogeneous model, which was also fitted using the cumulative mortality data for wave C, was also rigorously analysed. Unlike for the case of the one-group model, it was shown, for the two-group model, that vaccine-derived herd immunity can indeed be achieved during Wave C of the pandemic if at least 61% of the populace is fully vaccinated. Thus, this study shows that adding age heterogeneity into a SARS-CoV-2 vaccination model with homogeneous mixing significantly reduces the level of vaccination coverage needed to achieve vaccine-derived herd immunity (specifically, for the heterogeneous model, herd-immunity can be attained during Wave C if a moderate proportion of susceptible individuals are fully vaccinated). The consequence of this result is that vaccination models for SARS-CoV-2 that do not explicitly account for age heterogeneity may be overestimating the level of vaccine-derived herd immunity threshold needed to eliminate the SARS-CoV-2 pandemic.

Analysis of a diffusive epidemic model with a zero-infection zone

In this paper, we propose and analyze a diffusive epidemic model with a zero-infection zone for the susceptible population. A distinct feature of the model is that the infected people are assumed to not able to cross the boundary of the zero-infection zone while the susceptible people cross the boundary at a prescribed rate. We investigate the global existence and boundedness of the solutions of the model and the existence and uniqueness of the disease-free equilibrium. We define the basic reproduction number and show that it is a threshold parameter for the disease dynamics. Moreover, we study the impact of the population transfer rate at the zero-infection zone boundary on the magnitude of the basic reproduction number. Finally, numerical simulations are performed as a supplement to our theoretical results.

On Theoretical and Numerical Results of Serum Hepatitis Disease Using Piecewise Fractal–Fractional Perspectives

In the present research, we consider a biological model of serum hepatitis disease. We carry out a detailed analysis of the mentioned model along with a class with asymptomatic carriers to explore its theoretical and numerical aspects. We initiate the study by using the piecewise fractal–fractional derivative (FFD) by which the crossover effects within the model are examined. We split the time interval into subintervals. In one subinterval, FFD with a power law kernel is taken, while in the second one, FFD with an exponential decay kernel of the proposed model is considered. This model is then studied for its disease-free equilibrium point, existence, and Hyers–Ulam (H-U) stability results. For numerical results of the model and a visual presentation, we apply the Lagrange interpolation method and an extended Adams–Bashforth–Moulton (ABM) method, respectively.

An infectious disease epidemic model with migration and stochastic transmission in deterministic and stochastic environments

Understanding population migration is essential for controlling highly infectious diseases. This paper studies the global dynamics of an infectious disease epidemic model incorporating population migration and a stochastic transmission rate. Our findings demonstrate that in deterministic and stochastic environments, the models exhibit global Lyapunov stability near the disease-free equilibrium point, determined by a threshold parameter. Furthermore, we analyze the effect of migration on infectious diseases. We discover that the number of infections and the peak value of the infection curve increase with a higher level of population migration. These results are supported by numerical illustrations that hold epidemiological relevance. Additionally, the disease-free equilibrium of the associated time delay model is linearly asymptotically stable, and the endemic equilibrium exhibits more bifurcation for larger time delay values.

A spatial epidemic model with contact and mobility restrictions

ABSTRACT Several spatiotemporal epidemic models have described how contact and mobility restrictions have a dynamic effect on morbidity and mortality of fast transmitting pathogens in epidemics. Despite this, there have been rather limited contributions looking at policy optimization. This work combines a new spatiotemporal epidemic model of a heterogeneous mixed population located at different places with an optimal control approach to show the effects of contact and mobility restrictions under policy optimization. The objective of optimization not only includes epidemiological but also socio-economic implications of the restrictions. Several scenarios are numerically investigated, and the dependence of the optimal policy on some basic epidemiological parameters is analysed. The results illustrate the strong impact of spatial heterogeneity on optimal policy measures. An analysis of the stability of the disease-free equilibrium of the model is also presented.

Optimal control of a multi-scale immuno-influenza A transmission model with viral load-dependent infection

Influenza A is a highly contagious respiratory illness that spreads globally and results in millions of cases worldwide. The transmission rate and mortality rate of influenza in the population are determined by the load of the influenza A virus. In this study, we have developed a multi-scale immuno-influenza model considering incomplete vaccine immunity. We have calculated the basic reproduction numbers for both immunological and epidemiological models. Additionally, we have examined the stability of the disease-free equilibrium and the existence of the endemic equilibrium at the population level. To assess the cost-effectiveness of antiviral therapy at an individual level and vaccination at a population level, we have formulated an optimal control problem. The objective of this problem is to minimize both the number of influenza cases and the cost of vaccination. To solve this optimal control problem, we have employed the Ekeland variational principle. Furthermore, we have designed a global forward and backward sweep algorithm to estimate the impact of antiviral treatment on reducing influenza transmission among humans. Our findings indicate that the use of antiviral treatment significantly reduces viral loads in individuals. However, it may also increase the potential risk at the population level. Therefore, vaccination against influenza is the most effective option from a public health perspective.

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.

Dynamics of Hepatitis B Virus Disease with Infectious Latent and Vertical Transmission

Hepatitis B has become a major health threat because it is a life-threatening liver disease with an estimated 0.25 billion people suffering from this infectious disease worldwide. This paper presents a SLITR (Susceptible-Latent-Infectious-Treatment-Recovery) mathematical model that combines both vaccination and treatment as a means of controlling the hepatitis B virus (HBV). The nonlinear ordinary differential equations for the HBV transmission capacities were resolved and the basic reproduction number R0 computed using the next generation matrix method and simulated numerically using the Runge-Kutta fourth order scheme implemented using MatLab. The stability points for disease-free equilibrium state (DFE), endemic equilibrium state (EE), and basic reproduction number R0 were obtained and the results show that the disease-free equilibrium was both locally and globally asymptotically stable (R0&lt;1) . Similarly, treatment or vaccine administered was effective in alleviating the spread of HBV disease, and when both control strategies are combined, the diseases are quickly controlled and eventually eradicated.

Numerical analysis of age-structured HIV model with general transmission mechanism

In this paper, we discuss the numerical representation of the linearly implicit Euler method for an age-structured HIV infection model with a general transmission mechanism. We first define the basic reproduction number of the continuous model, and present the stability results of the equilibriums. For the numerical process, we establish the solvability of the system and the non-negativity and convergence of numerical solutions. In the analysis of the long-term dynamical behavior, this paper mainly focus on the existence of the infection equilibrium determined by the numerical reproduction number R0Δt. To overcome the difficulty caused by the complexity of epidemic transmission mechanisms, the 1-order convergence analysis of numerical basic reproduction numbers R0Δt is implemented by using the properties of the fundamental solution matrix. By a comparison principle, we show that the disease-free equilibrium is globally asymptotically stable if R0Δt<1. Moreover, for R0Δt>1, a unique numerical endemic equilibrium exists, which converges to the exact one, is locally asymptotically stable. Hence, numerical processes visually represent the dynamic properties of nonlinear age-structured HIV models. Finally, some numerical experiments demonstrate the verification and the efficiency of our results.

Global Stability Analysis of Onchocerciasis Transmission Dynamics with Vigilant Compartment in Two Interacting Populations

A deterministic compartmental model for the transmission dynamics of onchocerciasis with vigilant compartment in two interacting populations is studied. The model is qualitatively analyzed to investigate its global asymptotic behavior with respect to disease-free and endemic equilibria. It is shown, using a linear Lyapunov function, that the disease-free equilibrium is globally asymptotically stable when the associated basic reproduction number, R0&lt;1. When the basic reproduction number R0&gt;1, under some certain conditions on the model parameters, we prove that the endemic equilibrium is globally asymptotically stable with the aid of a suitable nonlinear Lyapunov function.

Mathematical Modeling and its Stability Analysis of an SEIR Model to Control Dengue by Segregating the Infective: An Approach for Efficient Resource Allocation

Objectives: Many research deals study on modeling the spread of dengue virus disease. The study dealing mathematical model to influence the concept of segregating infective is very important as it makes an organised practice of treatment which enlightens civilization. This article focuses on the same and frames a model with a proper analysis. Methods: BhirkoffRhota theorem helps in proving the boundedness of the model. By using the Next-Generation matrix the expression for Reproduction number is determined. The Routh-Hurwitz stability criterion is used to reveal the Local Stability of the proposed model. The aid of Lyapunov-LaSalle's principle proves the Global Stability. Findings: The proposed model is found to be positive and also bounded. Moreover, the equilibrium points exists and stable locally and globally. The numerical simulation shows that at the disease-free equilibrium and at the endemic equilibrium proving the stability of the model numerically. The comparative analysis of the model results in revealing the fact that the proposed model controls the infective better than the usual model. Novelty: The article differs from the usual model by incorporating the idea of categorizing the infected population into 3 phase namely febrile, critical and convalescent concluding that the use of proposed model helps in effective treatment and in efficient management of hospital resources in a systematic way. Keywords: Epidemic, Treatment Class, SEIR model, Jacobian matrix, Sensitivity index