Disease-free Steady State Research Articles

Analysis of a diffusive brucellosis model with partial immunity and stage structure in heterogeneous environment

In this paper, in order to study comprehensive effect of stage-structure, incomplete immunity and spatial diffusion on the transmission dynamics of sheep brucellosis, we formulate a reaction-diffusion brucellosis model with partial immunity and stage structure in heterogeneous environment. Firstly, the well-posedness of the system is investigated, including the existence of global solution and its ultimate boundedness, and then the basic reproduction number R0 is defined using the next generation operator. Further, the threshold criteria on the global dynamics of the model are established in terms of R0 in two special cases. That is, if R0<1, the disease-free steady state is globally asymptotically stable, while if R0>1 1$\\end{document}]]>, the model is uniformly persistent and there at least exists a endemic steady state. Furthermore, for the homogeneous space and heterogeneous diffusion model, by constructing suitable Lyapunov functions, we obtain the global asymptotic stability for the disease-free steady-state when R0≤1 and the global asymptotic stability endemic steady states when R0>11$\\end{document}]]>. Finally, two simulation examples are given to verify our theoretical results.

Dynamical analysis of vector–host epidemic model with age structure and asymptomatic infection

Due to the differences in viability, behaviour and immunity of hosts and vectors of different biological ages, and the prevalence of asymptomatic infected individuals, a vector–host infectious disease model with biological age and asymptomatic infections is proposed. The exact expression of the basic reproduction number R 0 is derived. Further, we prove that the disease-free steady state is globally asymptotically stable if R 0 < 1 , in this case, the disease does not depend on the initial value and always dies out. Further, if R 0 > 1 , model admits a unique endemic steady state which is local asymptotical stable under some conditions. Finally, the theoretical results are explained by some numerical simulations. Numerical simulations also show that the duration of asymptomatic infections has an important impact on the number of confirmed cases, and a long duration can significantly reduce the number of confirmed cases. The rational use of asymptomatic infections has a positive effect on the response to the transmission of vector-borne diseases.

Minimal wave speed for a two-group epidemic model with nonlocal dispersal and spatial-temporal delay

In this paper, a two-group SIR reaction-diffusion epidemic model with nonlocal dispersal and spatial-temporal delay based on within-group and inter-group transmission mechanisms is investigated. The basic reproduction number R0 is calculated using the method of next-generation matrix. The critical wave speed cm* is determined by analyzing the distribution of roots of the characteristic equation. When R0&amp;gt;1 and wave speed c⩾cm*, the existence of traveling waves connecting disease-free and endemic steady states is obtained by constructing sub- and super-solutions and a Lyapunov functional, and applying Schauder’s fixed-point theorem and a limit argument. When R0&amp;gt;1 and 0&amp;lt;c&amp;lt;cm*, the nonexistence of traveling waves connecting disease-free and endemic steady states is proven by contradiction and two-sided Laplace transform. This indicates that the critical wave speed cm* is exactly the minimum wave speed. Numerical simulations are carried out to illustrate theoretical results. The dependence of the minimal speed cm* on time delay, diffusion rates and contact rates is discussed, showing that the longer the latent period and the lower the diffusion rates of infected individuals and the inter-group transmission rates between groups, the slower the spread of disease.

Dynamics of an age-structured SIS epidemic model with local dispersal and general incidence functions

In this paper, an age-structured SIS epidemic model with local dispersal and general incidence functions is formulated. We first analytically derive the well-posedness of solutions, the basic reproduction number R0 and the existence of steady states. Then, we show that the disease-free steady state is global asymptotically stable if R0≤1 and disease in the model is uniformly persistent if R0>1. Moreover, we show global asymptotic stability of the endemic steady state under some cases. Finally, infection age, media coverage and spatial diffusion on the epidemic of infectious diseases are demonstrated by numerical simulations.

Analysis of the transmission dynamics and effects of containment strategies on the eradication of malaria

A debilitating disease such as malaria has been a global health challenge for a long time. Mathematical models have provided insights on how best to control, prevent, or eradicate the disease. Most of the models used parameters to represent control strategies. The current research aims at using control strategies as disease compartments for both vector control and human-protection. By this, the long-time evolution of containment measures can be assessed. Consequently, a model of nine disease compartments is constructed using first-order ordinary differential equations. A qualitative analysis of the malaria-free equilibrium was carried out. The results indicate that the malaria-free equilibrium state is locally and globally asymptotically stable when R0HV<1, which implies that the disease-free state will always be attained from any initial conditions. A bifurcation analysis also shows that the malaria-endemic and malaria-free equilibria cannot overlap. Numerical simulations show that vector control is more effective in the containment of malaria than human-protection, which confirms the findings of Ross 53 years ago. Numerical results also show that solutions attain a disease-free steady state with time, which agrees with the theoretical results. The implication of the findings is that more attention should be paid to vector control if malaria is to be eradicated.

A fractional perspective on the transmission dynamics of a parasitic infection, considering the impact of both strong and weak immunity.

Infectious disease cryptosporidiosis is caused by the cryptosporidium parasite, a type of parasitic organism. It is spread through the ingestion of contaminated water, food, or fecal matter from infected animals or humans. The control becomes difficult because the parasite may remain in the environment for a long period. In this work, we constructed an epidemic model for the infection of cryptosporidiosis in a fractional framework with strong and weak immunity concepts. In our analysis, we utilize the well-known next-generation matrix technique to evaluate the reproduction number of the recommended model, indicated by [Formula: see text]. As [Formula: see text], our results show that the disease-free steady-state is locally asymptotically stable; in other cases, it becomes unstable. Our emphasis is on the dynamical behavior and the qualitative analysis of cryptosporidiosis. Moreover, the fixed point theorem of Schaefer and Banach has been utilized to investigate the existence and uniqueness of the solution. We identify suitable conditions for the Ulam-Hyers stability of the proposed model of the parasitic infection. The impact of the determinants on the sickness caused by cryptosporidiosis is highlighted by the examination of the solution pathways using a novel numerical technique. Numerical investigation is conducted on the solution pathways of the system while varying various input factors. Policymakers and health officials are informed of the crucial factors pertaining to the infection system to aid in its control.

Mathematical Analysis of Hepatitis B Virus Model with Interventions in Taraba State, Nigeria

A new version of a mathematical model for the dynamics of the Hepatitis B Virus (HBV) with a combination of three interventions was proposed. The primary focus of the study was to combine treatment, vaccination, and media campaigns as control strategies for reducing the spread of HBV. The study established the existence of a disease-free steady state of the model. The disease-free steady state was both locally and globally asymptotically stable. Additionally, the numerical results revealed that combining the three interventions is the best control strategy for curbing the menace of HBV. The study suggested that more effective campaign efforts should be supplemented with other major control strategies to get substantial paybacks in curbing the menace of HBV. Hepatitis B Virus (HBV), Intervention Stability, Steady-State

Qualitative analysis on a reaction–diffusion SIS epidemic model with nonlinear incidence and Dirichlet boundary

In this paper, the dynamical behavior in a spatially heterogeneous reaction–diffusion SIS epidemic model with general nonlinear incidence and Dirichlet boundary condition is investigated. The well-posedness of solutions, including the global existence, nonnegativity, ultimate boundedness, as well as the existence of compact global attractor, are first established, then the basic reproduction number R0 is calculated by defining the next generation operator. Secondly, the threshold dynamics of the model with respect to R0 are studied. That is, when R0<1 the disease-free steady state is globally asymptotically stable, and when R0>1 the model is uniformly persistent and admits one positive steady state, and under some additional conditions the uniqueness of positive steady state is obtained. Furthermore, some interesting properties of R0 are established, including the calculating formula of R0, the asymptotic profiles of R0 with respect to diffusion rate dI, and the monotonicity of R0 with diffusion rate dI and domain Ω. In addition, the bang–bang-type configuration optimization of R0 also is obtained. This rare result in diffusive equation reveals that we can control disease diffusion at least at one peak. Finally, the numerical examples and simulations are carried out to illustrate the rationality of open problems proposed in this paper, and explore the influence of spatial heterogeneous environment on the disease spread and make a comparison on dynamics between Dirichlet boundary condition and Neumann boundary condition.

Dynamical analysis of healthcare policy effects in an integrated economic-epidemiological model

We study the static and dynamical properties of a model that describes the interaction between the economic and epidemiological domains. The epidemiological sphere is represented by a susceptible–infected–susceptible model, while the economic domain consists of an overlapping generations model in which the workers correspond to the non-infected population of adults. The productivity of the firms and the propensity to save for retirement of the households are negatively affected by the disease spread. A capital tax is levied and the collected resources are used to curb the spread of the outbreak. We show that multiple endemic steady states can arise from the interaction between the two domains, and different stable endemic attractors can coexist with the stable disease free steady state. We study analytically and numerically the complex dynamics and the evolution of the basins of attraction in the case of multistability. We show that the effect of taxation can be beneficial from both the epidemiological and the economic points of view, as it can give rise to new steady states characterized by reduced shares of infected people and increased capital level, it can simplify the dynamical behaviors and reduce the size of the basins of attraction of those outcomes in which large shares of infected people and low capital levels are observed.

Dynamics of an EIS spatially heterogeneous rabies model

This paper is devoted to the study of population dynamics of an EIS epidemic reaction-diffusion model in spatially heterogeneous environments, where the rates of disease-induced mortality and disease transmission are space dependent. For such a class of systems, we provide clear pictures on the global dynamics in terms of the basic reproduction number R0, which improves the local dynamics proposed in Wang and Zhao (2012) [20]. When R0≤1, the disease-free steady state is globally asymptotically stable. Different from earlier works for many epidemic models, we obtain the positive steady states by using bifurcation theory for R0>1, which can be easily generalized to the case where the reaction-diffusion system involves partial diffusion coefficients of zero.

Are zero-Covid policies optimal?

We analyze the impact of public health policy on the spread of a disease using a version of the SIR model that includes vital statistics, waning immunity, and vaccination. This model is rich enough to accommodate endemic steady states and disease-free steady states. We derive social distancing and vaccination policies that maximize an objective function that penalizes lost output resulting from social distancing, deaths resulting from the disease, and the cost of vaccination. Even though a disease-free steady state is attainable, optimal policy leads to an endemic steady state, albeit with a small number of deaths and negligible loss of output.

Threshold dynamics of a reaction-advection-diffusion waterborne disease model with seasonality and human behavior change

To investigate the effects of environmental pollution and human behavior change on waterborne diseases, we propose a reaction-advection-diffusion waterborne disease model with a general boundary condition, which incorporates human hosts and reservoir aquatic of pathogen. We identify the basic reproduction number [Formula: see text] and discuss its asymptotic properties. We prove its threshold role: if [Formula: see text], the disease-free periodic solution is globally attractive; if [Formula: see text], the model system is uniformly persistent; if [Formula: see text], the disease-free steady state is globally asymptotically stable without the advection term, in which the proof is quite challenging due to human behavior change.

Global analysis of a diffusive Cholera model with multiple transmission pathways, general incidence and incomplete immunity in a heterogeneous environment.

With the consideration of the complexity of the transmission of Cholera, a partially degenerated reaction-diffusion model with multiple transmission pathways, incorporating the spatial heterogeneity, general incidence, incomplete immunity, and Holling type Ⅱ treatment was proposed. First, the existence, boundedness, uniqueness, and global attractiveness of solutions for this model were investigated. Second, one obtained the threshold condition $ \mathcal{R}_{0} $ and gave its expression, which described global asymptotic stability of disease-free steady state when $ \mathcal{R}_{0} < 1 $, as well as the maximum treatment rate as zero. Further, we obtained the disease was uniformly persistent when $ \mathcal{R}_{0} > 1 $. Moreover, one used the mortality due to disease as a branching parameter for the steady state, and the results showed that the model undergoes a forward bifurcation at $ \mathcal{R}_{0} $ and completely excludes the presence of endemic steady state when $ \mathcal{R}_{0} < 1 $. Finally, the theoretical results were explained through examples of numerical simulations.

Bifurcation analysis of HIV infection model with cell-to-cell transmission and non-cytolytic cure

Abstract A mathematical model is proposed and discussed to study the effect of cell-to-cell transmission, the non-cytolytic process, and the effect of logistic growth on the dynamics of HIV in vivo. The model system consists of one disease-free steady state and another endemic steady state. The disease-free steady state is globally asymptotically stable and the disease eradicated if the basic reproduction number is smaller than one. However, the endemic steady state is globally stable under specific parametric conditions, when it exists. At R 0 = 1 {R}_{0}=1 , the forward transcritical bifurcation is obtained. Also, by considering proliferation rate as bifurcation parameter, we get Hopf and Hopf–Hopf bifurcations. We have performed numerical simulations using MATLAB to support our analytical results and show the effects of cell-to-cell infection, proliferation rate, and non-cytolytic cure on all three populations. In the end, we have performed data fitting and note the same behaviour of observed data with predicted data.

Optimal vaccination ages for emerging infectious diseases under limited vaccine supply.

Rational allocation of limited vaccine resources is one of the key issues in the prevention and control of emerging infectious diseases. An age-structured infectious disease model with limited vaccine resources is proposed to explore the optimal vaccination ages. The effective reproduction number [Formula: see text] of the epidemic disease is computed. It is shown that the reproduction number is the threshold value for eradicating disease in the sense that the disease-free steady state is globally stable if [Formula: see text] and there exists a unique endemic equilibrium if [Formula: see text]. The effective reproduction number is used as an objective to minimize the disease spread risk. Using the epidemic data from the early spread of Wuhan, China and demographic data of Wuhan, we figure out the strategies to distribute the vaccine to the age groups to achieve the optimal vaccination effects. These analyses are helpful to the design of vaccination schedules for emerging infectious diseases.

Dynamical analysis of a novel discrete fractional lumpy skin disease model

Lumpy skin disease is a viral disease that affects cattle and is caused by the lumpy skin disease virus. This work is devoted to presenting and analyzing the nonlinear dynamics of a novel discrete fractional model for lumpy skin disease. The equilibrium points of the proposed discrete fractional model are found. The stability analysis of equilibrium points is carried out. The influences of key parameters in the model are investigated, and then the stability regions of a disease-free steady state in the space of parameters are obtained. A proposed efficient control scheme is implemented to stabilize the disease-free equilibrium point when it is unstable. The influences of fractional-order parameters on the applied control scheme are explored. Finally, numerical simulations are performed to verify the theoretical findings obtained and confirm the effectiveness of the employed control scheme.

Dynamics on a degenerated reaction–diffusion Zika transmission model

In view of environmental transmission and spatial heterogeneity, a degenerated reaction–diffusion Zika transmission model is established and analyzed with more realistic factors. Besides the mosquito reproduction number Rm, a novel computation formula of the basic reproduction number of disease R0 in terms of the principal eigenvalue of an elliptic eigenvalue problem is provided. Dynamic analysis results show that the proposed system follows a threshold dynamics based on Rm or R0. Especially, for the critical case of R0=1, global attractiveness of the disease-free steady state is obtained by constructing a new Lyapunov function, which is usually one of challenges for some spatially heterogeneous models.

Modelling the Dynamics of Multi-strain COVID-19 Transmission

It is on record that rolling out COVID-19 vaccines has been one of the fastest for any vaccine production worldwide. Despite this prompt action taken to mitigate the transmission of COVID-19, the disease persists. One of the reasons for the persistence of the disease is that the vaccines do not confer immunity against the infections. Moreover, the virus-causing COVID-19 mutates, rendering the vaccines less effective on the new strains of the disease. This research addresses the multi-strains transmission dynamics and herd immunity threshold of the disease. Local stability analysis of the disease-free steady state reveals that the pandemic can be contained when the basic reproduction number, R 0 is brought below unity. The results of numerical simulations also agree with the theoretical results. The herd immunity thresholds for some of the vaccines against COVID-19 were computed to guide the management of the disease. This model can be applied to any strain of the disease.

Fractional Caputo and sensitivity heat map for a gonorrhea transmission model in a sex structured population

Gonorrhea is a disease that is spread by sexual contact, and it can potentially cause infections in the genital region, the rectum, and even the throat. Due to the shared history between infected individuals and their sexual partners, infected individuals will likely continue to have sexual relations with those same partners. As a result, this article aims to investigate how memory affects the transmission of gonorrhea in a structured population using the Caputo fractional derivative and sensitivity analysis. The model is shown to be positively invariant with a unique bound. The existence and uniqueness criteria of the fractional model are established using fixed-point theory. The stable nature of the model is obtained using the Ulam Hyers and Ulam Hyers Rassias ideas. To highlight the stability of the fractional model, the stability of solution trajectories to the disease-free and endemic steady states is graphically illustrated for the gonorrhea basic reproduction number, R0ϱ∗<1 and R0ϱ∗>1, respectively. We showed the sensitivities linked to the proposed model using the Latin hypercube sampling, singular value analysis, box plots, scatter plots, contour plots, three-dimensional plots, and sensitivity heat maps. We noticed that the transmission rate from females to males, βfm, is the most influential parameter in the spread of the disease. From the sensitivity heat maps, it is noticed that using the first four principal components analysis, the most sensitive state variables to the parameters in the model are symptomatic females, recovered males, susceptible females, and recovered females. In conjunction with the modified Adams–Bashforth method, the numerical trajectories of the fractional Caputo model are investigated. Finally, we noticed that memory changes impact the number of incubative females and incubative males.

Threshold dynamics of an age-structured vaccinated epidemic model with both direct and indirect routes of infections

Age-structured epidemic models play a crucial role in the study of epidemiological modeling. Motivated by this, in this article, we propose an epidemic model with age since infection and vaccination age of individuals, a coupled system of PDE and integro-differential equations (IDE). Here, two contagious routes, (a) direct human-to-human contact and (b) indirect environmentally contaminated surfaces or objects are considered. First, we establish the well-posedness of the model, followed by the basic reproduction number (R0) and the role of the threshold value of R0 in the asymptotic profile of the solution semi-flow is established. We observe the global stability of the disease-free steady state for R0<1, the uniform persistence of the disease and the existence of the endemic steady state for R0>1. This endemic steady state is also globally asymptotically stable for R0>1. We have further analyzed the influence of vaccination age and age since infection in the threshold parameter R0. Our analysis shows that the threshold parameter R0 does not depend explicitly on vaccination age, but it strictly decreases with the natural depletion rate of the contaminated environment. Finally, the model is discretized using the finite difference method to illustrate our theoretical results numerically.