Block Toeplitz Matrices: Multiplicative Properties

Given A, B, C, and D, block Toeplitz matrices, we will prove the necessary and sufficient condition for AB - CD = 0, and AB - CD to be a block Toeplitz matrix. In addition, with respect to change of basis, the characterization of normal block Toeplitz matrices with entries from the algebra of diagonal matrices is also obtained.

Backward Bifurcation and Hysteresis in a Mathematical Model of COVID19 with Imperfect Vaccine

Vaccination has been used as strategy to eradicate the spread of COVID-19. But imperfect vaccine has been reported to induce backward bifurcation and hysteresis in mathematical models of disease transmission. Backward bifurcation is a phenomenon whereby a stable endemic equilibrium exists contemporaneously with a stable disease-free equilibrium when the basic reproduction number is less than 1. This situation can cause difficulty in controlling an epidemic because the basic reproduction is no longer the only means of eradicating the disease. In this paper, we propose a mathematical model for the transmission of disease which includes imperfect vaccination. We show that our model is capable of capturing backward bifurcation under certain conditions. By using parameters that are relevant to COVID-19 transmission in Malaysia, our numerical analysis shows that low vaccine efficacy can trigger backward bifurcation.

Stability analysis and optimal control on a multi-strain coinfection model with amplification and vaccination

In this paper, a multi-strain coinfection model with amplification (or mutation) is established to characterize the interaction between common strain and amplified strain, as well as vaccination. The basic reproduction number ℛ0 is derived, from which the criteria on the existence and local (or global) stability of equilibria (including disease-free, dominant-strain and coexistence-strain) are established. By analyzing the effectiveness of vaccination, we find that a critical inoculation level could make the disease eliminate when ℛ0<1, while inefficient vaccines could cause backward bifurcation when ℛ0<1. Based on sensitivity analysis and realistic control policy, the optimal strategy of disease control is obtained. The theoretical results are illustrated by numerical simulation and clinical data of COVID-19 in Morocco.

Forward-backward and period doubling bifurcations in a discrete epidemic model with vaccination and limited medical resources.

A discrete epidemic model with vaccination and limited medical resources is proposed to understand its underlying dynamics. The model induces a nonsmooth two dimensional map that exhibits a surprising array of dynamical behavior including the phenomena of the forward-backward bifurcation and period doubling route to chaos with feasible parameters in an invariant region. We demonstrate, among other things, that the model generates the above described phenomena as the transmission rate or the basic reproduction number of the disease gradually increases provided that the immunization rate is low, the vaccine failure rate is high and the medical resources are limited. Finally, the numerical simulations are provided to illustrate our main results.

Transmission Dynamics of COVID-19 with Saturated Treatment: A Case Study of Spain

In this paper, an epidemic compartmental model with saturated type treatment function is presented to investigate the transmission dynamics of COVID-19 with a case study of Spain (in Europe). We obtain the basic reproduction number of the model which plays a very important role in disease spreading. We show that if the basic reproduction number is less than unity then the disease-free equilibrium point is locally asymptotically stable, but making the basic reproduction number less than unity is not sufficient to eradicate COVID-19 infection which is shown through backward bifurcation. The model is validated with the real COVID-19 data of Spain (in Europe), Algeria (in Africa), and India (in Asia) and also estimated important model parameters in all cases. The effect of an important model parameter for controlling the disease spreading is also investigated for the infection scenario of Spain only. We establish that the asymptomatic class plays a very important role for spreading this pandemic disease. The effective reproduction number has been estimated which varies in time in Spain. Finally, the model is reformulated as an optimal control problem which shows that the social distancing due to adapting a partial lockdown by some countries is highly effective for controlling COVID-19.

A bifurcation analysis and model of Covid-19 transmission dynamics with post-vaccination infection impact.

SARS COV-2 (Covid-19) has imposed a monumental socio-economic burden worldwide, and its impact still lingers. We propose a deterministic model to describe the transmission dynamics of Covid-19, emphasizing the effects of vaccination on the prevailing epidemic. The proposed model incorporates current information on Covid-19, such as reinfection, waning of immunity derived from the vaccine, and infectiousness of the pre-symptomatic individuals into the disease dynamics. Moreover, the model analysis reveals that it exhibits the phenomenon of backward bifurcation, thus suggesting that driving the model reproduction number below unity may not suffice to drive the epidemic toward extinction. The model is fitted to real-life data to estimate values for some of the unknown parameters. In addition, the model epidemic threshold and equilibria are determined while the criteria for the stability of each equilibrium solution are established using the Metzler approach. A sensitivity analysis of the model is performed based on the Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCCs) approaches to illustrate the impact of the various model parameters and explore the dependency of control reproduction number on its constituents parameters, which invariably gives insight on what needs to be done to contain the pandemic effectively. The foregoing notwithstanding, the contour plots of the control reproduction number concerning some of the salient parameters indicate that increasing vaccination coverage and decreasing vaccine waning rate would remarkably reduce the value of the reproduction number below unity, thus facilitating the possible elimination of the disease from the population. Finally, the model is solved numerically and simulated for different scenarios of disease outbreaks with the findings discussed.

Modeling SARS-CoV-2 and HBV co-dynamics with optimal control.

Clinical reports have shown that chronic hepatitis B virus (HBV) patients co-infected with SARS-CoV-2 have a higher risk of complications with liver disease than patients without SARS-CoV-2. In this work, a co-dynamical model is designed for SARS-CoV-2 and HBV which incorporates incident infection with the dual diseases. Existence of boundary and co-existence endemic equilibria are proved. The occurrence of backward bifurcation, in the absence and presence of incident co-infection, is investigated through the proposed model. It is noted that in the absence of incident co-infection, backward bifurcation is not observed in the model. However, incident co-infection triggers this phenomenon. For a special case of the study, the disease free and endemic equilibria are shown to be globally asymptotically stable. To contain the spread of both infections in case of an endemic situation, the time dependent controls are incorporated in the model. Also, global sensitivity analysis is carried out by using appropriate ranges of the parameter values which helps to assess their level of sensitivity with reference to the reproduction numbers and the infected components of the model. Finally, numerical assessment of the control system using various intervention strategies is performed, and reached at the conclusion that enhanced preventive efforts against incident co-infection could remarkably control the co-circulation of both SARS-CoV-2 and HBV.

The stability analysis of a co-circulation model for COVID-19, dengue, and zika with nonlinear incidence rates and vaccination strategies.

This paper aims to study the impacts of COVID-19 and dengue vaccinations on the dynamics of zika transmission by developing a vaccination model with the incorporation of saturated incidence rates. Analyses are performed to assess the qualitative behavior of the model. Carrying out bifurcation analysis of the model, it was concluded that co-infection, super-infection and also re-infection with same or different disease could trigger backward bifurcation. Employing well-formulated Lyapunov functions, the model's equilibria are shown to be globally stable for a certain scenario. Moreover, global sensitivity analyses are performed out to assess the impact of dominant parameters that drive each disease's dynamics and its co-infection. Model fitting is performed on the actual data for the state of Amazonas in Brazil. The fittings reveal that our model behaves very well with the data. The significance of saturated incidence rates on the dynamics of three diseases is also highlighted. Based on the numerical investigation of the model, it was observed that increased vaccination efforts against COVID-19 and dengue could positively impact zika dynamics and the co-spread of triple infections.

THE THRESHOLD VALUE OF THE NUMBER OF HOSPITAL BEDS IN A SEIHR EPIDEMIC MODEL

To investigate the impact of the number of hospital beds on the control of infectious diseases and help allocate the limited medical resources in a region, a SEIHR epidemic model including exposed and hospitalized classes is established. Different from available models, the hospitalization rate is expressed as a function of the number of empty beds. The existence and stability of the equilibria are analyzed, and it is proved that the system undergoes backward bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation of codimension 2 under certain conditions by using the center manifold theory and normal form theory. In particular, our results show that there is a threshold value for the capacity of hospital beds in a region. If the capacity of hospital beds is lower than this threshold value, there will be a backward bifurcation, which means that even if the basic reproduction number, R0, is less than unity, it is not enough to prevent the outbreaks. Before taking disease control measures, one should compare the number of beds with the threshold value to avoid misjudgment and try to increase the capacity of hospital beds above this threshold value. The method to estimate the threshold value is also given. In addition, the impacts of the duration of the exposed period on the basic reproduction number and disease transmission are investigated. © 2023 American Institute of Mathematical Sciences. All rights reserved.

Global stability of a general HTLV-I infection model with Cytotoxic T-Lymphocyte immune response and mitotic transmission

Mathematical models have been considered as a robust tool to support biological and medical studies of human viral infections. The global stability of viral infection models remains an important and largely open research problem. Such results are necessary to evaluate treatment strategies for infections and to establish thresholds for treatment rates. Human T-lymphotropic virus class I (HTLV-I) is a retrovirus which infects the CD4+T cells and causes chronic and deadly diseases. In this article, we developed a general nonlinear system of ODEs which describes the within-host dynamics of HTLV-I under the effect Cytotoxic T-Lymphocytes (CTLs) immunity. The mitotic division of actively infected cells are modeled. We consider general nonlinear functions for the generation, proliferation and clearance rates for all types of cells. The incidence rate of infec-tion is also modeled by a general nonlinear function. These general functions are assumed to satisfy a set of suitable conditions and include several forms presented in the literature. We determine a bounded domain for the system's solutions. We prove the existence of the system's equilibrium points and determine two threshold numbers, the basic reproductive number R0 and the CTL immunity stimulation number R1. We establish the global stability of all equilibrium points by con-structing Lyapunov function and applying Lyapunov-LaSalle asymptotic stability theorem. Under certain conditions it is shown that if R0 <= 1, then the infection-free equilibrium point is globally asymptotically stable (GAS) and the HTLV-I infection is cleared. If R1 < 1 < R0, then the infected equilibrium point without CTL immunity is GAS and the HTLV-I infection becomes chronic with no sustained CTL immune response. If R1 > 1, then the infected equilibrium point with CTL immu-nity is GAS and the infection becomes chronic with persistent CTL immune response. We present numerical simulations for the system by choosing special shapes of the general functions. The effect of Crowley-Martin functional response and mitotic division of actively infected cells on the HTLV-I progression are studied. Our results cover and improve several ones presented in the literature.(c) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).

An SEQAIHR model to study COVID-19 transmission and optimal control strategies in Hong Kong, 2022.

During the COVID-19 pandemic, one of the major concerns was a medical emergency in human society. Therefore it was necessary to control or restrict the disease spreading among populations in any fruitful way at that time. To frame out a proper policy for controlling COVID-19 spreading with limited medical facilities, here we propose an SEQAIHR model having saturated treatment. We check biological feasibility of model solutions and compute the basic reproduction number ( R 0 ). Moreover, the model exhibits transcritical, backward bifurcation and forward bifurcation with hysteresis with respect to different parameters under some restrictions. Further to validate the model, we fit it with real COVID-19 infected data of Hong Kong from 19th December, 2021 to 3rd April, 2022 and estimate model parameters. Applying sensitivity analysis, we find out the most sensitive parameters that have an effect on R 0 . We estimate R 0 using actual initial growth data of COVID-19 and calculate effective reproduction number for same period. Finally, an optimal control problem has been proposed considering effective vaccination and saturated treatment for hospitalized class to decrease density of the infected class and to minimize implemented cost.

COVID-19 outbreak: a predictive mathematical study incorporating shedding effect.

In this paper, a modified SEIR epidemic model incorporating shedding effect is proposed to analyze transmission dynamics of the COVID-19 virus among different individuals' classes. The direct impact of pathogen concentration over susceptible populations through the shedding of COVID-19 virus into the environment is investigated. Moreover, the threshold value of shedding parameters is computed which gives information about their significance in decreasing the impact of the disease. The basic reproduction number ( R 0 ) is calculated using the next-generation matrix method, taking shedding as a new infection. In the absence of disease, the condition for the equilibrium point to be locally and globally asymptotically stable with R 0 < 1 are established. It has been shown that the unique endemic equilibrium point is globally asymptotically stable under the condition R 0 > 1 . Bifurcation theory and center manifold theorem imply that the system exhibit backward bifurcation at R 0 = 1 . The sensitivity indices of R 0 are computed to investigate the robustness of model parameters. The numerical simulation is demonstrated to illustrate the results.

Bifurcation analysis and optimal control of COVID-19 with exogenous reinfection and media coverages

In this paper, a SEIR epidemic model related to media coverage and exogenous reinfections is established to explore the transmission dynamics of COVID-19. The basic reproduction number is calculated using the next generation matrix method. First, the existence of equilibrium points is investigated, and different kinds of equilibrium points indicate that the disease may disappear, or exist that result in different quantity of susceptible individuals, pre-symptomatic infected individuals and symptomatic infected individuals. The stability of the equilibria is discussed by a geometric approach, and it is found that controlling reproduction number to be lower than 1 is not sufficient for eradication of COVID-19. Second, transcritical bifurcation is explored, and it is found that improving the ratio of exogenous reinfection may lead to backward bifurcation under poor medical conditions, which indicates that two endemic equilibrium points appear. Third, to investigate the influence of parameters on the basic reproduction, sensitivity analysis is done to choose relatively sensitive parameters, and the parameters for treatment and media coverage are selected. An optimal control model is established to balance the treatment and media awareness. By exploring the existence and the uniqueness of the optimal control solution, the optimal control strategies are given. Finally, we run numerical simulations to verify the theoretical analysis on actual data of China, and the data from the four different states of India is used for forecasting the situation of infected individuals in a short period. It is found by the simulation that the co-function of treatment and media coverage results in the reduced number of infectious individuals. [ FROM AUTHOR]

Backward bifurcation, basic reinfection number and robustness of an SEIRE epidemic model with reinfection

Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose an SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds R-0 (the basic reproduction number) and R-c (a threshold less than one). Our investigations show that when R-0 > 1, the system has an endemic equilibrium, which is globally asymptotically stable. When R-c < R-0 < 1, the epidemic system exhibits bistable dynamics. That is, the system has backward bifurcation and the disease cannot be eradicated. In order to eradicate the disease, we must ensure that the basic reproduction number R0 is less than Rc. The basic reinfection number is obtained to measure the reinfection force, which turns out to be a new tipping point for disease dynamics. We also give definition of robustness, a new concept to measure the difficulty of completely eliminating the disease for a bistable epidemic system. Numerical simulations are carried out to verify the conclusions.

Early-confinement strategy to tackling COVID-19 in Morocco, a mathematical modelling study

Morocco is among the countries that started setting up confinement in the early stage of the COVID-19 spread. Comparing the number of cumulative cases in various countries, a partial lock-down has delayed the exponential outbreak of COVID-19 in Morocco. Using a compartmental model, we attempt to estimate the mean proportion of correctly confined sub-population in Morocco as well as its effect on the continuing spread of COVID-19. A fitting to Moroccan data is established. Furthermore, we have highlighted some COVID-19 epidemic scenarios that could have happened in Morocco after the deconfinement onset while considering a different combination of preventive measures. © The authors. Published by EDP Sciences, ROADEF, SMAI 2022.

A Mathematical Model of Vaccinations Using New Fractional Order Derivative.

Purpose: This paper studies a simple SVIR (susceptible, vaccinated, infected, recovered) type of model to investigate the coronavirus's dynamics in Saudi Arabia with the recent cases of the coronavirus. Our purpose is to investigate coronavirus cases in Saudi Arabia and to predict the early eliminations as well as future case predictions. The impact of vaccinations on COVID-19 is also analyzed. Methods: We consider the recently introduced fractional derivative known as the generalized Hattaf fractional derivative to extend our COVID-19 model. To obtain the fitted and estimated values of the parameters, we consider the nonlinear least square fitting method. We present the numerical scheme using the newly introduced fractional operator for the graphical solution of the generalized fractional differential equation in the sense of the Hattaf fractional derivative. Mathematical as well as numerical aspects of the model are investigated. Results: The local stability of the model at disease-free equilibrium is shown. Further, we consider real cases from Saudi Arabia since 1 May−4 August 2022, to parameterize the model and obtain the basic reproduction number R0v≈2.92. Further, we find the equilibrium point of the endemic state and observe the possibility of the backward bifurcation for the model and present their results. We present the global stability of the model at the endemic case, which we found to be globally asymptotically stable when R0v>1. Conclusion: The simulation results using the recently introduced scheme are obtained and discussed in detail. We present graphical results with different fractional orders and found that when the order is decreased, the number of cases decreases. The sensitive parameters indicate that future infected cases decrease faster if face masks, social distancing, vaccination, etc., are effective.

Deciphering the transmission dynamics of COVID-19 in India: optimal control and cost effective analysis.

In this paper we assess the effectiveness of different non-pharmaceutical interventions (NPIs) against COVID-19 utilizing a compartmental model. The local asymptotic stability of equilibria (disease-free and endemic) in terms of the basic reproduction number have been determined. We find that the system undergoes a backward bifurcation in the case of imperfect quarantine. The parameters of the model have been estimated from the total confirmed cases of COVID-19 in India. Sensitivity analysis of the basic reproduction number has been performed. The findings also suggest that effectiveness of face masks plays a significant role in reducing the COVID-19 prevalence in India. Optimal control problem with several control strategies has been investigated. We find that the intervention strategies including implementation of lockdown, social distancing, and awareness only, has the highest cost-effectiveness in controlling the infection. This combined strategy also has the least value of average cost-effectiveness ratio (ACER) and associated cost.

Bifurcation analysis and optimal control of COVID-19 with exogenous reinfection and media coverages

In this paper, a SEIR epidemic model related to media coverage and exogenous reinfections is established to explore the transmission dynamics of COVID-19. The basic reproduction number is calculated using the next generation matrix method. First, the existence of equilibrium points is investigated, and different kinds of equilibrium points indicate that the disease may disappear, or exist that result in different quantity of susceptible individuals, pre-symptomatic infected individuals and symptomatic infected individuals. The stability of the equilibria is discussed by a geometric approach, and it is found that controlling reproduction number to be lower than 1 is not sufficient for eradication of COVID-19. Second, transcritical bifurcation is explored, and it is found that improving the ratio of exogenous reinfection may lead to backward bifurcation under poor medical conditions, which indicates that two endemic equilibrium points appear. Third, to investigate the influence of parameters on the basic reproduction, sensitivity analysis is done to choose relatively sensitive parameters, and the parameters for treatment and media coverage are selected. An optimal control model is established to balance the treatment and media awareness. By exploring the existence and the uniqueness of the optimal control solution, the optimal control strategies are given. Finally, we run numerical simulations to verify the theoretical analysis on actual data of China, and the data from the four different states of India is used for forecasting the situation of infected individuals in a short period. It is found by the simulation that the co-function of treatment and media coverage results in the reduced number of infectious individuals.

An SIRS model with nonmonotone incidence and saturated treatment in a changing environment.

Nonmonotone incidence and saturated treatment are incorporated into an SIRS model under constant and changing environments. The nonmonotone incidence rate describes the psychological or inhibitory effect: when the number of the infected individuals exceeds a certain level, the infection function decreases. The saturated treatment function describes the effect of infected individuals being delayed for treatment due to the limitation of medical resources. In a constant environment, the model undergoes a sequence of bifurcations including backward bifurcation, degenerate Bogdanov-Takens bifurcation of codimension 3, degenerate Hopf bifurcation as the parameters vary, and the model exhibits rich dynamics such as bistability, tristability, multiple periodic orbits, and homoclinic orbits. Moreover, we provide some sufficient conditions to guarantee the global asymptotical stability of the disease-free equilibrium or the unique positive equilibrium. Our results indicate that there exist three critical values [Formula: see text] and [Formula: see text] for the treatment rate r: (i) when [Formula: see text], the disease will disappear; (ii) when [Formula: see text], the disease will persist. In a changing environment, the infective population starts along the stable disease-free state (or an endemic state) and surprisingly continues tracking the unstable disease-free state (or a limit cycle) when the system crosses a bifurcation point, and eventually tends to the stable endemic state (or the stable disease-free state). This transient tracking of the unstable disease-free state when [Formula: see text] predicts regime shifts that cause the delayed disease outbreak in a changing environment. Furthermore, the disease can disappear in advance (or belatedly) if the rate of environmental change is negative and large (or small). The transient dynamics of an infectious disease heavily depend on the initial infection number and rate or the speed of environmental change.

THE THRESHOLD VALUE OF THE NUMBER OF HOSPITAL BEDS IN A SEIHR EPIDEMIC MODEL

To investigate the impact of the number of hospital beds on the control of infectious diseases and help allocate the limited medical resources in a region, a SEIHR epidemic model including exposed and hospitalized classes is established. Different from available models, the hospitalization rate is expressed as a function of the number of empty beds. The existence and stability of the equilibria are analyzed, and it is proved that the system undergoes backward bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation of codimension 2 under certain conditions by using the center manifold theory and normal form theory. In particular, our results show that there is a threshold value for the capacity of hospital beds in a region. If the capacity of hospital beds is lower than this threshold value, there will be a backward bifurcation, which means that even if the basic reproduction number, R0, is less than unity, it is not enough to prevent the outbreaks. Before taking disease control measures, one should compare the number of beds with the threshold value to avoid misjudgment and try to increase the capacity of hospital beds above this threshold value. The method to estimate the threshold value is also given. In addition, the impacts of the duration of the exposed period on the basic reproduction number and disease transmission are investigated.