Assessing the Time Evolution of COVID-19 Effective Reproduction Number in Brazil.

In this paper, we use a Bayesian method to estimate the effective reproduction number ( R ( t ) ), in the context of monitoring the time evolution of the COVID-19 pandemic in Brazil at different geographic levels. The focus of this study is to investigate the similarities between the trends in the evolution of such indicators at different subnational levels with the trends observed nationally. The underlying question addressed is whether national surveillance of such variables is enough to provide a picture of the epidemic evolution in the country or if it may hide important localized trends. This is particularly relevant in the scenario where health authorities use information obtained from such indicators in the design of non-pharmaceutical intervention policies to control the epidemic. A comparison between R ( t ) estimates and the moving average (MA) of daily reported infections is also presented, which is another commonly monitored variable. The analysis carried out in this paper is based on the data of confirmed infected cases provided by a public repository. The correlations between the time series of R ( t ) and MA in different geographic levels are assessed. Comparing national with subnational trends, higher degrees of correlation are found for the time series of R ( t ) estimates, compared to the MA time series. Nevertheless, differences between national and subnational trends are observed for both indicators, suggesting that local epidemiological surveillance would be more suitable as an input to the design of non-pharmaceutical intervention policies in Brazil, particularly for the least populated states.

A novel simulation-based analysis of a stochastic HIV model with the time delay using high order spectral collocation technique.

The economic impact of Human Immunodeficiency Virus (HIV) goes beyond individual levels and it has a significant influence on communities and nations worldwide. Studying the transmission patterns in HIV dynamics is crucial for understanding the tracking behavior and informing policymakers about the possible control of this viral infection. Various approaches have been adopted to explore how the virus interacts with the immune system. Models involving differential equations with delays have become prevalent across various scientific and technical domains over the past few decades. In this study, we present a novel mathematical model comprising a system of delay differential equations to describe the dynamics of intramural HIV infection. The model characterizes three distinct cell sub-populations and the HIV virus. By incorporating time delay between the viral entry into target cells and the subsequent production of new virions, our model provides a comprehensive understanding of the infection process. Our study focuses on investigating the stability of two crucial equilibrium states the infection-free and endemic equilibriums. To analyze the infection-free equilibrium, we utilize the LaSalle invariance principle. Further, we prove that if reproduction is less than unity, the disease free equilibrium is locally and globally asymptotically stable. To ensure numerical accuracy and preservation of essential properties from the continuous mathematical model, we use a spectral scheme having a higher-order accuracy. This scheme effectively captures the underlying dynamics and enables efficient numerical simulations.

Global analysis of an age-structured tuberculosis model with an application to Jiangsu, China.

Diagnostic delay for TB infected individuals and the lack of TB vaccines for adults are the main challenges to achieve the goals of WHO by 2050. In order to evaluate the impacts of diagnostic delay and vaccination for adults on prevalence of TB, we propose an age-structured model with latent age and infection age, and we incorporate Mycobacterium TB in the environment and vaccination into the model. Diagnostic delay is indicated by the age of infection before receiving treatment. The threshold dynamics are established in terms of the basic reproduction number R 0 . When R 0 < 1 , the disease-free equilibrium is globally asymptotically stable, which means that TB epidemic will die out; When R 0 = 1 , the disease-free equilibrium is globally attractive; there exists a unique endemic equilibrium and the endemic equilibrium is globally attractive when R 0 > 1 . We estimate that the basic reproduction number R 0 = 0.5320 (95% CI (0.3060, 0.7556)) in Jiangsu Province, which means that TB epidemic will die out. However, we find that the annual number of new TB cases by 2050 is 1,151 (95%CI: (138, 8,014)), which means that it is challenging to achieve the goal of WHO by 2050. To this end, we evaluate the possibility of achieving the goals of WHO if we start vaccinating adults and reduce diagnostic delay in 2025. Our results demonstrate that when the diagnostic delay is reduced from longer than four months to four months, or 20% adults are vaccinated, the goal of WHO in 2050 can be achieved, and 73,137 (95%CI: (23,906, 234,086)) and 54,828 (95%CI: (15,811, 206,468)) individuals will be prevented from being infected from 2025 to 2050, respectively. The modeling approaches and simulation results used in this work can help policymakers design control measures to reduce the prevalence of TB.

Evaluation of age-structured vaccination strategies for curbing the disease spread.

Age structure is one of the crucial factors in characterizing the heterogeneous epidemic transmission. Vaccination is regarded as an effective control measure for prevention and control epidemics. Due to the shortage of vaccine capacity during the outbreak of epidemics, how to design vaccination policy has become an urgent issue in suppressing the disease transmission. In this paper, we make an effort to propose an age-structured SVEIHR model with the disease-caused death to take account of dynamics of age-related vaccination policy for better understanding disease spread and control. We present an explicit expression of the basic reproduction number R 0 , which determines whether or not the disease persists, and then establish the existence and stability of endemic equilibria under certain conditions. Numerical simulations are illustrated to show that the age-related vaccination policy has a tremendous influence on curbing the disease transmission. Especially, vaccination of people over 65 is better than for people aged 21-65 in terms of rapid eradication of the disease in Italy.

Fractional epidemic model of coronavirus disease with vaccination and crowding effects.

Most of the countries in the world are affected by the coronavirus epidemic that put people in danger, with many infected cases and deaths. The crowding factor plays a significant role in the transmission of coronavirus disease. On the other hand, the vaccines of the covid-19 played a decisive role in the control of coronavirus infection. In this paper, a fractional order epidemic model (SIVR) of coronavirus disease is proposed by considering the effects of crowding and vaccination because the transmission of this infection is highly influenced by these two factors. The nonlinear incidence rate with the inclusion of these effects is a better approach to understand and analyse the dynamics of the model. The positivity and boundedness of the fractional order model is ensured by applying some standard results of Mittag Leffler function and Laplace transformation. The equilibrium points are described analytically. The existence and uniqueness of the non-integer order model is also confirmed by using results of the fixed-point theory. Stability analysis is carried out for the system at both the steady states by using Jacobian matrix theory, Routh-Hurwitz criterion and Volterra-type Lyapunov functions. Basic reproductive number is calculated by using next generation matrix. It is verified that disease-free equilibrium is locally asymptotically stable if R 0 < 1 and endemic equilibrium is locally asymptotically stable if R 0 > 1 . Moreover, the disease-free equilibrium is globally asymptotically stable if R 0 < 1 and endemic equilibrium is globally asymptotically stable if R 0 > 1 . The non-standard finite difference (NSFD) scheme is developed to approximate the solutions of the system. The simulated graphs are presented to show the key features of the NSFD approach. It is proved that non-standard finite difference approach preserves the positivity and boundedness properties of model. The simulated graphs show that the implementation of control strategies reduced the infected population and increase the recovered population. The impact of fractional order parameter α is described by the graphical templates. The future trends of the virus transmission are predicted under some control measures. The current work will be a value addition in the literature. The article is closed by some useful concluding remarks.

Modeling outbreaks of COVID-19 in China: The impact of vaccination and other control measures on curbing the epidemic.

This study aims to examine the development trend of COVID-19 in China and propose a model to assess the impacts of various prevention and control measures in combating the COVID-19 pandemic. Using COVID-19 cases reported by the National Health Commission of China from January 2, 2020, to January 2, 2022, we established a Susceptible-Exposed-Infected-Asymptomatic-Quarantined-Vaccinated-Hospitalized-Removed (SEIAQVHR) model to calculate the COVID-19 transmission rate and Rt effective reproduction number, and assess prevention and control measures. Additionally, we built a stochastic model to explore the development of the COVID-19 epidemic. We modeled the incidence trends in five outbreaks between 2020 and 2022. Some important features of the COVID-19 epidemic are mirrored in the estimates based on our SEIAQVHR model. Our model indicates that an infected index case entering the community has a 50%-60% chance to cause a COVID-19 outbreak. Wearing masks and getting vaccinated were the most effective measures among all the prevention and control measures. Specifically targeting asymptomatic individuals had no significant impact on the spread of COVID-19. By adjusting prevention and control parameters, we suggest that increasing the rates of effective vaccination and mask-wearing can significantly reduce COVID-19 cases in China. Our stochastic model analysis provides a useful tool for understanding the COVID-19 epidemic in China.

Global dynamics of discrete mathematical models of tuberculosis.

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R0 which is based on the disease-free equilibrium, and a new net reproduction number R0(E∗) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R0≤ 1 and unstable if R0>1. Moreover, the endemic equilibrium is locally asymptotically stable if R0(E∗)<1

Modelling the impact of human behavior using a two-layer Watts-Strogatz network for transmission and control of Mpox.

PURPOSE: This study aims to evaluate the effectiveness of mitigation strategies and analyze the impact of human behavior on the transmission of Mpox. The results can provide guidance to public health authorities on comprehensive prevention and control for the new Mpox virus strain in the Democratic Republic of Congo as of December 2023. METHODS: We develop a two-layer Watts-Strogatz network model. The basic reproduction number is calculated using the next-generation matrix approach. Markov chain Monte Carlo (MCMC) optimization algorithm is used to fit Mpox cases in Canada into the network model. Numerical simulations are used to assess the impact of mitigation strategies and human behavior on the final epidemic size. RESULTS: Our results show that the contact transmission rate of low-risk groups and susceptible humans increases when the contact transmission rate of high-risk groups and susceptible humans is controlled as the Mpox epidemic spreads. The contact transmission rate of high-risk groups after May 18, 2022, is approximately 20% lower than that before May 18, 2022. Our findings indicate a positive correlation between the basic reproduction number and the level of heterogeneity in human contacts, with the basic reproduction number estimated at 2.3475 (95% CI: 0.0749-6.9084). Reducing the average number of sexual contacts to two per week effectively reduces the reproduction number to below one. CONCLUSION: We need to pay attention to the re-emergence of the epidemics caused by low-risk groups when an outbreak dominated by high-risk groups is under control. Numerical simulations show that reducing the average number of sexual contacts to two per week is effective in slowing down the rapid spread of the epidemic. Our findings offer guidance for the public health authorities of the Democratic Republic of Congo in developing effective mitigation strategies.

Viral infection dynamics with immune chemokines and CTL mobility modulated by the infected cell density.

We study a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a standing assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells, while the diffusion rate of CTL is a decreasing function of the density of infected cells. We first establish the global existence and ultimate boundedness of the solution via a priori energy estimates. We then define the basic reproduction number of viral infection R 0 and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state E 0 is globally asymptotically stable if R 0 < 1 . When R 0 > 1 , then E 0 becomes unstable, and another basic reproduction number of CTL response R 1 becomes the dynamic threshold in the sense that if R 1 < 1 , then the CTL-inactivated steady state E 1 is globally asymptotically stable; and if R 1 > 1 , then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state E 2 is globally asymptotically stable. To establish the global stability results, we need to prove point dissipativity, obtain uniform persistence, construct suitable Lyapunov functions, and apply the LaSalle invariance principle.

Global dynamics of a time-delayed nonlocal reaction-diffusion model of within-host viral infections.

In this paper, we study a time-delayed nonlocal reaction-diffusion model of within-host viral infections. We introduce the basic reproduction number R 0 and show that the infection-free steady state is globally asymptotically stable when R 0 ≤ 1 , while the disease is uniformly persistent when R 0 > 1 . In the case where all coefficients and reaction terms are spatially homogeneous, we obtain an explicit formula of R 0 and the global attractivity of the positive constant steady state. Numerically, we illustrate the analytical results, conduct sensitivity analysis, and investigate the impact of drugs on curtailing the spread of the viruses.

Impacts of optimal control strategies on the HBV and COVID-19 co-epidemic spreading dynamics.

Different cross-sectional and clinical research studies investigated that chronic HBV infected individuals' co-epidemic with COVID-19 infection will have more complicated liver infection than HBV infected individuals in the absence of COVID-19 infection. The main objective of this study is to investigate the optimal impacts of four time dependent control strategies on the HBV and COVID-19 co-epidemic transmission using compartmental modeling approach. The qualitative analyses of the model investigated the model solutions non-negativity and boundedness, calculated all the models effective reproduction numbers by applying the next generation operator approach, computed all the models disease-free equilibrium point (s) and endemic equilibrium point (s) and proved their local stability, shown the phenomenon of backward bifurcation by applying the Center Manifold criteria. By applied the Pontryagin's Maximum principle, the study re-formulated and analyzed the co-epidemic model optimal control problem by incorporating four time dependent controlling variables. The study also carried out numerical simulations to verify the model qualitative results and to investigate the optimal impacts of the proposed optimal control strategies. The main finding of the study reveals that implementation of protections, COVID-19 vaccine, and treatment strategies simultaneously is the most effective optimal control strategy to tackle the HBV and COVID-19 co-epidemic spreading in the community.

Stability analysis of a SAIR epidemic model on scale-free community networks.

The presence of asymptomatic carriers, often unrecognized as infectious disease vectors, complicates epidemic management, particularly when inter-community migrations are involved. We introduced a SAIR (susceptible-asymptomatic-infected-recovered) infectious disease model within a network framework to explore the dynamics of disease transmission amid asymptomatic carriers. This model facilitated an in-depth analysis of outbreak control strategies in scenarios with active community migrations. Key contributions included determining the basic reproduction number, $ R_0 $, and analyzing two equilibrium states. Local asymptotic stability of the disease-free equilibrium is confirmed through characteristic equation analysis, while its global asymptotic stability is investigated using the decomposition theorem. Additionally, the global stability of the endemic equilibrium is established using the Lyapunov functional theory.

Analysis and comparative study of a deterministic mathematical model of SARS-COV-2 with fractal-fractional operators: a case study.

In this paper, we investigate a fractal-fractional-order mathematical model with the influence of hospitalized patients and the impact of vaccination with fractal-fractional operators. The respective derivatives are considered in the Caputo, Caputo Fabrizio, and Atangana-Baleanu senses of fractional order α and fractal dimension τ . For the proposed problem, some results regarding basic reproduction number and stability are given. Using the next-generation matrix approach, we have investigated the global and local stability of several types of equilibrium points. We provide a detailed analysis of the existence and uniqueness of the solution. Moreover, we fit the model with the real data of Pakistan from June 01, 2020, till March 24, 2021. Then, we use the fractal-fractional derivative to find a numerical solution for the model. MATLAB software is used for numerical illustration. Graphical presentations corresponding to different parameteric values are given as well.

R 0 May Not Tell Us Everything: Transient Disease Dynamics of Some SIR Models Over Patchy Environments.

This paper examines the short-term or transient dynamics of SIR infectious disease models in patch environments. We employ reactivity of an equilibrium and amplification rates, concepts from ecology, to analyze how dispersals/travels between patches, spatial heterogeneity, and other disease-related parameters impact short-term dynamics. Our findings reveal that in certain scenarios, due to the impact of spatial heterogeneity and the dispersals, the short-term disease dynamics over a patch environment may disagree with the long-term disease dynamics that is typically reflected by the basic reproduction number. Such an inconsistence can mislead the public, public healthy agencies and governments when making public health policy and decisions, and hence, these findings are of practical importance.

Final epidemic size of a two-community SIR model with asymmetric coupling.

Communities are commonly not isolated but interact asymmetrically with each other, allowing the propagation of infectious diseases within the same community and between different communities. To reveal the impact of asymmetrical interactions and contact heterogeneity on disease transmission, we formulate a two-community SIR epidemic model, in which each community has its contact structure while communication between communities occurs through temporary commuters. We derive an explicit formula for the basic reproduction number R 0 , give an implicit equation for the final epidemic size z, and analyze the relationship between them. Unlike the typical positive correlation between R 0 and z in the classic SIR model, we find a negatively correlated relationship between counterparts of our model deviating from homogeneous populations. Moreover, we investigate the impact of asymmetric coupling mechanisms on R 0 . The results suggest that, in scenarios with restricted movement of susceptible individuals within a community, R 0 does not follow a simple monotonous relationship, indicating that an unbending decrease in the movement of susceptible individuals may increase R 0 . We further demonstrate that network contacts within communities have a greater effect on R 0 than casual contacts between communities. Finally, we develop an epidemic model without restriction on the movement of susceptible individuals, and the numerical simulations suggest that the increase in human flow between communities leads to a larger R 0 .

Transmission dynamics of symptom-dependent HIV/AIDS models.

In this study, we proposed two, symptom-dependent, HIV/AIDS models to investigate the dynamical properties of HIV/AIDS in the Fujian Province. The basic reproduction number was obtained, and the local and global stabilities of the disease-free and endemic equilibrium points were verified to the deterministic HIV/AIDS model. Moreover, the indicators $ R_0/ $ and $ R_0^e $ were derived for the stochastic HIV/AIDS model, and the conditions for stationary distribution and stochastic extinction were investigated. By using the surveillance data from the Fujian Provincial Center for Disease Control and Prevention, some numerical simulations and future predictions on the scale of HIV/AIDS infections in the Fujian Province were conducted.

The role of long-lived plasma cells in viral clearance.

The adaptive immune system has two types of plasma cells (PC), long-lived plasma cells (LLPC) and short-lived plasma cells (SLPC), that differ in their lifespan. In this paper, we propose that LLPC is crucial to the clearance of viral particles in addition to reducing the viral basic reproduction number in secondary infections. We use a sequence of within-host mathematical models to show that, CD8 T cells, SLPC and memory B cells cannot achieve full viral clearance, and the viral load will reach a low positive equilibrium level because of a continuous replenishment of target cells. However, the presence of LLPC is crucial for viral clearance.

Mathematical modelling, analysis and numerical simulation of social media addiction and depression.

We formulate a mathematical model of social media addiction and depression (SMAD) in this study. Key aspects, such as social media addiction and depression disease-free equilibrium point (SMADDFEP), social media addiction and depression endemic equilibrium point (SMADEEP), and basic reproduction number (R0), have been analyzed qualitatively. The results indicate that if R0 < 1, the SMADDFEP is locally asymptotically stable. The global asymptotic stability of the SMADDFEP has been established using the Castillo-Chavez theorem. On the other hand, if R0 > 1, the unique endemic equilibrium point (SMADEEP) is locally asymptotically stable by Lyapunov theorem, and the model exhibits a forward bifurcation at R0 = 1 according to the Center Manifold theorem. To examine the model's sensitivity, we calculated the normalized forward sensitivity index and conducted a Partial Rank Correlation Coefficient (PRCC) analysis to describe the influence of parameters on the SMAD. The numerical results obtained using the Fourth-order Runge-Kutta (RK-4) scheme show that increasing the number of addicted individuals leads to an increase in the number of depressed individuals.

On cognitive epidemic models: spatial segregation versus nonpharmaceutical interventions.

Fick's law and the Fokker-Planck law of diffusion are applied to manifest the cognitive dispersal of individuals in two reaction-diffusion SEIR epidemic models, where the disease transmission is illustrated by nonlocal infection mechanisms in heterogeneous environments. Building upon the well-posedness of solutions, threshold dynamics are discussed in terms of the basic reproduction numbers for the two cognitive epidemic models. The numerical investigation reveals that the Fokker-Planck law can better describe the diffusion of individuals by taking different dispersal strategies of exposed individuals in our cognitive epidemic models, and provides some insights on spatial segregation and nonpharmaceutical interventions: (i) spatial segregation occurs in the random diffusion model when the nonlocal infection radius is small, while it appears in the symmetric diffusion model when the radius is large; (ii) nonpharmaceutical interventions on restricting the dispersal of exposed and infected individuals do not contribute to reducing the infection proportion, but rather eliminate the disease in a region, which expands as the nonlocal infection radius increases. We additionally find that the final infection size in the random diffusion model is significantly smaller than that in the symmetric diffusion model and decreases as the nonlocal infection radius increases.

The role of natural recovery category in malaria dynamics under saturated treatment.

In the process of malaria transmission, natural recovery individuals are slightly infectious compared with infected individuals. Our concern is whether the infectivity of natural recovery category can be ignored in areas with limited medical resources, so as to reveal the epidemic pattern of malaria with simpler analysis. To achieve this, we incorporate saturated treatment into two-compartment and three-compartment models, and the infectivity of natural recovery category is only reflected in the latter. The non-spatial two-compartment model can admit backward bifurcation. Its spatial version does not possess rich dynamics. Besides, the non-spatial three-compartment model can undergo backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. For spatial three-compartment model, due to the complexity of characteristic equation, we apply Shengjin's Distinguishing Means to realize stability analysis. Further, the model exhibits Turing instability, Hopf bifurcation and Turing-Hopf bifurcation. This makes the model may admit bistability or even tristability when its basic reproduction number less than one. Biologically, malaria may present a variety of epidemic trends, such as elimination or inhomogeneous distribution in space and periodic fluctuation in time of infectious populations. Notably, parameter regions are given to illustrate substitution effect of two-compartment model for three-compartment model in both scenarios without or with spatial movement. Finally, spatial three-compartment model is used to present malaria transmission in Burundi. The application of efficiency index enables us to determine the most effective method to control the number of cases in different scenarios.