Reproduction Number R0 Research Articles

Dynamics of a Fokker–Planck type diffusion epidemic model with general incidence and relapse

In this paper, a spatially heterogeneous SIRI epidemic model with general morbidity is studied, which conforms to the Fokker–Planck type diffusion law. The basic reproduction number R0 of the model is introduced, and the threshold dynamics based on R0 is discussed. In particular, with the help of elliptic eigenvalue theory, we determine the asymptotic profiles of the endemic equilibrium when the diffusion rate of susceptible individuals approaches zero or infinity.

Explosive epidemic transitions induced by quarantine fatigue

Quarantine measures are one of the first lines of defense against the spread of infectious diseases. However, maintaining these measures over extended periods can be challenging due to a phenomenon known as quarantine fatigue. In this paper, we investigate the impact of quarantine fatigue on the spread of infectious diseases by using an epidemic model on random networks with cliques. In our model, susceptible individuals can be quarantined up to n times, after which they stop complying with quarantine orders due to fatigue. Our results show that quarantine fatigue may induce a regime in which increasing the probability of detecting and isolating infected individuals (along with their close contacts) could subsequently increase the expected number of cases at the end of an outbreak. Moreover, we observe that quarantine fatigue can trigger an abrupt phase transition at the critical reproduction number R0=1. Finally, we explore a scenario where a non-negligible number of individuals are infected at the beginning of an epidemic, and our results show that, depending on the value of n, an abrupt transition between a controlled epidemic and a large epidemic event can occur for R0<1.

LOCAL AND GLOBAL STABILITY ANALYSIS OF MEASLES EPIDEMIC MODEL AT DISEASE-FREE EQUILIBRIUM

In this study, a continuous mathematical model for the dynamics of Measles (rubeola) outbreak at constant recruitment rate was formulated. In the model, we partitioned the population into Susceptible (S), Vaccinated (V), exposed (E), Infected (I) and recovered (R) individuals. We analyzed a SVEIR compartmental nonlinear deterministic mathematical model of measles epidemic in a community with constant population. Analytical studies were carried out on the model using the method of linearized stability. The basic reproductive number R0 that governs the disease transmission is obtained from the largest eigenvalue of the next-generation matrix. The disease-free equilibrium is computed and proved to be locally and globally asymptotically stable if R0&lt;1 and unstable if R0 &gt;1 respectively. Finally, we simulate the model system in MATLAB and obtained the graphical behavior of each compartment. From the simulation, we observed that the measles infection was eradicated in the environment when R0&lt;1.

Analyzing dynamics and stability of single delay differential equations for the dengue epidemic model

This paper introduces a mathematical model that simulates the transmission of the dengue virus in a population over time. The model takes into account aspects such as delays in transmission, the impact of inhibitory effects, the loss of immunity, and the presence of partial immunity. The model has been verified to ensure the positivity and boundedness. The basic reproduction number R0 of the model is derived using the advanced next-generation matrix approach. An analysis is conducted on the stability criteria of the model, and equilibrium points are investigated. Under appropriate circumstances, it was shown that there is local stability in both the virus-free equilibrium and the endemic equilibrium points when there is a delay. Analyzing the global asymptotic stability of equilibrium points is done by using the appropriate Lyapunov function. In addition, the model exhibits a backward bifurcation, in which the virus-free equilibrium coexists with a stable endemic equilibrium. By using a sensitivity analysis technique, it has been shown that some factors have a substantial influence on the behavior of the model. The research adeptly elucidates the ramifications of its results by effortlessly validating theoretical concepts with numerical examples and simulations. Furthermore, our research revealed that augmenting the rate of inhibition on infected vectors and people leads to a reduction in the equilibrium point, suggesting the presence of an endemic 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.

Estimation of Parameters for the Mathematical Model of the Spread of Hepatitis B in Burkina Faso Using Grey Wolf Optimizer

In this paper, we developed a mathematical model of differential susceptibility, taking into account vaccination and treatment, to simulate the transmission of the hepatitis B virus in the population of Burkina Faso. The existence and uniqueness of non-negative solutions are proved. The model has a globally asymptotically stable disease free equilibrium when the basic reproduction number R0 &lt; 1 and an endemic equilibrium when R0 &gt; 1. We estimated the parameters of the model based on hepatitis B cases from 1997 to 2020 by using a Grey Wolf Optimizer Algorithm (GWO). The results demonstrated the efficacy of the GWO algorithm in estimating the model parameters. A sensitivity analysis was carried out to determine the determining factors in the spread of hepatitis B in Burkina Faso. The estimated parameters were used to simulate the spread of hepatitis B in Burkina Faso from 1997 to 2020.

Impact of infectious density-induced additional screening and treatment saturation on COVID-19: Modeling and cost-effective optimal control

This study introduces a novel SI2HR model, where “I2” denotes two infectious classes representing asymptomatic and symptomatic infections, aiming to investigate and analyze the cost-effective optimal control measures for managing COVID-19. The model incorporates a novel concept of infectious density-induced additional screening (IDIAS) and accounts for treatment saturation. Furthermore, the model considers the possibility of reinfection and the loss of immunity in individuals who have previously recovered. To validate and calibrate the proposed model, real data from November–December 2022 in Hong Kong are utilized. The estimated parameters obtained from this calibration process are valuable for prediction purposes and facilitate further numerical simulations. An analysis of the model reveals that delays in screening, treatment, and quarantine contribute to an increase in the basic reproduction number R0, indicating a tendency towards endemicity. In particular, from the elasticity of R0, we deduce that normalized sensitivity indices of baseline screening rate (θ), quarantine rates (γ, αs), and treatment rate (α) are negative, which shows that delaying any of these may cause huge surge in R0, ultimately increases the disease burden. Further, by the contour plots, we note the two-parameter behavior of the infectives (both symptomatic and asymptomatic). Expanding upon the model analysis, an optimal control problem (OCP) is formulated, incorporating three control measures: precautionary interventions, boosted IDIAS, and boosted treatment. The Pontryagin's maximum principle and the forward-backward sweep method are employed to solve the OCP. The numerical simulations highlight that enhanced screening and treatment, coupled with preventive interventions, can effectively contribute to sustainable disease control. However, the cost-effectiveness analysis (CEA) conducted in this study suggests that boosting IDIAS alone is the most economically efficient and cost-effective approach compared to other strategies. The CEA results provide valuable insights into identifying specific strategies based on their cost-efficacy ranking, which can be implemented to maximize impact while minimizing costs. Overall, this research offers significant insights for policymakers and healthcare professionals, providing a framework to optimize control efforts for COVID-19 or similar epidemics in the future.

Role of ART and PrEP treatments in a stochastic HIV/AIDS epidemic model

In this paper, a stochastic HIV/AIDS epidemic model is presented to study the synthetic effect of ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments among MSM ( men who have sex with men). Firstly, we give the global stability of disease-free equilibrium and the endemic equilibrium in terms of basic reproduction number R0 for deterministic model. And then the existence of global positive solutions and the existence of unique ergodic stationary distribution under R0S>1 for stochastic model are given. Further, the long-time stochastic dynamic of the model is investigated, including the criteria on the extinction and persistence in mean for the stochastic model. Finally, we give some numerical simulations to illustrate our theoretical results, and the sensitive analysis shows that ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments can effectively control the spread of AIDS among MSM population.

Mathematical model with sensitivity analysis and control strategies for marijuana consumption

The current study focuses on reducing the use of marijuana in the population. Marijuana seems to represent a serious threat to public health in developing countries since it is an illegal narcotic with various harmful health effects. The novelty of this manuscript is to modify the non-user, experimental users, recreational users, and addict's (NERA) model for marijuana consumption by incorporating a new class of addicted users under treatment, called the hospitalized class. This real-world issue is expressed in mathematical terms using first-order non-liner ordinary differential equations, and as a result, a mathematical model for marijuana consumption is established. The general population is divided into two main groups: marijuana users and non-users. Subsequently, four subgroups of abusers are created, each reflecting a different stage or degree of substance misuse. The invariant region and basic reproduction number R0 are those parts that are used for the validation of the proposed modified model and to find out the initial transmission rate of marijuana smoking in the general population, respectively. The essential factors influencing cannabis growth were identified based on the results of sensitivity analysis. Additionally, strategies were constructed in the form of prevention objectives to prevent individuals from experiencing increasing effects of marijuana. The effectiveness of these control strategies was subsequently confirmed through numerical simulations, and the numerical simulations were carried out with the help of MATLAB using the RK-4 method.

Modelling the unexpected dynamics of COVID-19 in Manaus, Brazil

In late March 2020, SARS-CoV-2 arrived in Manaus, Brazil, and rapidly developed into a large-scale epidemic that collapsed the local health system and resulted in extreme death rates. Several key studies reported that ∼76% of residents of Manaus were infected (attack rate AR≃76%) by October 2020, suggesting protective herd immunity had been reached. Despite this, an unexpected second wave of COVID-19 struck again in November and proved to be larger than the first, creating a catastrophe for the unprepared population. It has been suggested that this could be possible if the second wave was driven by reinfections. However, it is widely reported that reinfections were at a low rate (before the emergence of Omicron), and reinfections tend to be mild. Here, we use novel methods to model the epidemic from mortality data without considering reinfection-caused deaths and evaluate the impact of interventions to explain why the second wave appeared. The method fits a “flexible” reproductive number R0(t) that changes over the epidemic, and it is demonstrated that the method can successfully reconstruct R0(t) from simulated data. For Manaus, the method finds AR≃34% by October 2020 for the first wave, which is far less than required for herd immunity yet in-line with seroprevalence estimates. The work is complemented by a two-strain model. Using genomic data, the model estimates transmissibility of the new P.1 virus lineage as 1.9 times higher than that of the non-P.1. Moreover, an age class model variant that considers the high mortality rates of older adults show very similar results. These models thus provide a reasonable explanation for the two-wave dynamics in Manaus without the need to rely on large reinfection rates, which until now have only been found in negligible to moderate numbers in recent surveillance efforts.

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.

Wording the trajectory of the three-year COVID-19 epidemic in a general population – Belgium

The trajectory of COVID-19 epidemic waves in the general population of Belgium was analysed by defining quantitative criteria for epidemic waves from March 2020 to early 2023. Peaks and starting/ending times characterised nine waves numerated I to IX based on the daily reported incidence number (symbol INCID) and three “endemic” interval periods between the first four waves. The SIR compartmental model was applied to the first epidemic wave by fitting the daily prevalence pool (symbol I) calculated as the sum of the daily incidence rate and estimated number of subjects still infectious from the previous days. The basic reproductive number R0 was calculated based on the exponential growth rate during the early phase and on medical literature knowledge of the time of generation of SARS-CoV-2 infection. The first COVID-19 wave was well fitted by an open SIR model. According to this approach, dampened recurrent epidemic waves evolving through an endemic state would have been expected. This was not the case with the subsequent epidemic waves being characterised by new variants of concern (VOC). Evidence-based observations: 1) each epidemic wave affected less than a fifth of the general population; 2) the Vth epidemic wave (VOC Omicron) presented the greatest amplitude. The lack of recurrence of the same VOC during successive epidemic waves strongly suggests that a VOC has a limited persistence, disappearing from the population well before the expected proportion of the theoretical susceptible cohort being maximally infected. Fitting the theoretical SIR model, a limited persistence of VOCs in a population could explain that new VOCs replace old ones, even if the new VOC has a lower transmission rate than the preceding one. In conclusion, acquisition of potential defective mutations in VOC during an epidemic wave is a potential factor explaining the absence of resurgence of a same VOC during successive waves. Such an hypothesis is open to discussion and to rebuttal. A modified SIR model with epidemic waves of variable amplitude related not only to R0 and public health measures but also to acquisition of defective fitting in virus within a population should be tested.

Existence of Traveling Waves of a Diffusive Susceptible–Infected–Symptomatic–Recovered Epidemic Model with Temporal Delay

The aim of this article is to investigate the existence of traveling waves of a diffusive model that represents the transmission of a virus in a determined population composed of the following populations: susceptible (S), infected (I), asymptomatic (A), and recovered (R). An analytical study is performed, where the existence of solutions of traveling waves in a bounded domain is demonstrated. We use the upper and lower coupled solutions method to achieve this aim. The existence and local asymptotic stability of the endemic (Ee) and disease-free (E0) equilibrium states are also determined. The constructed model includes a discrete-time delay that is related to the incubation stage of a virus. We find the crucial basic reproduction number R0, which determines the local stability of the steady states. We perform numerical simulations of the model in order to provide additional support to the theoretical results and observe the traveling waves. The model can be used to study the dynamics of SARS-CoV-2 and other viruses where the disease evolution has a similar behavior.

Effectiveness evaluation of mosquito suppression strategies on dengue transmission under changing temperature and precipitation

Widespread resurgence of dengue outbreaks has seriously threatened the global health. Due to lack of treatments and vaccines, one key strategy in dengue control is to reduce the vector population size. As an environment-friendly mosquito control approach, releasing male mosquitoes transinfected with specific Wolbachia strain into the field to suppress the wild mosquito population size has become wildly accepted. The current study evaluates the effectiveness of this suppression strategy on dengue control under changing temperature and precipitation profiles. We formulate a mathematical model which includes larval intra-specific competition, the maturation period for mosquitoes, the extrinsic incubation period (EIP) and intrinsic incubation period (IIP). The persistence of mosquitoes and disease is discussed in terms of two basic reproduction numbers (RM and R0) and the release ratio pw. Further numerical simulations are carried out to not only validate theoretical results, but also provide interesting quantitative observations. Sensitivity analysis on the reproduction numbers, peak size, peak time and the final epidemic size is performed with respect to model parameters, which highlights effective control measures against dengue transmission. Moreover, by assuming temperature and precipitation dependent mosquito-related parameters, the model can be used to project the effectiveness of releasing Wolbachia-carrying males under climatic variations. It is shown that the effectiveness of various control strategies is highly dependent on the changing temperature and precipitation profiles. In particular, the model projects that it is most challenging to control the disease at the favorable temperature (around 27∼30∘C) and precipitation (5∼8mm/day) range, during which the basic reproduction number R0 is very high and more Wolbachia-infected males should be released.

Analysis of a Delayed Multiscale AIDS/HIV-1 Model Coupling Between-Host and Within-Host Dynamics

Taking into account the effects of the immune response and delay, and complexity on HIV-1 transmission, a multiscale AIDS/HIV-1 model is formulated in this paper. The multiscale model is described by a within-host fast time model with intracellular delay and immune delay, and a between-host slow time model with latency delay. The dynamics of the fast time model is analyzed, and includes the stability of equilibria and properties of Hopf bifurcation. Further, for the coupled slow time model without an immune response, the basic reproduction number R0h is defined, which determines whether the model may have zero, one, or two positive equilibria under different conditions. This implies that the slow time model demonstrates more complex dynamic behaviors, including saddle-node bifurcation, backward bifurcation, and Hopf bifurcation. For the other case, that is, the coupled slow time model with an immune response, the threshold dynamics, based on the basic reproduction number R˜0h, is rigorously investigated. More specifically, if R˜0h&lt;1, the disease-free equilibrium is globally asymptotically stable; if R˜0h&gt;1, the model exhibits a unique endemic equilibrium that is globally asymptotically stable. With regard to the coupled slow time model with an immune response and stable periodic solution, the basic reproduction number R0 is derived, which serves as a threshold value determining whether the disease will die out or lead to periodic oscillations in its prevalence. The research results suggest that the disease is more easily controlled when hosts have an extensive immune response and the time required for new immune particles to emerge in response to antigenic stimulation is within a certain range. Finally, numerical simulations are presented to validate the main results and provide some recommendations for controlling the spread of HIV-1.

A fractional modeling approach to a new Hepatitis B model in light of asymptomatic carriers, vaccination and treatment

Hepatitis B virus (HBV) is a much more threatening and prolonged disease than hepatitis A. It can start as an acute disease, but in roughly 5 to 10% of situations, it can progress to a chronic condition that permanently damages the liver. After getting or coming into contact with the Hepatitis B virus (HBV), the said symptoms normally last from 10 days to 6 months. In the present paper, to have a firm understanding of the dynamics of HBV, we derived a new fractional model in the Caputo sense with asymptomatic carriers, vaccination, and treatment classes. We obtain the basic reproduction number R0 for the diagnosis and recurrence of the disease. We determine the stability of the proposed model locally and globally for R0 values less than and greater than 1. We use the Lyapunov function theory in fractional environments to establish the global stability of the fractional Hepatitis B virus (HBV) model. We demonstrate the existence and uniqueness of the fractional-order model. To look at the fractional order model’s solution graphically, we used a useful numerical strategy to get more detailed numerical results for the factors that have a big effect on getting rid of diseases. Additionally, the inclusion of the treatment class in the population allows for the determination of the impact of alternative medicines used to treat infected populations. Numerical simulation of the proposed fractional order (FO) Caputo model is performed, allowing for the analysis of graphical representations and the significance of fractional order derivatives to illustrate the impact of our theoretical findings.

Mathematical Analysis for a Zika Virus Dynamics in a Seasonal Environment

We propose a mathematical model for the Zika virus (ZIKV) spread under the influence of a seasonal environment. The basic reproduction number R0 was calculated for both cases, the fixed and seasonal environment permitting the characterisation of the extinction and the persistence of the disease for both cases. We proved that the virus-free steady state is globally asymptotically stable if R0≤1, while the disease will be persist if R0&gt;1. Finally, extensive numerical simulations are given to confirm the theoretical findings.

Modelling the preventive treatment under media impact on tuberculosis: A comparison in four regions of China

Preventive treatment for people with latent Tuberculosis infection (LTBI) has aroused our great interest. In this paper, we propose and analyze a novel mathematical model of TB considering preventive treatment with media impact. The basic reproduction number R0 is defined by the next generation matrix method. In the case without media impact, we prove that the disease-free equilibrium is globally asymptotically stable (unstable) if R0<1(R0>1). Furthermore, we obtain that a unique endemic equilibrium exists when R0>1, which is globally asymptotically stable in the case of permanent immunity and no media impact. We fit the model to the newly reported TB cases data from 2009 to 2019 of four regions in China and estimate the parameters. And we estimated R0=0.5013<1 in Hubei indicating that TB in Hubei will be eliminated in the future. However, the estimated R0=1.015>1 in Henan, R0=1.282>1 in Jiangxi and R0=1.930>1 in Xinjiang imply that TB will continue to persist in these three regions without further prevention and control measures. Besides, sensitivity analysis is carried out to illustrate the role of model parameters for TB control. Our finding reveals that appropriately improving the rate of timely treatment for actively infected people and increasing the rate of individuals with LTBI seeking preventive treatment could achieve the goal of TB elimination. In addition, another interesting finding shows that media impact can only reduce the number of active infections to a limited extent, but cannot change the prevalence of TB.

Stability Analysis of SIRS Model considering Pulse Vaccination and Elimination Disturbance

It is well known that many natural phenomena and human activities do exhibit impulsive effects in the fields of epidemiology. At the same time, compared with a single control strategy, it is obvious that the multiple control strategies are more beneficial to restrain the spread of infectious diseases. In this paper, we consider pulse vaccination and pulse elimination strategies at the same time and establish an SIRS epidemic model with standard incidence. Firstly, according to the stroboscopic mapping method of the discrete dynamical system, the disease-free T periodic solution of the model under the condition of pulse vaccination and pulse elimination is obtained. Secondly, the basic reproductive number R0 is defined, and the local asymptotic stability of the disease-free T periodic solution is proved by Floquet theory for R0&lt;1. Finally, based on the impulsive differential inequality theory, the global asymptotic stability of the disease-free T periodic solution is given for R0&lt;1, and the disease dies out eventually. The results show that in order to stop the disease epidemic, it is necessary to choose the appropriate vaccination rate and elimination rate and the appropriate impulsive period.

Dynamics and optimal control for a spatial heterogeneity model describing respiratory infectious diseases affected by air pollution

Air pollution has a serious impact on the spread of respiratory infectious diseases. However, the concentration of air pollution is not the same in different places. In order to characterize the spatial heterogeneity and the diffusion of pollutants, this paper proposes a spatial heterogeneity model, where the transmission rate is closely related to air pollution. The dynamics are established in terms of the basic reproduction number R0: if R0≤1, the disease-free equilibrium is globally asymptotically stable; if R0>1, there exists at least one endemic equilibrium and the disease is uniformly persistent. Our findings suggest that spatial heterogeneity, increased pollutants emission, and limited movement of infected individuals will all contribute to the spread of the respiratory infectious diseases. Subsequently, the corresponding strategies are introduced into the model to control the prevalence of the diseases. Under the minimization of the objective function, the necessary condition for the optimal control is acquired with the help of Pontryagin’s maximum principle. Numerical simulations are applied to validate the theoretical results of threshold dynamics and to demonstrate the influences of inflow and clearance rates of pollutants on threshold dynamics and optimal control.