Search

An epidemic outbreak largely disrupts supply chains (SCs) worldwide through plummeting business confidence, especially when it becomes a pandemic; its unpredictable re-emergence and spreadability may lead to inappropriate decision-making, in turn causing severe economic shocks. In March 2020, the coronavirus disease 2019 (COVID-19) outbreak attained a pandemic level, and many millions of cases were confirmed globally. Many countries reported an increasing number of active cases and formulated long-term lockdown guidelines, which resulted in an unexpected disruption of SCs. A key challenge in this scenario is that the rising number of confirmed COVID-19 cases does not necessarily reflect the already infected or asymptomatic cases. It is thus critical to understand the impact of asymptomatic carriers on the SC, as they may be the key driver of the novel virus spread, disrupting long-term SCs. This paper generalised the susceptible-exposed-infected-recovered (S-E-I-R) approach to create a mathematical model for which the impact of a proposed asymptomatic situation on the SC is evaluated through the basic reproduction number (R0), considered the main driver of SC disruption and the equilibrium status of infection over time. This paper presents an action plan for reducing disruption in the SC based on the R0 of the model. Overall, the current study as validated through a case study suggests that the asymptomatic-situation-based model is more convenient for critically understanding as well as forecasting the outbreak’s impact on SCs. This study also highlights different perspectives of SCs for managing such types of pandemics using modelling approaches. [ABSTRACT FROM AUTHOR]

In this paper, following previous works of ours, we deal with a mathematical model of hepatitis B virus (HBV) infection. We assume spatial diffusion of free HBV particles, logistic growth for both healthy and infected hepatocytes, and use the standard incidence function for viral infection. Moreover, one time delay is introduced to account for actual virus production. Another time delay is used to account for virus maturation. The existence, uniqueness, positivity and boundedness of solutions are established. Analyzing the model qualitatively and using a Lyapunov functional, we establish the existence of a threshold $ \mathcal {T}_0 $ T 0 such that, if the basic reproduction number $ \mathcal {R}_0 $ R 0 verifies $ \mathcal {R}_0\leq \mathcal {T}_0{ \lt }1 $ R 0 ≤ T 0 < 1 , the infection-free equilibrium is globally asymptotically stable. When $ \mathcal {R}_0 $ R 0 is greater than one, we discuss the local asymptotic stability of the unique endemic equilibrium and the occurrence of a Hopf bifurcation. Also, when $ \mathcal {R}_0 $ R 0 is greater than one, the system is uniformly persistent, which means that the HBV infection is endemic. Finally, we carry out some relevant numerical simulations to clarify and interpret the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction–diffusion dengue model on a varying domain that monotonically increases in time and gradually approaches saturation arising from environmental change. By the upper and lower solutions, comparison principle, asymptotic autonomous semiflows and the technique of Lyapunov function, we investigate the stabilities of equilibria in terms of the basic reproduction number $ \Re _0^{\rho } $ ℜ 0 ρ . The results show that (i) if $ \Re _0^{\rho } \gt 1 $ ℜ 0 ρ > 1 , the nontrivial solutions starting from the upper and lower solutions of the model approach to the set formulated by the maximal and minimal solutions of its related elliptic problem; (ii) the disease-free equilibrium is globally asymptotically stable when $ \Re _0^{\rho } \lt 1 $ ℜ 0 ρ < 1. Comparing our problem in different settings including growing domain, fixed domain and without spatial structure, our results demonstrate that the disease can spread in the growing domain, while vanish in the fixed domain; and the spatial model decreases the transmission risk compared with the system without spatial structure. [ABSTRACT FROM AUTHOR]

In this paper, we investigated a new heroin–cocaine epidemic model which incorporates spatial heterogeneity and nonlinear incidence rate. The main project of this paper is to explore the threshold dynamics in terms of the basic reproduction number $ \mathcal {R}_0 $ R 0 , which was defined by applying the next-generation operator. The threshold type results shown that if $ \mathcal {R}_0 \lt 1 $ R 0 < 1 , then the drug-free steady state is globally asymptotically stable. If $ \mathcal {R}_0 \gt 1 $ R 0 > 1 , then heroin–cocaine spread is uniformly persistent. Furthermore, the globally asymptotic stability of the drug-free steady state has been established for the critical case of $ \mathcal {R}_0=1 $ R 0 = 1 by analysing the local asymptotic stability and global attractivity. [ABSTRACT FROM AUTHOR]

This article is a dedication to Professor Jim M. Cushing, a respected mathematician and researcher in the field of difference equations and their applications to population dynamics. Professor Cushing has had a prolific career spanning over 47 years, during which he has made foundational contributions to the field. His work includes advancements in asymptotic theory, bifurcation dynamics, and the application of difference equations to various biologically significant settings. Professor Cushing has also been a mentor to many students, fostering inclusiveness and mathematical rigor. His research has had a profound impact on the field and has inspired many researchers. [Extracted from the article]

Several efforts have been recently devoted to the studies on epidemic mathematical models based on fractional-order operators, by virtue of their capability to take into account memory effects and nonlocal features. The aim of this paper is to make a contribution to the topic by introducing a novel Covid-19 model described by non-integer-order difference equations. By conducting a stability analysis, the paper shows that the conceived system has two fixed points at most, i.e. a disease-free fixed point and an endemic fixed point. In particular, a theorem is proved, which assures the global stability of the disease-free fixed point, indicating that the pandemic will disappear when a simple condition on the system parameters is satisfied. Finally, simulation results are carried out with the aim to highlight the capability of the conceived approach. [ABSTRACT FROM AUTHOR]

HIV/AIDS-COVID-19 co-infection is a major public health concern especially in developing countries of the world. This paper presents HIV/AIDS-COVID-19 co-infection to investigate the impact of interventions on its transmission using ordinary differential equation. In the analysis of the model, the solutions are shown to be non-negative and bounded, using next-generation matrix approach the basic reproduction numbers are computed, sufficient conditions for stabilities of equilibrium points are established. The sensitivity analysis showed that transmission rates are the most sensitive parameters that have direct impact on the basic reproduction numbers and protection and treatment rates are more sensitive and have indirect impact to the basic reproduction numbers. Numerical simulations shown that some parameter effects on the transmission of single infections as well as co-infection, and applying the protection rates and treatment rates have effective roles to minimize and also to eradicate the HIV/AIDS-COVID-19 co-infection spreading in the community. [ABSTRACT FROM AUTHOR]

In this paper, an optimal control model for the transmission dynamics of COVID-19 is investigated. We established important model properties like nonnegativity and boundedness of solutions, and also the region of invariance. Further, an expression for the basic reproduction number is computed and its sensitivity w.r.t model parameters is carried out to identify the most sensitive parameter. Based on sensitivity analysis, optimal control strategies were presented to reduce the disease burden and related costs. It is demonstrated that optimal control does exist and is unique. The characterization of optimal trajectories is analytically studied via Pontryagin's Minimum Principle. Moreover, various simulations were performed to support analytical results. The simulation results showed that the proposed controls significantly influence the disease burden compared to the absence of control cases. Further, it reveals that the applied control strategies are effective throughout the intervention period in reducing COVID-19 diseases in the community. Besides, the simulation results of the optimal control suggested that concurrently applying all controlling strategies outperform in mitigating the spread of COVID-19 compared to any other preventive measures. [ABSTRACT FROM AUTHOR]

Coffee is the most critical stimulant beverage in the world and represents a significant source of income in many tropical and subtropical countries. In this paper, a deterministic mathematical model has been formulated to describe the infestation dynamic of coffee berry borer (CBB) using a system of non-linear ordinary differential equations with farmers' awareness and optimal control. The system has two equilibrium points, namely the CBB free equilibrium point and the endemic equilibrium point which exist conditionally. The basic reproduction number, which plays a vital role in mathematical epidemiology, was derived. The qualitative analysis of the model revealed the scenario for both CBB free equilibrium and endemic equilibrium points. The local stability of the equilibria is established via the Jacobian matrix and Routh-Hurwitz criteria, while the global stability of the equilibria is proven by using an appropriate Lyapunov function. The normalized sensitivity analysis has also been performed to observe the impact of different parameters on the basic reproduction number. We extended the proposed model into an optimal control problem by incorporating two control variables, the effort made to reduce the colonizing females based on chemicals, traps, and biological control using entomopathogenic fungi such as Beauveria bassiana, that are applied to the surface of the coffee berries and kill the colonizing females of CBB when they drill an entry hole into the coffee berry and the effort made to increase the awareness of farmers through media campaign and education for the coffee farmer. Then the optimal control strategy is found by minimizing the number of CBB individuals considering the cost of implementation. The existence of optimal controls is examined using Pontryagin's minimum principle. Finally, the numerical simulations show agreement with the analytical results. [ABSTRACT FROM AUTHOR]

The world is grappled with an unprecedented challenges due to Corona virus. We are all battling this epidemic together, but we have not been able to defeat this epidemic yet. A new variant of this virus, named 'Omicron' is spreading these days. The fractional differential equations are providing us with better tools to study the mathematical model with memory effects. In this paper, we will consider an extended SER mathematical model with quarantined and vaccinated compartment to speculate the Omicron variant. This extended Susceptible Exposed Infected Recovered SER model involves equations that associate with the group of individuals those are susceptible (S), exposed (E): this class includes the individuals who are infected but not yet infectious, infectious (W): this class includes the individuals who are infected but not yet Quarantined, quarantined (Q): this class includes those group of people who are infectious, confirmed and quarantined, recovered (R) this class includes the group of individuals who have recovered, and vaccinated (V): this class includes the group of individuals who have been vaccinated. The non-negativity and of the extended SER model is analysed, the equilibrium points and the basic reproduction number are also calculated. The proposed model is then extended to the mathematical model using AB derivative operator. Proof for the existence and the uniqueness for the solution of fractional mathematical model in sense of AB fractional derivative is detailed and a numerical method is detailed to obtain the numerical solutions. Further we have discussed the efficiency of the vaccine against the Omicron variant via graphical representation. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of social media addiction that pose a major threat to the human population especially children and teenagers. It is well known that Cognitive Behavioral Therapy (CBT) is an effective treatment to treat the addict individuals and delay in the treatment leads the patient to worst stage even to death. Therefore, it is important to identify the individuals who has addiction symptoms at early stage and to provide proper counseling. We propose and analyze a nonlinear mathematical model for social media addiction problem using case detection strategy to reduce the addiction. The basic reproduction number and equilibria of the model are computed. Further, the deterministic model is extended to delay differential equation model by incorporating transmission delay and treatment delay in the system. The local stability of different equilibria is discussed in detail. Additionally, the model is converted to stochastic model and numerical simulation is carried out to compare the results of both deterministic and stochastic model. Numerical result shows that the introduction of time delays can destabilize the model system and Hopf bifurcation occurs due to periodic oscillations when certain equilibrium point crosses the delay threshold limit. Our results of stochastic model show a smaller number of social media users and addict population when compared with deterministic model. Also, our results reveal that detection and counseling parameters play a vital role in reducing addiction population. Presented results clearly suggest that there is a need to use effective detection strategy and suitable counseling program to reduce the social media addiction level. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical properties of a reaction-diffusion susceptible infected susceptible (SIS) epidemic model under a linear external source. A numerical scheme is constructed with a finite difference scheme for the space discretization and an Implicit-Explicit (IMEX) method in time integration. A threshold value, numerical basic reproduction number, is proposed in the long-time stability analysis of numerical solutions. Differently from previous works on the same model, the numerical basic reproduction number can preserve the behaviours of the basic reproduction number of the model, towards which it converges when the spatial stepsize vanishes. Moreover, it plays a role for the discrete dynamics similar to the one played by its continuous counterpart. Some numerical experiments are given in the end to confirm the conclusions and detect the conjecture on the stability of endemic equilibrium (EE) in general case. [ABSTRACT FROM AUTHOR]

Motivated by the recent paper [M.R.S. Kulenović, M. Nurkanović, and A.A. Yakubu, Asymptotic behaviour of a discrete-time density-dependent SI epidemic model with constant recruitment, J. Appl. Math. Comput. 67 (2021), pp. 733–753. DOI:], in this paper, we consider the class of the SI epidemic models with recruitment where the Poisson function, a decreasing exponential function of the population of infectious individuals, is replaced by a general probability function that satisfies certain conditions. We compute the basic reproduction number R 0. We establish the global asymptotic stability of the disease-free equilibrium (GAS) for R 0 < 1. We use the Lyapunov function method developed in [P. van den Driessche and A.-A. Yakubu, Disease extinction versus persistence in discrete-time epidemic models, Bull. Math. Biol. 81 (2019), pp. 4412–4446], to demonstrate the GAS of the disease-free equilibrium and uniform persistence of the considered class of models. We show that the considered type of model is permanent for R 0 > 1. For R 0 = 1 , the transcritical bifurcation appears. For R 0 > 1 , we prove the global attractivity result for endemic equilibrium and instability of the disease-free equilibrium. We apply theoretical results to specific escape functions of the susceptibles from infectious individuals. For each case, we compute the basic reproduction number R 0 . [ABSTRACT FROM AUTHOR]

AbstractThis paper presents a new nonlinear epidemic model for the spread of SARS-CoV-2 that incorporates the effect of double dose vaccination. The model is analyzed using qualitative, stability, and sensitivity analysis techniques to investigate the impact of vaccination on the spread of the virus. We derive the basic reproduction number and perform stability analysis of the disease-free and endemic equilibrium points. The model is also subjected to sensitivity analysis to identify the most influential model parameters affecting the disease dynamics. The values of the parameters are estimated with the help of the least square curve fitting tools. Finally, the model is simulated numerically to assess the effectiveness of various control strategies, including vaccination and quarantine, in reducing the spread of the virus. Optimal control techniques are employed to determine the optimal allocation of resources for implementing control measures. Our results suggest that increasing the vaccination coverage, adherence to quarantine measures, and resource allocation are effective strategies for controlling the epidemic. The study provides valuable insights into the dynamics of the pandemic and offers guidance for policymakers in formulating effective control measures. [ABSTRACT FROM AUTHOR]

In this paper, we study mathematically a two strains epidemic model taking into account non-monotonic incidence rates and vaccination strategy. The model contains seven ordinary differential equations that illustrate the interaction between the susceptible, the vaccinated, the exposed, the infected and the removed individuals. The model has four equilibrium points, namely, disease free equilibrium, endemic equilibrium with respect to the first strain, endemic equilibrium with respect to the second strain and the endemic equilibrium with respect to both strains. The global stability of the equilibria has been demonstrated using some suitable Lyapunov functions. The basic reproduction number is found depending on the first strain reproduction number R 0 1 and the second reproduction number R 0 2. We have shown that the disease dies out when the basic reproduction number is less than unity. It was remarked that the global stability of the endemic equilibria depends, on the strain basic reproduction number and on the strain inhibitory effect reproduction number. We have also observed that the strain with high basic reproduction number will dominate the other strain. Finally, the numerical simulations are presented in the last part of this work to support our theoretical results. We notice that our suggested model has some limitations and does not predicting the long-term dynamics for some reproduction numbers cases. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number R 0 and show that if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable, whereas if R 0 > 1 , then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the influence of the waning immunity on the dynamics of Tilapia Lake Virus (TiLV) transmission in wild and farmed tilapia within freshwater. We formulate a model for which susceptible individuals can contract the disease in two ways: (i) direct mode caused by contact with infected individuals; (ii) indirect mode due to the presence of pathogenic agents in the water. We obtain an age-structured model which combines both age since infection and age since recovery. We derive an explicit formula for the reproductive number R 0 and show that the disease-free equilibrium is locally asymptotically stable when, R 0 < 1. We discuss on the form of the waning immunity parameter and show numerically that a Hopf bifurcation may occur for suitable immunity parameter values, which means that there is a periodic solution around the endemic equilibrium when, R 0 > 1. [ABSTRACT FROM AUTHOR]

In this paper, a reaction-diffusion SIR epidemic model via environmental driven infection in heterogeneous space is proposed. To reflect the prevention and control measures of disease in allusion to the susceptible in the model, the nonlinear incidence function E f (S) is applied to describe the protective measures of susceptible. In the general spatially heterogeneous case of the model, the well-posedness of solutions is obtained. The basic reproduction number R 0 is calculated. When R 0 ≤ 1 the global asymptotical stability of the disease-free equilibrium is obtained, while when R 0 > 1 the model is uniformly persistent. Furthermore, in the spatially homogeneous case of the model, when R 0 > 1 the global asymptotic stability of the endemic equilibrium is obtained. Lastly, the numerical examples are enrolled to verify the open problems. [ABSTRACT FROM AUTHOR]

Epidemiological models with the identical basic reproduction number R 0 may behave differently on both short time scale and long time scale. In this paper, we compare the predicted final sizes for several deterministic epidemic models and estimate the probabilities of a minor/major outbreak for continuous-time Markov chain (CTMC) models, all epidemic models have the identical R 0 . It is proved that the final size predicted by the epidemic model with homogeneous mixing is larger than with heterogeneous mixing. For CTMC models with heterogeneous mixing, the probabilities of a minor outbreak initiated by superspreaders and non-superspreaders are calculated and compared. For both deterministic modelling and stochastic modelling, numerical simulations are performed to support the mathematical analysis. [ABSTRACT FROM AUTHOR]

The outbreak of COVID-19 was first experienced in Wuhan City, China, during December 2019 before it rapidly spread over globally. This paper has proposed a mathematical model for studying its transmission dynamics in the presence of face mask wearing and hospitalization services of human population in Tanzania. Disease-free and endemic equilibria were determined and subsequently their local and global stabilities were carried out. The trace-determinant approach was used in the local stability of disease-free equilibrium point while Lyapunov function technique was used to determine the global stability of both disease-free and endemic equilibrium points. Basic reproduction number, R 0 , was determined in which its numerical results revealed that, in the presence of face masks wearing and medication services or hospitalization as preventive measure for its transmission, R 0 = 0.698 while in their absence R 0 = 3.8. This supports its analytical solution that the disease-free equilibrium point E 0 is asymptotically stable whenever R 0 < 1 , while endemic equilibrium point E ∗ is globally asymptotically stable for R 0 > 1. Therefore, this paper proves the necessity of face masks wearing and hospitalization services to COVID-19 patients to contain the disease spread to the population. [ABSTRACT FROM AUTHOR]

Brucellosis is one of the major infective and contagious bacterial diseases among animals in pastoral areas of some countries. In this paper, we introduce the effect of environment white noise in the spatial propagation process of brucellosis, and consider a stochastic two-patch brucellosis model. On one hand, we get existence and uniqueness of the global positive solution to the stochastic systems. On the other hand, by using the stochastic Lyapunov function theory we obtain a series of stochastic threshold dynamics results, incorporating extinction of the disease, existence of a unique ergodic stationary distribution of the positive solutions to systems in both patch 1 and patch 2. Furthermore, we find that stochastic perturbation is contribute to extinction of the disease to some extent by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we have calculated the basic reproduction number (R0) and doubling time (Td) for the novel Coronavirus disease 2019 (COVID-19). The calculation is performed for March 2020 from the data provided by worldometer. We have investigated the data for Germany and Bangladesh. The calculation of R0 is performed based on SIR model. The parameter Td is estimated based on the new cases of each day. Since Td and R0 in use to judge the lockdowns and other measures to prevent spreading of the virus, we have provided simple approximation of both parameters. [ABSTRACT FROM AUTHOR]

Novel coronavirus has altered the socio-economic condition of the whole world through its devastating effects on the human population. Mathematical models and computation techniques may play an important role in understanding this epidemic and contribute a lot in policy making to control the infection in a more systematic and effective way. In this paper, we have proposed a deterministic mathematical model for the Covid-19 pandemic taking into account the different epidemiological status of individuals of a given geographical region and analyzed it with respect to the basic reproduction number. Uncertainty is obvious in the case of a growing epidemic and it multiplies if the disease etiology is unknown. Taking into account the uncertainty in the epidemiological parameters, we extended the deterministic system into a stochastic system through random parameter perturbations in three epidemiological parameters. Analyzing the model, we determined the disease persistence and eradication conditions. The asymptotic behavior of the stochastic solution around the coexistence equilibrium of the deterministic model was also presented. As a case study, we considered the Covid-19 pandemic of India and estimated the model parameters from the epidemic data. We demonstrated different analytical results and predicted the course of the epidemic. Our simulation results indicate that the epidemic in India may continue up to third week of July 2021 and the cumulative confirmed Covid-19 cases may vary from 12.13 × 10 6 to 13.41 × 10 6. Such results may be useful from management and policy development viewpoints. [ABSTRACT FROM AUTHOR]

This paper concerns with a reaction–diffusion susceptible-vaccinated-infectious-recovered model with a fixed latent period. The model is formulated as a non-local and time-delayed reaction–diffusion model due to the fact that an individual infected by the disease in one place may not stay at the same space in the domain due to the movement of human during the incubation period. We then derive the basic reproduction number ℜ 0 as the spectral radius of the next infection operator and show that it serves as a threshold role in predicting whether the disease will spread. Further, the explicit formula of basic reproduction number is obtained when all model parameters to be positive constants and the domain to the one-dimensional case. Moreover, we demonstrate the differences in the form of basic reproduction number between the standard incidence and bilinear incidence rate. [ABSTRACT FROM AUTHOR]

In meta-population models for infectious diseases, the basic reproduction number can be as much as 70% larger in the case of preferential mixing than that in homogeneous mixing [J.W. Glasser, Z. Feng, S.B. Omer, P.J. Smith, and L.E. Rodewald, The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: A modelling study, Lancet ID 16 (2016), pp. 599–605. doi: ]. This suggests that realistic mixing can be an important factor to consider in order for the models to provide a reliable assessment of intervention strategies. The influence of mixing is more significant when the population is highly heterogeneous. In this paper, another quantity, the final epidemic size () of an outbreak, is considered to examine the influence of mixing and population heterogeneity. Final size relation is derived for a meta-population model accounting for a general mixing. The results show that can be influenced by the pattern of mixing in a significant way. Another interesting finding is that, heterogeneity in various sub-population characteristics may have the opposite effect on and. [ABSTRACT FROM AUTHOR]

Since mathematical models play a key role in investigating the dynamics of infectious diseases, many mathematical models for these diseases are developed. In this paper, it is aimed to determine the dynamics of Tuberculosis (TB) in Turkey, how much it will affect the future and the impact of vaccine therapy on the disease. For this purpose, three mathematical models (SIR, SEIR and BSEIR) in the literature are considered for the case of Turkey. The model parameters are obtained with TB reported data from 2005 to 2015 by using the least square method. The obtained results revealed that the basic reproduction ratio for all three models is less than 1. Moreover, the stability analysis of the models and sensitivity analysis of the model parameters are presented and discussed. Finally, the accuracy of results for all three models is compared and the effect of the vaccination rate is discussed. [ABSTRACT FROM AUTHOR]

In this paper, with eclipse stage in consideration, we propose an HIV-1 infection model with a general incidence rate and CTL immune response. We first study the existence and local stability of equilibria, which is characterized by the basic infection reproduction number R 0 and the basic immunity reproduction number R 1 . The local stability analysis indicates the occurrence of transcritical bifurcations of equilibria. We confirm the bifurcations at the disease-free equilibrium and the infected immune-free equilibrium with transmission rate and the decay rate of CTLs as bifurcation parameters, respectively. Then we apply the approach of Lyapunov functions to establish the global stability of the equilibria, which is determined by the two basic reproduction numbers. These theoretical results are supported with numerical simulations. Moreover, we also identify the high sensitivity parameters by carrying out the sensitivity analysis of the two basic reproduction numbers to the model parameters. [ABSTRACT FROM AUTHOR]

Self-medication is an important initial response to illness in Africa. This mode of medication is often done with the help of African traditional medicines. Because of the misconception that African traditional medicines can cure/prevent all diseases, some Africans may opt for COVID-19 prevention and management by self-medicating. Thus to efficiently predict the dynamics of COVID-19 in Africa, the role of the self-medicated population needs to be taken into account. In this paper, we formulate and analyse a mathematical model for the dynamics of COVID-19 in Cameroon. The model is represented by a system of compartmental age-structured ODEs that takes into account the self-medicated population and subdivides the human population into two age classes relative to their current immune system strength. We use our model to propose policy measures that could be implemented in the course of an epidemic in order to better handle cases of self-medication. [ABSTRACT FROM AUTHOR]

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number R 01 and immune reproduction number R 02 are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when R 01 < 1 , the virus will be cleared out eventually and the CTL immune response will also disappear; when R 02 < 1 < R 01 , the virus persists within the host, but the CTL immune response disappears eventually; when R 02 > 1 , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given. [ABSTRACT FROM AUTHOR]

In this paper, an SIRS epidemic model with immunity age is investigated, where the constant treatment rate and the loss of the acquired immunity are incorporated. The well-posedness of the model is verified by changing it into an abstract non-densely defined Cauchy problem, and the conditions for the existence of disease-free equilibrium and the endemic equilibria are found. The theoretic analysis showed that the disease-free equilibrium is globally asymptotically stable as the basic reproduction number is less than unity, and the numerical simulation illustrated that it is asymptotically stable as the number is greater than unity. Combining numerical simulations, the instability and the local stability of different endemic equilibrium, and the existence of saddle-node bifurcation, and Hopf bifurcation are analyzed. Again, we think it is possible that the Bogdanov–Takens bifurcation may occur for the model under some conditions. Both non-periodic and periodic behaviors are shown when the disease persists in population, where the duration that the recovered individual stays in the recovery class plays an important role in the spread of the disease. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive epidemic model with latent period and vaccination strategy. The well-posedness of solutions is first investigated for such a system. Then, in terms of the basic reproduction number and the critical wave speed, we establish a complete threshold result which reveals the existence and nonexistence of the strong traveling waves connecting the disease-free equilibrium to the endemic steady state. Finally, some numerical simulations are performed to illustrate our theoretical analysis. Our results implies that (i) the diffusion ability of the infected individuals can accelerate the minimal wave speed; and (ii) the latent period and vaccination strategy would slow down the spread of the disease. [ABSTRACT FROM AUTHOR]

In this paper, we study a more general diffusive spatially dependent vaccination model for infectious disease. In our diffusive vaccination model, we consider both therapeutic impact and nonlinear incidence rate. Also, in this model, the number of compartments of susceptible, vaccinated and infectious individuals are considered to be functions of both time and location, where the set of locations (equivalently, spatial habitats) is a subset of R n with a smooth boundary. Both local and global stability of the model are studied. Our study shows that if the threshold level R 0 ≤ 1 , the disease-free equilibrium E 0 is globally asymptotically stable. On the other hand, if R 0 > 1 then there exists a unique stable disease equilibrium E ∗ . The existence of solutions of the model and uniform persistence results are studied. Finally, using finite difference scheme, we present a number of numerical examples to verify our analytical results. Our results indicate that the global dynamics of the model are completely determined by the threshold value R 0 . [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyse a delayed HIV-1 model with both viral and cellular transmissions and virus waning. We obtain the threshold dynamics of the proposed model, characterized by the basic reproduction number R 0 . If R 0 < 1 , the infection-free steady state is globally asymptotically stable; whereas if R 0 > 1 , the system is uniformly persistent. When the delays are positive, we show that the intracellular delays in both viral and cellular infections may lead to stability switches of the infected steady state. Both analytical and numerical results indicate that if the effect of cell-to-cell transmission is ignored, then the risk of HIV-1 infection will be underestimated. Moreover, the viral load of model without virus waning is higher than the one of model with virus waning. These results highlight the important role of two ways of viral transmission and virus waning on HIV-1 infection. [ABSTRACT FROM AUTHOR]

Scarlet fever is an acute respiratory infectious disease and the incidence rate is increasing from 2011 throughout the world. In this paper, the mathematical models are proposed, which incorporate both direct transmissions and indirect transmissions of scarlet fever. The threshold conditions for disease invasion are obtained in terms of the basic reproduction number. The peak value, final size and epidemic time in a seasonal prevalence are investigated numerically. Furthermore, the effects of seasonal fluctuations on disease outbreak are also studied on the basis of real data in China. [ABSTRACT FROM AUTHOR]

In this paper, we apply optimal control theory to a novel coronavirus (COVID-19) transmission model given by a system of non-linear ordinary differential equations. Optimal control strategies are obtained by minimizing the number of exposed and infected population considering the cost of implementation. The existence of optimal controls and characterization is established using Pontryagin's Maximum Principle. An expression for the basic reproduction number is derived in terms of control variables. Then the sensitivity of basic reproduction number with respect to model parameters is also analysed. Numerical simulation results demonstrated good agreement with our analytical results. Finally, the findings of this study shows that comprehensive impacts of prevention, intensive medical care and surface disinfection strategies outperform in reducing the disease epidemic with optimum implementation cost. [ABSTRACT FROM AUTHOR]

Backward bifurcation is an important property of infectious disease models. A centre manifold method has been developed by Castillo-Chavez and Song for detecting the presence of backward bifurcation and deriving a necessary and sufficient condition for its occurrence in Ordinary Differential Equations (ODE) models. In this paper, we extend this method to partial differential equation systems. First, we state a main theorem. Next we illustrate the application of the new method on a chronological age-structured Susceptible-Infected-Susceptible (SIS) model with density-dependent recovery rate, an age-since-infection structured HIV/AIDS model with standard incidence and an age-since-infection structured cholera model with vaccination. [ABSTRACT FROM AUTHOR]

This paper proposes a malaria transmission model to describe the dynamics of malaria transmission in the human and mosquito populations. This model emphasizes the impact of limited resource on malaria transmission. We derive a formula for the basic reproductive number of infection and investigate the existence of endemic equilibria. It is shown that this model may undergo backward bifurcation, where the locally stable disease-free equilibrium co-exists with an endemic equilibrium. Furthermore, we determine conditions under which the disease-free equilibrium of the model is globally asymptotically stable. The global stability of the endemic equilibrium is also studied when the basic reproductive number is greater than one. Finally, numerical simulations to illustrate our findings and brief discussions are provided. [ABSTRACT FROM AUTHOR]

Vector-borne infectious diseases may involve both horizontal transmission between hosts and transmission from infected vectors to susceptible hosts. In this paper, we incorporate these two transmission modes into a vector-borne disease model that includes general nonlinear incidence rates and the age of infection for both hosts and vectors. We show the existence, uniqueness, nonnegativity, and boundedness of solutions for the model. We study the existence and local stability of steady states, which is shown to be determined by the basic reproduction number. By showing the existence of a global compact attractor and uniform persistence of the system, we establish the threshold dynamics using the Fluctuation Lemma and the approach of Lyapunov functionals. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable and otherwise the disease will be established when there is initial infection force for the hosts. We also study a model with the standard incidence rate and discuss the effect of different incidence rates on the disease dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an SIR epidemic model with infection age and spatial diffusion in the case of Neumann boundary condition. The original model is constructed as a nonlinear age structured system of reaction–diffusion equations. By using the method of characteristics, we reformulate the model into a system of a reaction–diffusion equation and a Volterra integral equation. For the reformulated system, we define the basic reproduction number R 0 by the spectral radius of the next generation operator, and show that if R 0 < 1 , then the trivial disease-free steady state is globally attractive, whereas if R 0 > 1 , then the disease in the system is persistent. Moreover, under an additional assumption that there exists a finite maximum age of infectiousness, we show the global attractivity of a constant endemic steady state for R 0 > 1. [ABSTRACT FROM AUTHOR]

A discrete SIRS epidemic model with vaccination and a general infection probability function is investigated in this paper. The basic reproduction number ℜ 0 of the model is calculated. It is proven that the disease-free equilibrium is locally asymptotically stable if ℜ 0 < 1 and globally asymptotically stable for some additional conditions, and the endemic equilibrium is locally asymptotically stable if ℜ 0 > 1. Numerical simulations are presented to illustrate the stability results, and the sensitivity analysis of the basic reproduction number ℜ 0 shows that the proportionality coefficient of unvaccinated susceptible, the probability that infected individuals do not recover, and the probability of survival have much more effect on ℜ 0 . [ABSTRACT FROM AUTHOR]

This paper studies the global stability of discrete-time viral infection models with humoural immunity. We consider both latently and actively infected cells. We study also a model with general production and clearance rates of all compartments as well as general incidence rate of infection. We use nonstandard finite difference method to discretize the continuous-time models. The positivity and boundedness of solutions of the discrete models are established. We establish by using Lyapunov method, the global stability of equilibria in terms of the basic reproduction number R 0 and the humoural immune response activation number R 1 . The results signify that the infection dies out if R 0 ≤ 1. Moreover, the infection persists with inactive immune response if R 1 ≤ 1 < R 0 and with active immune response if R 1 > 1. We illustrate our theoretical results by using numerical simulations. [ABSTRACT FROM AUTHOR]

A more realistic alcoholism model on scale-free networks with demographic and nonlinear infectivity is introduced in this paper. The basic reproduction number R 0 is derived from the next-generation method. Global stability of the alcohol-free equilibrium is obtained. The persistence of our model is also derived. Furthermore, the SAITS model with nonlinear infectivity is also investigated. Stability of all the equilibria and persistence are also obtained. Some numerical simulations are also presented to verify and extend our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the impact of heterogeneous susceptibility on disease transmission dynamics, using a simple mathematical model formulated by a system of ordinary differential equations. Our model describes a time-varying immunity level, which is enhanced by reinfection and diminished by waning immunity. Through mathematical analysis, we show that unexpected outbreaks, called delayed outbreaks, occur even when the basic reproduction number is less than one. A reallocation of susceptibility at the individual level, by repeated infections, in the host population induces delayed outbreaks. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model describing tuberculosis transmission with fast and slow progression and age-dependent latency and infection is investigated. It is assumed in the model that infected individuals can develop tuberculosis by either of two pathogenic mechanisms: direct progression or endogenous reactivation. It is shown that the transmission dynamics of the disease is fully determined by the basic reproduction number. By analyzing corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state of the model is established. By using the persistence theory for infinite dimensional system, it is proved that the system is uniformly persistent when the basic reproduction number is greater than unity. By constructing suitable Lyapunov functionals and using LaSalle's invariance principle, it is verified that the global dynamics of the system is completely determined by the basic reproduction number. [ABSTRACT FROM AUTHOR]

A delayed SEIR epidemic model with saturated incidence and saturated treatment function is considered in this paper. Sufficient conditions for the existence of local Hopf bifurcation are established by regarding the possible combination of the two delays as the bifurcation parameter. General formula for the direction, period and stability of the bifurcated periodic solutions are derived by using the normal form method and the centre manifold theory. Finally, some numerical simulations are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an age-structured HIV infection model with logistic growth for target cell. We rewrite the model as an abstract non-densely defined Cauchy problem and obtain the condition which guarantees the existence of the unique positive steady state. By linearizing the model at steady state and analysing the associated characteristic transcendental equations, we study the local asymptotic stability of the steady state. Furthermore, by using Hopf bifurcation theorem in Liu et al., we show that Hopf bifurcation occurs at the positive steady state when bifurcating parameter crosses some critical values. Finally, we perform some numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number R 0 is obtained such that R 0 ≤ 1 indicates the possibility of dysentery eradication in the community while R 0 > 1 represents uniform persistence of the disease. The Lyapunov–LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for R 0 > 1. Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation. [ABSTRACT FROM AUTHOR]

In this paper, we establish a mathematical model with two delays to reflect the intrinsic and extrinsic incubation periods of virus in dengue transmission. The basic reproduction number R 0 of the model is defined. It is proved that the disease-free equilibrium is stable when R 0 < 1 and the positive equilibrium is stable when R 0 > 1. Next, we derive an estimation formula for the reproduction number R t when the human population is partially susceptible to dengue. As an application, the R t values of dengue transmission in Singapore in the years 2013–2015 are estimated. Our estimation method can be applied to estimating R t of other infectious diseases, especially when the human population is not completely susceptible to the disease. [ABSTRACT FROM AUTHOR]

In this paper, an age-structured cholera model with multiple transmissions, saturation incidence and imperfect vaccination is proposed. In the model, we consider both the infection age of infected individuals and the biological age of Vibrio cholerae in the aquatic environment. Asymptotic smoothness is verified as a necessary argument. By analysing the characteristic equations, the local stability of disease-free and endemic steady states is established. By using Lyapunov functionals and LaSalle's invariance principle, it is proved that the global dynamics of the model can be completely determined by basic reproduction number. The study of optimal control helps us seek cost-effective solutions of time-dependent vaccination strategy against cholera outbreaks. Numerical simulations are carried out to illustrate the corresponding theoretical results. [ABSTRACT FROM AUTHOR]