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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]

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 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]

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]

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, 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]

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]

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 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]

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, 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]

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 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, 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]

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, 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, 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]

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 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]

This paper deals with the global property of a drinking model with public health educational campaigns. With the help of Lyapunov function, global stability of equilibria of the model is derived. The alcohol-free equilibrium is globally asymptotically stable and the alcohol problems are eliminated from population if R0 ≤ 1. A unique alcohol present equilibrium is globally asymptotically stable if R0 > 1. Furthermore, the basic reproductive R0 for the model is compared with the basic reproductive number R1 for the absence of public health educational campaigns. We conclude that the public health educational campaigns of drinking individuals can slow down the drinking dynamics. Some numerical simulations are also given to explain our conclusions. [ABSTRACT FROM AUTHOR]

This paper investigates consequences of vaccine implementation strategies for infectious diseases by a mathematical model. For an infectious disease, the degree of infection may vary widely among the individuals. Reports show that individuals belonging to certain groups possess considerably higher risk to infection. Incorporating this phenomenon into vaccination strategies, the host is categorized into different groups to measure the outcome of the vaccination. A mathematical model is proposed and analysed to evaluate this measure. Our results suggest that vaccinating a group with certain priority may lead to elimination of the disease effectively. The strategy is cost-effective as well. [ABSTRACT FROM AUTHOR]

This paper investigates consequences of vaccine implementation strategies for infectious diseases by a mathematical model. For an infectious disease, the degree of infection may vary widely among the individuals. Reports show that individuals belonging to certain groups possess considerably higher risk to infection. Incorporating this phenomenon into vaccination strategies, the host is categorized into different groups to measure the outcome of the vaccination. A mathematical model is proposed and analysed to evaluate this measure. Our results suggest that vaccinating a group with certain priority may lead to elimination of the disease effectively. The strategy is cost-effective as well. [ABSTRACT FROM AUTHOR]