Global dynamics of a tuberculosis model with age-dependent latency and time delays in treatment.

Since there exists heterogeneity in incubation periods of tuberculosis and a time lag between treatment and recovery. In this study, we develop a tuberculosis model that takes into account age-dependent latency and time delays in treatment to describe the transmission of tuberculosis. We first show that the solution semi-flow of the model is well-posed and has a global attractor [Formula: see text] within an infinite dimensional space [Formula: see text]. Then we define the basic reproduction number [Formula: see text] and prove that it determines the global dynamics of the model. If [Formula: see text], the global attractor [Formula: see text] reduces to the disease-free equilibrium state, indicating that the disease-free equilibrium state is globally asymptotically stable. When [Formula: see text], the semi-flow generated by the model is uniformly persistent, and there exists an interior global attractor [Formula: see text] for this uniformly persistent model. By constructing a suitable Lyapunov function and applying LaSalle's Invariance Principle, we show that the global attractor [Formula: see text] is reduced to the endemic equilibrium state, which means that the endemic equilibrium state is globally asymptotically stable. Based on the tuberculosis data in China from 2007 to 2020, we simulate the parameters and initial values of the proposed model. Furthermore, we calculate the sensitivity of [Formula: see text] to the parameters and find the most sensitive parameters to [Formula: see text]. Finally, we present an improved strategy to achieve the WHO's goal of reducing the incidence of tuberculosis by 90% by 2035 compared to 2015.

Analysis of an age-structured tuberculosis model with treatment and relapse.

A new tuberculosis model consisting of ordinary differential equations and partial differential equations is established in this paper. The model includes latent age (i.e., the time elapsed since the individual became infected but not infectious) and relapse age (i.e., the time between cure and reappearance of symptoms of tuberculosis). We identify the basic reproduction number [Formula: see text] for this model, and show that the [Formula: see text] determines the global dynamics of the model. If [Formula: see text], the disease-free equilibrium is globally asymptotically stable, which means that tuberculosis will disappear, and if [Formula: see text], there exists a unique endemic equilibrium that attracts all solutions that can cause the spread of tuberculosis. Based on the tuberculosis data in China from 2007 to 2018, we use Grey Wolf Optimizer algorithm to find the optimal parameter values and initial values of the model. Furthermore, we perform uncertainty and sensitivity analysis to identify the parameters that have significant impact on the basic reproduction number. Finally, we give an effective measure to reach the goal of WHO of reducing the incidence of tuberculosis by 80% by 2030 compared to 2015.

Modeling the Effects of Meteorological Factors and Unreported Cases on Seasonal Influenza Outbreaks in Gansu Province, China.

Influenza usually breaks out seasonally in temperate regions, especially in winter, infection rates and mortality rates of influenza increase significantly, which means that dry air and cold temperatures accelerate the spread of influenza viruses. However, the meteorological factors that lead to seasonal influenza outbreaks and how these meteorological factors play a decisive role in influenza transmission remain unclear. During the epidemic of infectious diseases, the neglect of unreported cases leads to an underestimation of infection rates and basic reproduction number. In this paper, we propose a new non-autonomous periodic differential equation model with meteorological factors including unreported cases. First, the basic reproduction number is obtained and the global asymptotic stability of the disease-free periodic solution is proved. Furthermore, the existence of periodic solutions and the uniformly persistence of the model are demonstrated. Second, the best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the influenza data in Gansu province, China. We also estimate that the basic reproduction number is 1.2288 (95% CI:(1.2287, 1.2289)). Then, to determine the key parameters of the model, uncertainty and sensitivity analysis are explored. Finally, our results show that influenza is more likely to spread in low temperature, low humidity and low precipitation environments. Temperature is a more important factor than relative humidity and precipitation during the influenza epidemic. In addition, our results also show that there are far more unreported cases than reported cases.

Dynamics and optimal control of tuberculosis model with the combined effects of vaccination, treatment and contaminated environments.

Tuberculosis has affected human beings for thousands of years, and until today, tuberculosis still ranks third among 29 infectious diseases in China. However, most of the existing mathematical models consider a single factor, which is not conducive to the study of tuberculosis transmission dynamics. Therefore, this study considers the combined effects of vaccination, treatment, and contaminated environments on tuberculosis, and builds a new model with seven compartments of $ SVEITRW $ based on China's tuberculosis data. The study shows that when the basic reproduction number $ R_{0} $ is less than 1, the disease will eventually disappear, but when $ R_{0} $ is greater than 1, the disease may persist. In the numerical analysis part, we use Markov-chain Monte-Carlo method to obtain the optimal parameters of the model. Through the next generation matrix theory, we calculate that the $ R_{0} $ value of tuberculosis in China is $ 2.1102 $, that is, if not controlled, tuberculosis in China will not disappear over time. At the same time, through partial rank correlation coefficients, we find the most sensitive parameter to the basic reproduction number $ R_{0} $. On this basis, we combine the actual prevalence of tuberculosis in China, apply Pontryagin's maximum principle, and perform cost-effectiveness analysis to obtain the conditions required for optimal control. The analysis shows that four control strategies could effectively reduce the prevalence of TB, and simultaneously controlling $ u_{2}, u_{3}, u_{4} $ is the most cost-effective control strategy.

Modelling mosquito population suppression based on competition system with strong and weak Allee effect.

Mosquito-borne diseases are threatening half of the world's population. To prevent the spread of malaria, dengue fever, or other mosquito-borne diseases, a new disease control strategy is to reduce or eradicate the wild mosquito population by releasing sterile mosquitoes. To study the effects of sterile insect technique on mosquito populations, we developed a mathematical model of constant release of sterile Aedes aegypti mosquitoes with strong and weak Allee effect and considered interspecific competition with Anopheles mosquitoes. We calculated multiple release thresholds and investigated the dynamical behavior of this model. In order to get closer to reality, an impulsive differential equation model was also introduced to study mosquito suppression dynamics under the strategy of releasing $ c $ sterile male mosquitoes at each interval time $ T $. Finally, the relationship between the releasing amount or the waiting period and the number of days required to suppress mosquitoes was illustrated by numerical simulations.

Dynamics and optimal control of a Zika model with sexual and vertical transmissions.

A new transmission model of Zika virus with three transmission routes including human transmission by mosquito bites, sexual transmission between males and females and vertical transmission is established. The basic reproduction number $ R_{0} $ is derived. When $ R_{0} < 1 $, it is proved that the disease-free equilibrium is globally stable. Furthermore, the optimal control and mitigation methods for transmission of Zika virus are deduced and explored. The MCMC method is used to estimate the parameters and the reasons for the deviation between the actual infection cases and the simulated data are discussed. In addition, different strategies for controlling the spread of Zika virus are simulated and studied. The combination of mosquito control strategies and internal human control strategies is the most effective way in reducing the risk of Zika virus infection.

On threshold dynamics for periodic and time-delayed impulsive systems and application to a periodic disease model.

The basic reproduction ratio R0 of more general periodic and time-delayed impulsive model which the period of model coefficients is different from that of fixed impulsive moments, is developed. That R0 is the threshold parameter for the stability of the zero solution of the associated linear system is also shown. The developed theory is further applied to a swine parasitic disease model with pulse therapy. Threshold results on its global dynamics in terms of R0 are obtained. Some numerical simulation results are also given to support our main results.

Global dynamics of a delayed alcoholism model with the effect of health education.

An alcohol consumption model with health education and three time delays is formulated and analyzed. The alcoholism generation number is defined. Two steady states of the model are found. At the same time, the corresponding global dynamics of the model are analyzed respectively in four cases with different time delays. Then, the effects of health education and three time delays in controlling the alcohol problem are discussed. Some numerical simulation results are also given to support our theoretical predictions.

Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK.

A new COVID-19 epidemic model with media coverage and quarantine is constructed. The model allows for the susceptibles to the unconscious and conscious susceptible compartment. First, mathematical analyses establish that the global dynamics of the spread of the COVID-19 infectious disease are completely determined by the basic reproduction number R0. If R0 ≤ 1, then the disease free equilibrium is globally asymptotically stable. If R0 > 1, the endemic equilibrium is globally asymptotically stable. Second, the unknown parameters of model are estimated by the MCMC algorithm on the basis of the total confirmed new cases from February 1, 2020 to March 23, 2020 in the UK. We also estimate that the basic reproduction number is R0 = 4.2816(95%CI: (3.8882, 4.6750)). Without the most restrictive measures, we forecast that the COVID-19 epidemic will peak on June 2 (95%CI: (May 23, June 13)) (Figure 3a) and the number of infected individuals is more than 70% of UK population. In order to determine the key parameters of the model, sensitivity analysis are also explored. Finally, our results show reducing contact is effective against the spread of the disease. We suggest that the stringent containment strategies should be adopted in the UK.

Global dynamics for a multi-group alcoholism model with public health education and alcoholism age.

A new multi-group alcoholism model with public health education and alcoholism age is considered. The basic reproduction number R0 is defined and mathematical analyses show that dynamics of model are determined by the basic reproduction number. The alcohol-free equilibrium P0 of the model is globally asymptotically stable if R0≤1 while the alcohol-present equilibrium P* of the model exists uniquely and is globally asymptotically stable if R0>1. The Lyapunov functionals for the globally asymptotically stable of the multi-group model are constructed by using the theory of non-negative matrices and a graph-theoretic approach. Meanwhile, the combined effects of the public health education and the alcoholism age on alcoholism dynamics are displayed. Our main results show that strengthening public health education and decreasing the age of the alcoholism are very helpful for the control of alcoholism.

Dynamics of an edge-based SEIR model for sexually transmitted diseases.

A new edge-based sexually transmitted SEIR model on the contact network is introduced in this paper. The contact infection between the opposite sex and no infectivity during the latent period on bipartite networks are included. The basic reproduction number and the equations of the final size of epidemic are derived. The dynamics of our model with arbitrary initial conditions are further studied. Sensitivity analysis on several parameters and numerical results of the model are derived. We show that the length of the latent period has an effect on arrival time and size of disease peak, but does not affect the final epidemic size and the basic reproduction number of the disease.

Hopf bifurcation of an age-structured HIV infection model with logistic target-cell growth.

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.

Dynamics of tuberculosis with fast and slow progression and media coverage.

A new tuberculosis model with fast and slow progression and media coverage is formulated and analyzed. The basic reproductive number R0 is derived, and the existence and stability of all the equilibria are discussed. The occurrences of forward and backward bifurcation are obtained by using center manifold theory. Numerical simulations are also given to support our theoretical results. Sensitivity analysis on a few parameters is also carried out. Our results show that media coverage can encourage people to take measures to avoid potential infections and control the spread of tuberculosis.

Global dynamics of an age-structured malaria model with prevention.

In this paper, we formulate a new age-structured malaria model, which incorporates the age of prevention period of susceptible people, the age of latent period of human and the age of latent period of female Anopheles mosquitoes. We show that there exists a compact global attractor and obtain a sufficient condition for uniform persistence of the solution semiflow. We obtain the basic reproduction number R0 and show that R0 completely determines the global dynamics of the model, that is, if R0 < 1, the disease-free equilibrium is globally asymptotically stable, if R0 > 1, there exists a unique endemic equilibrium that attracts all solutions for which malaria transmission occurs. Finally, we perform some numerical simulations to illustrate our theoretical results and give a brief discussion.

Analysis of the SAITS alcoholism model on scale-free networks with demographic and nonlinear infectivity.

A more realistic alcoholism model on scale-free networks with demographic and nonlinear infectivity is introduced in this paper. The basic reproduction number [Formula: see text] 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.

Modelling and analysis of an alcoholism model with treatment and effect of Twitter.

A new alcoholism model with treatment and effect of Twitter is introduced. The stability of all equilibria which is determined by the basic reproductive number ro is obtained. The occurrence of backward and forward bifurcation for a certain defined range of ro are established by the center manifold theory. Numerical results and sensitivity analysis on several parameters are conducted. Our results show that Twitter may be a good indicator of alcoholism model and affect the emergence and spread of drinking behavior.

Dynamics analysis of a predator-prey system with harvesting prey and disease in prey species.

In this paper, a predator-prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.

Bifurcation analysis of an age-structured alcoholism model.

In this paper, we investigate a new alcoholism model in which alcoholics have age structure. 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 analyzing 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. (Z. Angew. Math. Phys. 62 (2011) 191-222), 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 theoretical results and give a brief conclusion.

Stability of a binge drinking model with delay.

A more realistic binge drinking model with time delay is introduced. Time delay is used to represent the time lag of the immunity against drinking in our model. For the model without the time delay, using Routh-Hurwitz criterion, we obtain that the alcohol-free equilibrium is locally asymptotically stable if [Formula: see text]. We also obtain that the unique alcohol-present equilibrium is locally asymptotically stable if [Formula: see text]. For the model with time delay, the local stability of all the equilibria is derived. Furthermore, regardless of the time delay length, using comparison theorem and iteration technique, we obtain that the alcohol-free equilibrium is globally asymptotically stable under certain conditions. Numerical simulations are also conducted to illustrate and extend our analytic results.

Modelling the effect of immigration on drinking behaviour.

A drinking model with immigration is constructed. For the model with problem drinking immigration, the model admits only one problem drinking equilibrium. For the model without problem drinking immigration, the model has two equilibria, one is problem drinking-free equilibrium and the other is problem drinking equilibrium. By employing the method of Lyapunov function, stability of all kinds of equilibria is obtained. Numerical simulations are also provided to illustrate our analytical results. Our results show that alcohol immigrants increase the difficulty of the temperance work of the region.