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 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.

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.