Synchronization plays a significant role in information transfer and decision-making by neurons and brain neural networks. The development of control strategies for synchronizing a network of chaotic neurons with time delays, different direction-dependent coupling (unidirectional and bidirectional), and noise, particularly under external disturbances, is an essential and very challenging task. Researchers have extensively studied the synchronization mechanism of two coupled time-delayed neurons with bidirectional coupling and without incorporating the effect of noise, but not for time-delayed neural networks. To overcome these limitations, this study investigates the synchronization problem in a network of coupled FitzHugh–Nagumo (FHN) neurons by incorporating time delays, different direction-dependent coupling (unidirectional and bidirectional), noise, and ionic and external disturbances in the mathematical models. More specifically, this study investigates the synchronization of time-delayed unidirectional and bidirectional ring-structured FHN neuronal systems with and without external noise. Different gap junctions and delay parameters are used to incorporate time-delay dynamics in both neuronal networks. We also investigate the influence of the time delays between connected neurons on synchronization conditions. Further, to ensure the synchronization of the time-delayed FHN neuronal networks, different adaptive control laws are proposed for both unidirectional and bidirectional neuronal networks. In addition, necessary and sufficient conditions to achieve synchronization are provided by employing the Lyapunov stability theory. The results of numerical simulations conducted for different-sized multiple networks of time-delayed FHN neurons verify the effectiveness of the proposed adaptive control schemes.

On March 15, 2020, 61.3% of the confirmed cases of COVID-19 infection in South Korea are associated with the worship service that was organized on February 9 in the Shincheonji Church of Jesus in Daegu. We aim to evaluate the effects of mass infection in South Korea and assess the preventive control intervention. Using openly available data of daily cumulative confirmed cases and deaths, the basic and effective reproduction numbers was estimated using a modified susceptible-exposed-infected-recovered-type epidemic model. The basic reproduction number was estimated to be R0=1.77. The effective reproduction number increased approximately 20 times after the mass infections from the 31 st patient, which was confirmed on February 9 in the Shincheonji Church of Jesus, Daegu. However, the effective reproduction number decreased to less than unity after February 28 owing to the implementation of high-level preventive control interventions in South Korea, coupled with voluntary prevention actions by citizens. Preventive action and control intervention were successfully established in South Korea.

The paper considers a model for the transmission dynamics of a vector‐borne disease with nonlinear incidence rate. It is proved that the global dynamics of the disease are completely determined by the basic reproduction number. In order to assess the effectiveness of disease control measures, the sensitivity analysis of the basic reproductive number R₀ and the endemic proportions with respect to epidemiological and demographic parameters are provided. From the results of the sensitivity analysis, the model is modified to assess the impact of three control measures; the preventive control to minimize vector human contacts, the treatment control to the infected human, and the insecticide control to the vector. Analytically the existence of the optimal control is established by the use of an optimal control technique and numerically it is solved by an iterative method. Numerical simulations and optimal analysis of the model show that restricted and proper use of control measures might considerably decrease the number of infected humans in a viable way.

The mathematical modeling of malaria disease has a crucial role in understanding the insights of the transmission dynamics and corresponding appropriate prevention strategies. In this study, a novel nonlinear mathematical model for malaria disease has been proposed. To prevent the disease, we divided the infected population into two groups, unaware and aware infected individuals. The growth rate of awareness programs impacting the population is assumed to be proportional to the unaware infected individuals. It is further assumed that, due to the effect of awareness campaign, the aware infected individuals avoid contact with mosquitoes. The positivity and the boundedness of solutions have been derived through the completing differential process. Local and global stability analysis of disease-free equilibrium has been investigated via basic reproductive number R₀, if R₀<1, the system is stable otherwise unstable. The existence of the unique endemic equilibrium has been also determined under certain conditions. The solution to the proposed model is derived through an iterative numerical technique, the RungeKutta method. The proposed model is simulated for different numeric values of the population of humans and anopheles in each class. The results show that a significant increase in the population of susceptible humans is achieved in addition to the decrease in the population of the infected mosquitoes.

Malaria has persisted as an endemic near the Demilitarized Zone in the Republic of Korea since the re-emergence of Plasmodium vivax malaria in 1993. The number of patients affected by malaria has increased recently despite many controls tools, one of the reasons behind which is the relapse of malaria via liver hypnozoites. Tafenoquine, a new drug approved by the United States Food and Drug Administration in 2018, is expected to reduce the rate of relapse of malaria hypnozoites and thereby decrease the prevalence of malaria among the population. In this work, we have developed a new transmission model for Plasmodium vivax that takes into account a more realistic intrinsic distribution from existing literature to quantify the current values of relapse parameters and to evaluate the effectiveness of the anti-relapse therapy. The model is especially suitable for estimating parameters near the Demilitarized Zone in Korea, in which the disease follows a distinguishable seasonality. Results were shown that radical cure could significantly reduce the prevalence level of malaria. However, eradication would still take a long time (over 10 years) even if the high-level treatment were to persist. In addition, considering that the vector’s behavior is manipulated by the malaria parasite, relapse repression through vector control at the current level may result in a negative effect in containing the disease. We conclude that the use of effective drugs should be considered together with the increased level of the vector control to reduce malaria prevalence.

Mathematical modeling has become a valuable tool for the analysis of dynamics of infectious disease and for the support of control strategies development in recent years. This work highlights the conceptual ideas and mathematical tools needed for infectious diseases modeling. The main convergence of this was on the dynamics of infectious diseases, the analysis of transmission patterns in various populations, and methods to assess the effectiveness of control strategies such as HIV, childhood infections, influenza, and vector borne infections. It was concerned with qualitative behaviors of infectious disease model. The qualitative behavior of model includes positivity, uniqueness, local stability, global stability, bifurcation analysis, control of diseases, and existence of solutions. This study provided a platform for the discussion of the major research challenges and achievements on qualitative behaviors of infectious diseases and their control. Due to the availability of a lot of applications of this study, many authors contributed. G. R. Phaijoo and D. B. Gurung demonstrated that dengue is spreading in new areas due to people movement. They considered a multipatch model to assess the impact of temperature and human movement on the transmission dynamics of dengue disease. Dynamics of vector and host populations are investigated with different human movement rates and different temperature levels. N. Pipatsart et al. discussed adaptive random network models to describe human behavioral change during epidemics and performed stochastic simulations of SIR epidemic models on adaptive random networks. C. Burgess et al. derived risk-based immunization by deployment to polio-endemic regions, which is sufficient to prevent transmission among both deployed and nondeployed US military populations. T. Tilahun et al. analyzed a compartmental nonlinear deterministic mathematical model for the typhoid fever outbreak and optimal control strategies in a community with varying population. L. Worden et al. extended an approximation technique of the long-term behavior of a supercritical stochastic epidemic model, using the WKB approximation and a Hamiltonian phase space, to the subcritical case. S. Anis et al. discussed some of the genetic properties on the basis of algebra. Finally, A. Miao et al. proposed a stochastic SIR model with vertical transmission and vaccination. They showed that large noise can lead to the extinction of infectious diseases, which is conducive to epidemic diseases control.