<abstract><p>Consistently, influenza has become a major cause of illness and mortality worldwide and it has posed a serious threat to global public health particularly among the immuno-compromised people all around the world. The development of medication to control influenza has become a major challenge now. This work proposes and analyzes a structured model based on two geographical areas, in order to study the spread of influenza. The overall underlying population is separated into two sub populations: urban and rural. This geographical distinction is required as the immunity levels are significantly higher in rural areas as compared to urban areas. Hence, this paper is a novel attempt to proposes a linear and non-linear mathematical model with adaptive immunity and compare the host immune response to disease. For both the models, disease-free equilibrium points are obtained which are locally as well as globally stable if the reproduction number is less than 1 (<italic>R</italic><sub>01</sub> < 1 & <italic>R</italic><sub>02</sub> < 1) and the endemic point is stable if the reproduction number is greater then 1 (<italic>R</italic><sub>01</sub> > 1 & <italic>R</italic><sub>02</sub> > 1). Next, we have incorporated two treatments in the model that constitute the effectiveness of antidots and vaccination in restraining viral creation and slow down the production of new infections and analyzed an optimal control problem. Further, we have also proposed a spatial model involving diffusion and obtained the local stability for both the models. By the use of local stability, we have derived the Turing instability condition. Finally, all the theoretical results are verified with numerical simulation using MATLAB.</p></abstract>
This paper focuses on the dynamics of the mathematical model by considering the effect of human conduct and the role of immunity on cholera contamination. Since, spreading of infection is commonly modeled with the help of classes of the SIR model, the model is formulated as a SIR model inaugurated with one more compartment B which showcases the concentration of bacteria in the polluted water. We have obtained the basic reproductive rate R0 and showcased local stability as well as global stability through geometrical approach. Without infection, equilibrium point (E0 ) is stable locally in conjunction with globally whenever 0 R <1. Similarly, the stability condition for endemic equilibrium E* is 0 R ≥ 1. Hence, our aim is basically to explore the outcome as to how the alertness of mankind can play an effective role in the dynamics of such infections.
Flu, a common respiratory disease is caused mainly by the influenza virus. The Avian influenza (H5N1) outbreaks, as well as the 2009 H1N1 pandemic, have heightened global concerns about the emergence of a lethal influenza virus capable of causing a catastrophic pandemic. During the early stages of an epidemic a favourable change in the behaviour of people can be of utmost importance. An economic status-based (higher and lower economic class) structured model is formulated to examine the behavioural effect in controlling influenza. Following that, we have introduced controls into the model to analyse the efficacy of antiviral treatment in restraining infections in both economic classes and examined an optimal control problem. We have obtained the reproduction number along with the final epidemic size for both the strata and the relation between reproduction number and epidemic size. Through numerical simulation and global sensitivity analysis, we have shown the importance of the parameters and on reproduction number. Our result shows that by increasing , and by decreasing , and , we can reduce the infection in both the economic group. As a result of our analysis, we have found that the reduction of infections and their level of adversity is directly influenced by positive behavioural patterns or changes as without control susceptible population is increased by , the infective population is decreased by and the recovered population is increased by in the higher economic group who opted changed behaviour as compared to the lower the economic group (people living with normal behaviour). Thus normal behaviour contributes to the spread and growth of viruses and adds to the hassle. We also examined how antiviral drug control impacts both economic strata and found that in the higher economic strata, the susceptible population increased by , the infective population decreased by and the recovered population improved by as compared to the lower economic group, the susceptible population has increased by , the infective population is decreased by and the recovered population is improved by . Our results enlighten the role that how different behaviour in separate socio-economic class plays an important role in changing the dynamics of the system and also affects the basic reproduction number. The results of our study suggest that it is important to adopt a modified behaviour like social distancing, wearing masks accompanying the time-dependent controls in the form of an antiviral drug’s effectiveness to combat infections and increasing the proportion of the susceptible population.
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