Search

In this paper, we have considered an SIS epidemic model with arbitrary disease transmission function and arbitrary treatment control function. We have analyzed the model considering the disease transmission function as an increasing and decreasing function separately. To introduce heterogeneity in the system we have taken both the disease transmission function and treatment function as fuzzy numbers. Fuzzy expected value of the infected individual is defined and determined. Then fuzzy basic reproduction number is computed. At last, a threshold value is determined in both the cases where the system undergoes a transcritical and backward bifurcation.

COVID-19 has become one of the most severe health issues of the twenty-first century. More than 30 million people have been affected by this deadly disease and more than nine lakh people have been died due to this disease worldwide. Government authorities are forced to impose lockdown in their respective countries. In a word, COVID-19 has stopped the natural flow of life in worldwide. In this situation, scientists and researchers of different fields are trying to find out appropriate mitigating measures for COVID-19. However, there is no approved pharmaceutical measures like treatment or vaccination available to the health workers. So, the governments are strengthening on the non-pharmaceutical measures like maintaining social distance, use of face mask and sanitizer and on lockdown policy. In this scenario, determining the dynamics of the disease can help governments, health workers, scientists to decide future action to be taken in order to tackle the disease. In this chapter, we have proposed and analyzed an SEAIR type fractional order epidemic model in order to investigate the transmission mechanisms of the COVID-19. We have studied the non-negativity and boundedness of the solutions, and the existence and uniqueness of the solutions of the system. The explicit expression of the most important parameter, namely the basic reproduction number is derived and the further investigations of the system are performed in terms of the basic reproduction number. Apart from studying the local dynamics of the fractional order system, we have considered the environmental fluctuations in the deterministic system. We have studied the stochastic stability of the infected equilibrium points. Finally, important results are illustrated through numerical simulation works.

In the present study, we develop and analyze an SEIR epidemic model to assess the consequences of media awareness program and treatment control. The disease transmission rate as well as the treatment function are taken in the saturated form. Different equilibrium points and their stability are discussed. The threshold parameter, basic reproduction number is obtained and it is seen that the system may posses a backward bifurcation. An optimal control problem is formulated with treatment and media awareness parameters as control parameters and solved it analytically. The cost-effectiveness analysis is performed to find out the best strategy to be applied to control the transmission of diseases. In addition to our analytical results, several numerical simulations are also illustrated. Finally, a brief discussion is given regarding the role of treatment and media awareness program.

In this paper, we proposed and analyzed a susceptible-infected-recovered (SIR) type epidemic model to investigate the effect of transport-related infectious diseases namely tuberculosis, measles, rubella, influenza, sexually transmitted diseases, etc. The existence and stability criteria of both the diseases include free equilibrium point and endemic equilibrium point which are established and the threshold parametric condition for which the system passes through a transcritical bifurcation is also obtained. Optimal control strategy for control parameters is formulated and solved both theoretically and numerically. Lastly, we not only illustrate our theoretical results through graphical illustrations but also computer simulation is used to show that our model would be a good model to study the SARS epidemic in 2003.

Spatial heterogeneity is an important aspect to be studied in infectious disease models. It takes two forms: one is local, namely diffusion in space, and other is related to travel. With the advancement of transportation system, it is possible for diseases to move from one place to an entirely separate place very quickly. In a developing country like India, the mass movement of large numbers of individuals creates the possibility of spread of common infectious diseases. This has led to the study of infectious disease model to describe the infection during transport. An SIRS-type epidemic model is formulated to illustrate the dynamics of such infectious disease propagation between two cities due to population dispersal. The most important threshold parameter, namely the basic reproduction number, is derived, and the possibility of existence of backward bifurcation is examined, as the existence of backward bifurcation is very unsettling for disease control and it is vital to know from modeling analysis when it can occur. It is shown that dispersal of populations would make the disease control difficult in comparison with nondispersal case. Optimal vaccination and treatment controls are determined. Further to find the best cost-effective strategy, cost-effectiveness analysis is also performed. Though it is not a case study, simulation work suggests that the proposed model can also be used in studying the SARS epidemic in Hong Kong, 2003.

In this article, we consider a susceptible–infectious–recovered SIR type epidemic model with some isolation to the susceptible, treatment for infectives, and vaccination to the newly recruited individuals. The basic reproduction number is obtained, and both the existence and stability of the disease-free and endemic equilibrium are discussed. We study the influence of isolation on infected individuals. We also study the existence of optimal control for both vaccination and treatment to minimize both the infected population and the costs required to control the disease. All the theoretical results are verified through numerical simulations and some of the key findings are given at the end of the article.

In this paper, we propose and analyze a fractional-order SIS epidemic model with the saturated treatment and disease transmission. The existence and uniqueness, nonnegativity and finiteness of solutions for our suggested model have been studied. Different equilibria of the model are found and their local and global stability analyses are also examined. Furthermore, the conditions for fractional backward and fractional Hopf bifurcation are also analyzed in both the commensurate and incommensurate fractional-order model. We study how the control parameter and the order of the fractional derivative play role in local as well as global stability of equilibrium points and Hopf bifurcation. We have demonstrated the analytical results of our proposed model system through several numerical simulations.

In this paper, we have proposed and formulated an epidemic model with two types of diseases-one is comparative weaker and the other is comparatively stronger with the assumption that both the diseases are active simultaneously in the system. The dynamical behavior of the model; equilibrium analyses with their existence criteria and local stability criteria have been discussed rigorously. With the help of second generation matrix method, we evaluate basic reproduction number of the proposed model. We propose an optimal control problem considering treatment as control parameter and solve it in order to minimize the compound loss due to the presence of infection. All the theoretical results are verified with some appropriate computer simulation works.

In the present paper we consider an SEIR type epidemic model with transport related infection between two cities. It is observed that transportation among regions has a strong impact on the dynamic evolution of a disease which can be eradicated in the absence of transportation. Transportation can lead to the incorporation of a positive risk probability. The epidemiological threshold, commonly known as the basic reproduction number, is derived and it is observed that when the basic reproduction number is less than unity the disease dies out, where as if it exceeds unity the disease may persist in the system. A thorough dynamical behavior of the constructed model is studied. We formulate and solve an optimal control problem using vaccination as a control tool. Extensive numerical simulations are carried out based on our analytical results. Finally we try to relate our work with a real world problem.

This paper describes a traditional SIR type epidemic model with saturated infection rate and treatment function. The dynamics of the model is studied from the point of view of stability and bifurcation. Basic reproduction number is obtained and it is shown that the model system may possess a backward bifurcation. The global asymptotic stability of the endemic equilibrium is studied with the help of a geometric approach. Optimal control problem is formulated and solved. Some numerical simulation works are carried out to validate our analytical results.

In this paper, we have formulated a simple SIS type epidemic model in the presence of treatment control, and we have discussed the dynamical behavior of the system. The system is modified by considering both the disease transmission rate and the treatment function as fuzzy numbers, and also the fuzzy expected value of the infected individuals is calculated. Furthermore, the fuzzy basic reproduction number is investigated and a threshold condition of pathogen is obtained at which the system undergoes a transcritical bifurcation.

In this paper we take a sound approach to construct a susceptible-exposed-infected-recovered-susceptible type epidemic model with explicit vaccination control. The disease transmission from susceptible to infected class is taken in saturated form. The system is investigated both for constant control and optimal control cases. Basic reproduction number and its consequences to the system has been established. Geometric approach is used to obtain the global asymptotic stability condition of the system. Finally, with the help of some numerical illustrations, we check the validity of our theoretical results.

This paper describes the dynamical behavior of an epidemic model in the presence of vaccination and treatment controls. In the first part of our analysis we study both the local as well as global dynamics of the deterministic model. Next in the second part of our analysis we modify the model with an additional perturbation around the endemic equilibrium and study the asymptotically mean square stability of the modified system. Some numerical simulations are carried out on both the deterministic and the modified models to support our analytical results.

This paper formulates and analyses a susceptible-exposed-infected-recovered (SEIR) type epidemic model with the effect of transport-related infection between two cities in the presence of treatment control. The dispersal of populations from one city to another city has an important impact on the dynamics of disease evolution. The basic reproduction number is calculated for all the different cases of the proposed model system. It is found out that the disease-free equilibrium is disease-free if the basic reproduction is less than unity, otherwise the disease may remain in the system. In addition, the optimal control problem is constructed and solved analytically and numerically by considering the treatment control as a control variable. Further, we present a numerical simulation to confirm the analytical results. Finally, we show a comparison of the result of our predicted model with the real data of severe acute respiratory syndrome (SARS) outbreak in 2003 in Hong Kong.

In this paper, we describe an SIS epidemic model where both the disease transmission rate and treatment function are considered in saturated forms. The dynamical behavior of the system is analyzed. The system is customized by considering the disease transmission rate and treatment control as fuzzy numbers and then fuzzy expected value of the infected individuals is determined. The fuzzy basic reproduction number is investigated and a threshold condition of pathogen is derived at which the system undergoes a backward bifurcation.

In this paper, we propose and analyze an epidemic problem which can be controlled by vaccination as well as treatment. In the first part of our analysis we study the dynamical behavior of the system with fixed control for both vaccination and treatment. Basic reproduction number is obtained in all possible cases and it is observed that the simultaneous use of vaccination and treatment control is the most favorable case to prevent the disease from being epidemic. In the second part, we take the controls as time dependent and obtain the optimal control strategy to minimize both the infected populations and the associated costs. All the analytical results are verified by simulation works. Some important conclusions are given at the end of the paper.