Search Results (8)

Measles is the most infectious disease with a high basic reproduction number ($R_0$). For measles, it is reported that $R_0$ lies between 12 and 18 in an endemic situation. In this paper, a fractional order mathematical model for measles disease is proposed to identify the dynamics of disease transmission following a declining memory process. In the proposed model, a fractional order differential operator is used to justify the effect and success rate of vaccination. The total population of the model is subdivided into five sub-compartments: susceptible (S), exposed (E), infected (I), vaccinated (V), and recovered (R). Here, we consider the first dose of measles vaccination and convert the model to a controlled system. Finally, we transform the control-induced model to an optimal control model using control theory. Both models are analyzed to find the stability of the system, the basic reproduction number, the optimal control input, and the adjoint equations with the boundary conditions. Also, the numerical simulation of the model is presented along with using the analytical findings. We also verify the effective role of the fractional order parameter alpha on the model dynamics and changes in the dynamical behavior of the model with $R_0=1$. Full article

The main objective of this study is to find out the influences of cooperation and intra-specific competition in the prey population on escaping predation through refuge and the effect of the two intra-specific interactions on the dynamics of prey–predator systems. For this purpose, two mathematical models with Holling type II functional response functions were proposed and analyzed. The first model includes cooperation among prey populations, whereas the second one incorporates intra-specific competition. The existence conditions and stability of different equilibrium points for both models were analyzed to determine the qualitative behaviors of the systems. Refuge through intra-specific competition has a stabilizing role, whereas cooperation has a destabilizing role on the system dynamics. Periodic oscillations were observed in both systems through Hopf bifurcation. From the analytical and numerical findings, we conclude that intra-specific competition affects the prey population and continuously controls the refuge class under a critical value, and thus, it never becomes too large to cause predator extinction due to food scarcity. Conversely, cooperation leads the maximal number of individuals to escape predation through the refuge so that predators suffer from low predation success. Full article

Infectious diseases continue to be a significant threat to human health and civilization, and finding effective methods to combat them is crucial. In this paper, we investigate the impact of awareness campaigns and optimal control techniques on infectious diseases without proper vaccines. Specifically, we develop an SIRS-type mathematical model that incorporates awareness campaigns through media and treatment for disease transmission dynamics and control. The model displays two equilibria, a disease-free equilibrium and an endemic equilibrium, and exhibits Hopf bifurcation when the bifurcation parameter exceeds its critical value, causing a switch in the stability of the system. We also propose an optimal control problem that minimizes the cost of control measures while achieving a desired level of disease control. By applying the minimum principle to the optimal control problem, we obtain analytical and numerical results that show how the infection rate of the disease affects the stability of the system and how awareness campaigns and treatment can maintain the stability of the system. This study highlights the importance of awareness campaigns in controlling infectious diseases and demonstrates the effectiveness of optimal control theory in achieving disease control with minimal cost. Full article

We formulate an integrated pest management model to control natural pests of the crop through the periodic application of biopesticide and chemical pesticides. In a theoretical analysis of the system pest eradication, a periodic solution is found and established. All the system variables are proved to be bounded. Our main goal is then to ensure that pesticides are optimized, in terms of pesticide concentration and pesticide application frequency, and that the optimum combination of pesticides is found to provide the most benefit to the crop. By using Floquet theory and the small amplitude perturbation method, we prove that the pest eradication periodic solution is locally and globally stable. The acquired results establish a threshold time limit for the impulsive release of various controls as well as some valid theoretical conclusions for effective pest management. Furthermore, after a numerical comparison, we conclude that integrated pest management is more effective than single biological or chemical controls. Finally, we illustrate the analytical results through numerical simulations. Full article

Malaria is a serious illness caused by a parasite, called Plasmodium, transmitted to humans through the bites of female Anopheles mosquitoes. The parasite infects and destroys the red blood cells in the human body leading to symptoms, such as fever, headache, and flu-like illness. Awareness campaigns that educate people about malaria prevention and control reduce transmission of the disease. In this research, a mathematical model is proposed to study the impact of awareness-based control measures on the transmission dynamics of malaria. Some basic properties of the proposed model, such as non-negativity and boundedness of the solutions, the existence of the equilibrium points, and their stability properties, have been studied using qualitative theory. Disease-free equilibrium is globally asymptotic when the basic reproduction number, ${\mathcal{R}}_0$, is less than the number of current cases. Finally, optimal control theory is applied to minimize the cost of disease control and solve the optimal control problem by applying Pontryagin’s minimum principle. Numerical simulations have been provided for the confirmation of the analytical results. Endemic equilibrium exists for ${\mathcal{R}}_0>1$, and a forward transcritical bifurcation occurs at ${\mathcal{R}}_0=1$. The optimal profiles of the treatment process, organizing awareness campaigns, and insecticide uses are obtained for the cost-effectiveness of malaria management. This research concludes that awareness campaigns through social media with an optimal control approach are best for cost-effective malaria management. Full article

Mathematical modeling is crucial to investigating tthe ongoing coronavirus disease 2019 (COVID-19) pandemic. The primary target area of the SARS-CoV-2 virus is epithelial cells in the human lower respiratory tract. During this viral infection, infected cells can activate innate and adaptive immune responses to viral infection. Immune response in COVID-19 infection can lead to longer recovery time and more severe secondary complications. We formulate a micro-level mathematical model by incorporating a saturation term for SARS-CoV-2-infected epithelial cell loss reliant on infected cell levels. Forward and backward bifurcation between disease-free and endemic equilibrium points have been analyzed. Global stability of both disease-free and endemic equilibrium is provided. We have seen that the disease-free equilibrium is globally stable for $R_0<1$, and endemic equilibrium exists and is globally stable for $R_0>1$. Impulsive application of drug dosing has been applied for the treatment of COVID-19 patients. Additionally, the dynamics of the impulsive system are discussed when a patient takes drug holidays. Numerical simulations support the analytical findings and the dynamical regimes in the systems. Full article

During the past several years, the deadly COVID-19 pandemic has dramatically affected the world; the death toll exceeds 4.8 million across the world according to current statistics. Mathematical modeling is one of the critical tools being used to fight against this deadly infectious disease. It has been observed that the transmission of COVID-19 follows a fading memory process. We have used the fractional order differential operator to identify this kind of disease transmission, considering both fear effects and vaccination in our proposed mathematical model. Our COVID-19 disease model was analyzed by considering the Caputo fractional operator. A brief description of this operator and a mathematical analysis of the proposed model involving this operator are presented. In addition, a numerical simulation of the proposed model is presented along with the resulting analytical findings. We show that fear effects play a pivotal role in reducing infections in the population as well as in encouraging the vaccination campaign. Furthermore, decreasing the fractional-order parameter $\alpha$ value minimizes the number of infected individuals. The analysis presented here reveals that the system switches its stability for the critical value of the basic reproduction number $R_0=1$. Full article

In this paper, the aim is to capture the global pandemic of COVID-19 with parameters that consider the interactions among individuals by proposing a mathematical model. The introduction of a parsimonious model captures both the isolation of symptomatic infected individuals and population lockdown practices in response to containment policies. Local stability and basic reproduction numbers are analyzed. Local sensitivity indices of the parameters of the proposed model are calculated, using the non-normalization, half-normalization, and full-normalization techniques. Numerical investigations show that the dynamics of the system depend on the model parameters. The infection transmission rate (as a function of the lockdown parameter) for both reported and unreported symptomatic infected peoples is a significant parameter in spreading the infection. A nationwide public lockdown decreases the number of infected cases and stops the pandemic’s peak from occurring. The results obtained from this study are beneficial worldwide for developing different COVID-19 management programs. Full article