The description of the variables and parameters of the model.

Description of the sensitivity indices and behaviour of parameters involved in and .

The parameter values of model Eq 1 for point of reference. For , are taken with suitable values of and [64].

A minimally parameterized mathematical model for low-dose metronomic chemotherapy is formulated that takes into account angiogenic signaling between the tumor and its vasculature and tumor inhibiting effects of tumor-immune system interactions. The dynamical equations combine a model for tumor development under angiogenic signaling formulated by Hahnfeldt et al. with a model for tumor-immune system interactions by Stepanova. The dynamical properties of the model are analyzed. Depending on the parameter values, the system encompasses a variety of medically realistic scenarios that range from cases when (i) low-dose metronomic chemotherapy is able to eradicate the tumor (all trajectories converge to a tumor-free equilibrium point) to situations when (ii) tumor dormancy is induced (a unique, globally asymp- totically stable benign equilibrium point exists) to (iii) multi-stable situations that have both persistent benign and malignant behaviors separated by the stable manifold of an unstable equilibrium point and finally to (iv) situations when tumor growth can- not be overcome by low-dose metronomic chemotherapy. The model forms a basis for a more general study of chemotherapy when the main components of a tumor’s microenvironment are taken into account

Efficiency indices values w.r.t. the strategies O1 and O2.

This study presents a reliable mathematical model to explain the spread of typhoid fever, covering stages of susceptibility, infection, carrying, and recovery, specifically in the Sheno town community. A detailed analysis is done to ensure the solutions are positive, stay within certain limits, and are stable for both situations where the disease is absent and where it is consistently present. The Routh-Hurwitz stability criterion has been used and applied for the purpose of stability analysis. Using the next-generation matrix, we determined the intrinsic potential for disease transmission. It showing that typhoid fever is spreading actively in Sheno town, with cases above a critical level. Our findings reveal the instability of the disease-free equilibrium point alongside the stability of the endemic equilibrium point. We identified two pivotal factors for transmission of the disease: the infectious rate, representing the speed of disease transmission, and the recruitment rate, indicating the rate at which new individuals enter the susceptible population. These parameters are indispensable for devising effective control measures. It is imperative to keep these parameters below specific thresholds to maintain a basic reproduction number favorable for disease control. Additionally, the study carefully examines how different factors affect the spread of typhoid fever, giving a detailed understanding of its dynamics. At the end, this study provides valuable insights and specific strategies for managing the disease in the Sheno town community.

Biological representation of models’ parameters and variables.

There are thousands of languages in the world, many of which are in danger of extinction due to language competition and evolution. Language is an aspect of culture, the rise, and fall of a language directly affects its corresponding culture. To preserve languages and prevent their mass extinction, it is crucial to develop a mathematical model of language coexistence. In this paper, we use a qualitative theory of ordinary differential equations to analyze the bilingual competition model, and obtain the trivial and non-trivial solutions of the bilingual competition model without sliding mode control, then analyze the stability of solutions and prove that solutions of the model have positive invariance. In addition, to maintain linguistic diversity and prevent mass extinction of languages, we propose a novel bilingual competition model with sliding control. The bilingual competition model is analyzed by proposing a sliding control policy to obtain a pseudo-equilibrium point. Meanwhile, numerical simulations clearly illustrate the effectiveness of the sliding mode control strategy. The results show that the likelihood of successful language coexistence can be increased by changing the status of languages and the value of monolingual-bilingual interaction, provides theoretical analysis for the development of policies to prevent language extinction.

The prevalence of the varicella-zoster virus (VZV) and its correlation underscore its impact on a significant segment of the population. Notably contagious, VZV serves as a risk factor for the manifestation of HIV/AIDS, with its reactivation often signaling the onset of immunodeficiency. Recognizing the concurrent existence of these two diseases, this study focuses on the co-infection dynamics through a deterministic mathematical model. The population is categorized into seven exclusive groups, considering the complexities arising from the interplay of HIV and Zoster. We establish the non-negativity and boundedness of solutions, examine equilibrium points, calculate basic reproduction numbers via the next-generation matrix approach, and analyze the existence and local stabilities of equilibriums using the Routh-Hurwitz stability criteria. The numerical simulations reveal that the model converges to an endemic equilibrium point when the reproduction number exceeds unity. The primary objectives of this study are to comprehensively understand the transmission dynamics of HIV and Zoster in a co-infected population and to provide valuable insights for developing effective intervention strategies. The findings emphasize the importance of addressing these co-infections to mitigate their impact on public health.

The gathered data about the total population and typhoid fever in sheno town.

Comparison of numerical scheme accuracy across different time levels N for various values of ϕ.

In this research, the ongoing COVID-19 disease by considering the vaccination strategies into mathematical models is discussed. A modified and comprehensive mathematical model that captures the complex relationships between various population compartments, including susceptible (Sα), exposed (Eα), infected (Uα), quarantined (Qα), vaccinated (Vα), and recovered (Rα) individuals. Using conformable derivatives, a system of equations that precisely captures the complex interconnections inside the COVID-19 transmission. The basic reproduction number (R0), which is an essential indicator of disease transmission, is the subject of investigation calculating using the next-generation matrix approach. We also compute the R0 sensitivity indices, which offer important information about the relative influence of various factors on the overall dynamics. Local stability and global stability of R0 have been proved at a disease-free equilibrium point. By designing the finite difference approach of the conformable fractional derivative using the Taylor series. The present methodology provides us highly accurate convergence of the obtained solution. Present research fills research addresses the understanding gap between conceptual frameworks and real-world implementations, demonstrating the vaccination therapy’s significant possibilities in the struggle against the COVID-19 pandemic.

Immune system parameter values.

Multiscale model parameter values.

There are thousands of languages in the world, many of which are in danger of extinction due to language competition and evolution. Language is an aspect of culture, the rise, and fall of a language directly affects its corresponding culture. To preserve languages and prevent their mass extinction, it is crucial to develop a mathematical model of language coexistence. In this paper, we use a qualitative theory of ordinary differential equations to analyze the bilingual competition model, and obtain the trivial and non-trivial solutions of the bilingual competition model without sliding mode control, then analyze the stability of solutions and prove that solutions of the model have positive invariance. In addition, to maintain linguistic diversity and prevent mass extinction of languages, we propose a novel bilingual competition model with sliding control. The bilingual competition model is analyzed by proposing a sliding control policy to obtain a pseudo-equilibrium point. Meanwhile, numerical simulations clearly illustrate the effectiveness of the sliding mode control strategy. The results show that the likelihood of successful language coexistence can be increased by changing the status of languages and the value of monolingual-bilingual interaction, provides theoretical analysis for the development of policies to prevent language extinction.

Sliding mode control language status and interaction values.

This study aims to investigate and analyze the dynamics of diarrhea infectious disease model. For this purpose, a classical diarrhea disease model is converted into the diffusive diarrhea epidemic model by including the diffusion terms in every compartment of the system. Basic assumptions of the proposed model are described for a vivid understanding of the model’s behavior. In addition, the pros and cons of the proposed model for short and long terms behavior of the diffusive system are presented. The system has two steady states, namely the disease-free equilibrium and endemic equilibrium points. The system is analyzed, analytically by ensuring the positivity, boundedness and local, and global stability at both the steady states. Moreover, the implicit nonstandard finite difference scheme is designed to extract the numerical solutions of the diffusive epidemic model. To ensure the reliability and efficacy of the numerical scheme, the positivity, consistency and both linear and nonlinear stabilities are presented by establishing some standard results. Simulated graphs are sketched to study the nonlinear behavior of the disease dynamics. All the graphs depict the positive, bounded and convergent behavior of the projected numerical scheme. Also, the numerical graphs reflect the role of the basic reproductive number, R0, in attaining the steady state. The article is closed by providing productive outcomes of the study.

Malaria is a mosquito-borne disease spread by an infected vector (infected female Anopheles mosquito) or through transfusion of plasmodium-infected blood to susceptible individuals. The disease burden has resulted in high global mortality, particularly among children under the age of five. Many intervention responses have been implemented to control malaria disease transmission, including blood screening, Long-Lasting Insecticide Bed Nets (LLIN), treatment with an anti-malaria drug, spraying chemicals/pesticides on mosquito breeding sites, and indoor residual spray, among others. As a result, the SIR (Susceptible—Infected—Recovered) model was developed to study the impact of various malaria control and mitigation strategies. The associated basic reproduction number and stability theory is used to investigate the stability analysis of the model equilibrium points. By constructing an appropriate Lyapunov function, the global stability of the malaria-free equilibrium is investigated. By determining the direction of bifurcation, the implicit function theorem is used to investigate the stability of the model endemic equilibrium. The model is fitted to malaria data from Benue State, Nigeria, using R and MATLAB. Estimates of parameters were made. Following that, an optimal control model is developed and analyzed using Pontryaging's Maximum Principle. The malaria-free equilibrium point is locally and globally stable if the basic reproduction number (R0) and the blood transfusion reproduction number (Rα) are both less or equal to unity. The study of the sensitive parameters of the model revealed that the transmission rate of malaria from mosquito-to-human (βmh), transmission rate from humans-to-mosquito (βhm), blood transfusion reproduction number (Rα) and recruitment rate of mosquitoes (bm) are all sensitive parameters capable of increasing the basic reproduction number (R0) thereby increasing the risk in spreading malaria disease. The result of the optimal control shows that five possible controls are effective in reducing the transmission of malaria. The study recommended the combination of five controls, followed by the combination of four and three controls is effective in mitigating malaria transmission. The result of the optimal simulation also revealed that for communities or areas where resources are scarce, the combination of Long Lasting Insecticide Treated Bednets (u2), Treatment (u3), and Indoor insecticide spray (u5) is recommended. Numerical simulations are performed to validate the model's analytical results.

ObjectiveThis study aims to analyze how wage structure and individual ability affect practitioners' participation in primary healthcare.MethodologyThis study employs a mixed moral hazard and adverse selection model to analyze the optimal performance-based wage for general practitioners under separating and pooling equilibrium. Subsequently, we utilize tripartite evolutionary games to analyze the dynamic process of participation strategies to the primary health care with high—and less experienced practitioners.FindingsOur study yields four main findings: (1) Under an effective separating equilibrium, high-quality practitioners receive information rent, and there is no distortion at the top, while less experienced practitioners face allocation distortion. (2) When the performance risk of less experienced practitioners is greater than or equal to that of high-quality practitioners, reducing the performance risk of less experienced practitioners is an effective method of increasing their performance wage. Conversely, when the performance risk of less experienced practitioners is less than that of high-quality practitioners, and they can transform into high-quality practitioners by increasing education costs, they will be incentivized to continue as general practitioners, provided there is a precise promotion mechanism. (3) If reforms are made to the wage structure in primary healthcare, an effective approach is to increase the proportion of the floating part. This would lead primary healthcare institutions to choose contracts under separating equilibrium, encouraging high-quality practitioners to participate in primary healthcare and less experienced practitioners to improve their abilities by increasing education costs. However, the proportion of the floating part in the wage structure should not be excessively large. (4) The effective cost of medical resource utilization influences the wage structure, and establishing reasonable upper and lower limits for performance wages can effectively increase the incentive for high-quality practitioners to participate in primary healthcare.SignificanceFor the first time, our study employs a tripartite evolutionary game model to analyze the development of the general practitioner system. We analyze how the reform of the wage structure can encourage more practitioners to participate in primary healthcare. Our findings can lay the theoretical foundation for subsequent empirical analyses. Moreover, our findings provide theoretical assistance for government decisions and healthcare institutions.

In this paper, we present two different approaches to represent/predict the gas hydrate phase equilibria for the carbon dioxide, methane, or ethane + pure water system in the presence of various types of porous media with different pore sizes. The studied porous media include silica gel, mesoporous silica, and porous silica glass. First, a correlation is presented, which estimates the hydrate suppression temperature due to the pore effects from the ice point depression (IPD). In the second place, several mathematical models are proposed using the least squares support vector machine (LSSVM) algorithm for the determination of the dissociation pressures of the corresponding systems. The results indicate that although the applied correlation based on the (IPD) leads to obtaining reliable results for the gas hydrate systems in the presence of porous silica glass media, the developed LSSVM models seem to be more general due to their predictive capability over all of the investigated systems.