Unique Equilibrium Research Articles

Mathematical analysis of the effects of precarious health system on COVID-19 transmission dynamics: case of Burkina Faso

n the early moments of the pandemic of coronavirus disease 2019 (COVID-19), the level of effectiveness in treatment and management of patients was limited due to the lack of effective medications and adapted infrastructures in several countries especially in most developing countries. In this study, we propose an SIR-pattern model to investigate the epidemiological effect of delay in hospitalization and case managements. After computing the basic reproduction number thanks to the Next Generation Matrix method, we proved that the model has a locally asymptotically stable disease-free equilibrium whenever the basic reproduction number is less than one, and condition which ensure the appearance of two endemic equilibria indicating the possibility of having backward bifurcation is determined. When the basic reproduction number exceeds one, Lyapunov functional technique is used to establish the global stability of the unique endemic equilibrium point. Also, we presented a local sensitivity analysis that gives better understanding of the role of each model parameter appearing in the basic reproduction number. Numerical simulation of the model is also performed to corroborate the analytical results. Finally, we fitted our model by using reported data for Burkina Faso, and the results showed that it can be used for predicting the disease outcomes.

A Dynamic Game Analysis of Hospitals’ Quality-Improving and Congestion-Reducing with Corresponding Knowledge Accumulations Effects

In this paper, we develop a dynamic control to investigate the hospitals’ quality improvement and congestion reduction, along with corresponding knowledge accumulations. The main features of our work are: (i) developing a dynamic analysis model of the hospitals’ quality improvement and congestion reduction, considering knowledge accumulations in health care markets; (ii) the demand function for medical treatment depends on treatment quality, congestion degree, price, and traveling cost; (iii) each hospital’s instantaneous costs of quality improvement and congestion reduction depend on treatment quality, quality-improving investment, corresponding knowledge accumulation as well as congestion degree, congestion-reducing investment and the corresponding knowledge accumulation; (iv) the hospitals’ dynamic competitions are not only in treatment quality and congestion degree but also in treatment prices. Our results show that (i) there exists a unique saddle-stable steady-state equilibrium under hospital optimum and social optimum; (ii) knowledge accumulation and the complementarity or substitutability affect the hospitals’ decision behavior; (iii) whether the price and the hospitals’ investments in quality improvement and congestion reduction are higher or lower under hospital optimum than that under social optimum depends on the parameter regions of complementarity or substitutability.

Lognormal (re)distribution: A macrofounded theory of inequality

We study how political competition over redistribution determines income inequality under one macrofounded premise: the income distribution is approximately log-normal before and after any policy intervention. The unique equilibrium features substantial income inequality and less than maximal redistribution, even if the voters’ median income is very low. Fitting our model to the US economy, we argue that either the efficiency cost of redistribution is higher than estimates in the literature, or else US’s redistributive policies are optimal for an agent richer than the median voter.

A metapopulation model with exit screening measure for the 2014–2016 West Africa Ebola virus outbreak

We construct a new metapopulation model for the transmission dynamics and control of the Ebola Virus Disease (EVD) in an environment characterized by considerable migrations and travels of people. It is an extended SEIR model modified by the addition of Quarantine and Isolated compartments to account for travelers who undergo the exit screening. The model is well-fitted by using the reported cases from the neighboring countries Guinea, Liberia and Sierra Leone where the 2014–2016 Ebola outbreak simultaneously arose. We show that the unique disease-free equilibrium (DFE) of the model is unstable or locally asymptotically stable (LAS) depending on whether the control reproduction number is larger or less than unity. In the latter case, we prove that the DFE is globally asymptotically stable (GAS) provided that the exit screening is 100% negative. We also prove the GAS of the DFE by introducing more explicit thresholds, thanks to which the existence of at least one boundary equilibrium is established. We design two new nonstandard finite difference (NSFD) schemes, which preserve the dynamics of the continuous model. Numerical simulations that support the theory highlight that exit screening is useful to mitigate the infection. They also suggest that the disease is controlled or the explicit threshold is less than unity provided that the migration and the exit screening parameters are above a critical value.

A dynamical optimal control theory and cost-effectiveness analyses of the HBV and HIV/AIDS co-infection model.

Studies have shown that the co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B Virus (HBV) poses a major threat to the public health due to their combined negative impacts on health and increased risk of complications. Even though, some scholars formulated and analyzed the HBV and HIV co-infection model they did not consider the compartment that contains protected individuals against both HBV and HIV infections. They incorporated the optimal control theory and cost-effectiveness analysis simultaneously. With this in mind, we are motivated to formulate and analyze the HBV and HIV co-infection model, considering the protected group and incorporating optimal control theory and cost-effectiveness. In this study, we have theoretically computed all of the models disease-free equilibrium points, all the models effective reproduction numbers and unique endemic equilibrium points. The two sub-models disease-free equilibrium points are locally as well as globally asymptotically stable whenever their associated effective reproduction numbers are less than one. We reformulated the optimal control problem by incorporating five time-dependent control measures and conducted its theoretical analysis by utilizing the Pontryagin's maximum principle. Using the fourth order Runge-Kutta numerical method and MATLAB ODE45, we performed the numerical simulations with various combinations of control efforts to verify the theoretical results and investigate the impacts of the suggested protection and treatment control strategies for both the HBV and HIV diseases. Also, we carried out a cost-effectiveness analysis of the proposed control strategies. Eventually, we compared our model results with other researcher similar model results whenever cost-effectiveness analysis is not carried out the findings of this particular study suggest that implementing each of the proposed control strategies simultaneously has a high potential to reduce and control the spread of HBV and HIV co-infections in the community. According to the cost-effectiveness analysis, implementing the HBV treatment and the HIV and HBV co-infection treatment measures has a high potential effect on reducing and controlling the HBV and HIV co-infection transmission problem in the community.

Arms Spending and War Casualties: Understanding the Strategic Trade-Offs in Warfare

ABSTRACT This paper formulates an armed conflict as a game-theoretic model, and relying on the empirical evidence that war casualties are costly for governments due to loss of public support, it investigates the strategic trade-off between arms spending and casualty costs. By characterizing the unique Nash equilibrium in our conflict model, we offer insights into how each side acts in equilibrium when one side’s war efforts are more destructive, or its citizens are more sensitive to casualties, or both. Emphasizing the pivotal role of public sentiment in shaping government actions during armed conflicts, our results reveal how disadvantageous it can be to incur relatively higher costs from war casualties in terms of winning probabilities in equilibrium.

Mathematical Model for Antimicrobial Resistance Transmission Dynamics in Hospitals

In this study, we formulate and analyze a mathematical model to describe the possible transmission routes of Antimicrobial Resistance (A.M.R) in a hospital setting. We have examined the significant means of transmission of resistance in hospitals and found that the significant means of transmission is the use of antibiotics and through the contamination by health care workers. It has been shown that the resistance free equilibrium point is locally asymptotically stable. We have also shown that the model has a unique positive endemic equilibrium point which is locally asymptotically stable.

An aggregative approach to pricing equilibrium among multi‐product firms with nested demand

AbstractI show equilibrium existence for a price‐setting game among multi‐product firms facing a nested logit/CES demand. As opposed to previous research I allow arbitrary firm/nest overlap, making the result relevant for applied work. Additionally, under easy‐to‐verify conditions, I show that there exist extreme equilibria, which are the most and least preferred by consumers, and provide an algorithm to find them. This allows researcher to numerically verify equilibrium uniqueness in applications, that is, if the extreme equilibria are equal to each other. As a by‐product, I show that inverting FOCs correctly identifies the marginal costs that rationalize observed prices.

Generalizing Determinacy under Monetary and Fiscal Policy Switches: The Case of the Zero Lower Bound

AbstractIn a fixed‐regime context, it has been established since the work of Leeper (1991) that a determinate and unique equilibrium can be achieved under both monetary dominance (characterized by an active monetary policy and a passive fiscal policy) and fiscal dominance (characterized by an active fiscal policy and a passive monetary policy) regimes yield a determinate unique equilibrium. In this paper, we generalize this well‐established policy‐mix taxonomy to regime‐switching models, and show that switches from monetary dominance to fiscal dominance regimes (and vice versa) render the economy indeterminate for standard parameter values. We apply our results to the important case of the exit from the zero lower bound (ZLB), where monetary policy is inherently passive. Contrary to the fixed‐regime prediction, the economy switching between the ZLB in a fiscally led regime and a monetary dominance regime is robustly indeterminate, making the coordination of agents' beliefs hard for anchoring expected inflation. Finally, we propose a direction toward which the perspective of the fiscal theory of the price level can be made consistent with a unique stable equilibrium.

A sequentially variant Blotto game with one-sided and incomplete information

We develop a sequentially variant Blotto game with one-sided and incomplete information to investigate strategic interactions between a defender and an attacker whose target site values are unknown. The defender first allocates defensive resources before the attacker decides a probability distribution over which site to attack between the target sites. The attacker perfectly observes the defender’s resource allocations. The attacker’s type is continuous, following the attacker’s private values of victoriously attacking each site. We find the game’s essentially unique subgame perfect equilibrium. In equilibrium, the site the attacker attacks with a higher probability is the site with a lower expected loss for the defender when the defender defends both sites. We present numerical examples to examine (1) the impacts of the informational uncertainty concerning the attacker’s site values, (2) the impacts of the site values of the defender, (3) the impacts of the site values of the attacker, and (4) the impacts of the defender’s defense efficiency on the equilibrium behavior.

Host genetic and immune factors drive evasion of HIV-1 pathogenesis in viremic non-progressors.

Viremic non-progressors (VNPs) represent an exceptional and uncommon subset of people with HIV-1, characterized by the remarkable preservation of normal CD4+ Tcell counts despite uncontrolled viral replication-a trait reminiscent of natural hosts of simian immunodeficiency virus. The mechanisms orchestrating evasion from HIV-1 pathogenesis in human VNPs remain elusive, primarily due to the absence of integrative studies. We implemented a novel single-cell and multiomics approach to comprehensively characterize viral, genomic, transcriptomic, and metabolomic factors driving this exceedingly rare disease phenotype in 16 VNPs and 29 HIV+ progressors. Genetic predisposition to the VNP phenotype was evidenced by a higher prevalence of CCR5Δ32 heterozygosity, which was associated with lower levels of CCR5 expression and a lower frequency of infected cells in peripheral circulation. We also observed reduced levels of plasma markers of intestinal disruption and attenuated interferon responses in VNPs. These factors potentially drive the other phenotypic traits of immune preservation in this population, including the unaltered tryptophan metabolic profile, reduced activation of cytotoxic lymphocytes, and reduced bystander CD4+ Tcell apoptosis. In summary, our comprehensive analysis identified intricate factors collectively associated with the unique immunovirological equilibrium in VNPs, shedding light on potential avenues for therapeutic exploration in managing HIV pathogenesis. The work was supported by funding from the Spanish Ministry of Science and Innovation and the National Institutes of Health (NIH).

Three Duopoly Game-Theoretic Models for the Smart Grid Demand Response Management Problem

Demand response management (DRM) significantly influences the prospective advancement of electricity smart grids. This paper introduces three distinct game-theoretic duopoly models for the smart grid demand response management problem. It delineates several rational assumptions regarding the model variables, functions, and parameters. The first model adopts a Cournot duopoly form, offering a unique closed-form equilibrium solution. The second model adopts a Stackelberg duopoly structure, also providing a unique closed-form equilibrium solution. Following a comparison of the economic viability of the two model equilibria and an assessment of their sensitivity to parametric changes, the paper proposes a third model with a Cartel structure and discusses its advantages over the earlier models. Finally, the paper examines how demand forecasting affects the equilibrium quantities and pricing solutions of each model.

Untangling the mobility dynamics in a patchy environment: An SEIR epidemic model with vertical transmission

Congenital Rubella, AIDS, and other human diseases may pass from mother to fetus through the placenta. On the other side, insects and plants often transmit vertically by their eggs/seeds. Here, we study an SEIR epidemic model with vertical transmission in a multi‐patch context to explore the spread of disease in a population divided into distinct regions. The local and global stability of disease‐free equilibrium for the n‐patch model system has been determined in terms of the basic reproduction number . Moreover, we discuss the existence and stability of endemic equilibrium for different cases in two symmetric regions. Our findings indicate that when is less than one, the disease‐free equilibrium is globally asymptotically stable, whereas when is greater than one, it is unstable. As a result, for , the model system has an unique endemic equilibrium that is always locally asymptotically stable. The dynamics of the proposed model system depending on the distribution of infected individuals across various regions have been studied. The findings suggest that the movements from high disease risk regions to low disease risks may increase the number of the infected population in the community. The value of could be enhanced by increasing the values of any vertical transmission parameter. However, the changes in the values of are relatively minor. Our findings also imply that while migrations of all individuals across symmetric regions do not leave any impact on the equilibrium state of the infected population, they do affect the trajectory of the solution to reach a steady‐state. The flow of people between the two symmetrical regions could not alter the endemicity.

Dynamics of a pituitary–adrenal model with distributed time delays

This paper investigates the effects of distributed time delays on the dynamics of the pituitary–adrenal axis. Our two-dimensional model incorporates distributed time delays with general delay kernels, specifically illustrating interactions between adrenocorticotropic hormone (ACTH) and cortisol (CORT). We derive general theoretical results for general delay kernels, exemplified by Dirac and weak Gamma kernels, revealing stability transitions characterized by Hopf bifurcations. We establish conditions for the local asymptotic stability of the unique equilibrium and discuss the existence of periodic solutions. Numerical simulations demonstrate that periodic oscillations appear for appropriate values of the average time delays. Including an external input results in both ultradian and circadian rhythms, highlighting the system’s dynamic responsiveness to external stimuli.

Bipartite consensus of concatenated opinion dynamics for two antagonistic groups: A game theoretical perspective

Recent decades have witnessed a significant surge in scholarly attention towards formulating dynamic models for opinion formation on social networks. This paper considers the bipartite consensus problem over a series of issues for two antagonistic groups wherein two graphs are employed to model the interaction within groups and a noncooperative game to describe the interaction between groups. Firstly, we prove that the noncooperative game admits the unique Nash equilibrium. Secondly, we give sufficient conditions for achieving a bipartite consensus that both graphs are strongly connected. Additionally, we obtain that the game’s mechanism decides the convergence rate of bipartite consensus from one issue to the next. It is important to highlight that confrontation is key in elevating the most influential individuals to leadership positions within their groups. Thirdly, we construct the bipartite consensus state for a particular case where the networks of two groups are balanced. Finally, we present two examples to demonstrate the theoretical results.

Convergence of reputations under indirect reciprocity

Previous research has shown how indirect reciprocity can promote cooperation through evolutionary game theoretic models. Most work in this field assumes a separation of time-scales: individuals’ reputations equilibrate at a fast time scale for given frequencies of strategies while the strategies change slowly according to the replicator dynamics. Much of the previous research has focused on the behaviour and stability of equilibria for the replicator dynamics. Here we focus on the underlying reputational dynamics that occur on a fast time scale. We describe reputational dynamics as systems of differential equations and conduct stability analyses on their equilibria. We prove that reputations converge to a unique equilibrium under a solitary observer model for each of the five standard norms and whether assessments are public or private. These results confirm a crucial but previously understudied assumption underlying the theory of indirect reciprocity for the most studied set of norms.

Large Orders in Small Markets: Execution with Endogenous Liquidity Supply

Abstract We model the execution of a large uninformed sell order in the presence of strategic competitive market makers. We solve for the unique symmetric equilibrium of the model in closed form. Analysis of this equilibrium reveals that large orders unequivocally benefit market makers, while smaller investors stand to benefit only if the order trades with a sufficiently high intensity. The equilibrium results further provide a rationale for the empirically observed patterns of (i) shorter orders trading at higher intensities, and (ii) price pressures potentially subsiding before large orders stop executing.

Research on low-carbon supply chain emission reduction strategies based on blockchain technology

The difficulty of regulating carbon trading due to information asymmetry and low consumer trust in low-carbon products are key factors hindering companies from reducing emissions. This paper examines a manufacturer-led secondary low-carbon supply chain consisting of a single supplier and a retailer, focusing on the impact of blockchain technology on carbon transaction costs and consumers' low-carbon preferences. Utilizing Stackelberg game theory, the paper constructs a supply chain decision model for emission reduction, determining the payment matrix and analyzing the stable strategy for blockchain adoption through evolutionary game theory. The findings indicate that retailers' adoption of blockchain technology significantly promotes emission reduction within the supply chain, whereas manufacturers' adoption has minimal impact. Additionally, the study reveals that variations in blockchain adoption costs and carbon quotas result in multiple evolutionary stable strategies. Specifically, when blockchain adoption costs and carbon quotas are below certain thresholds, the system reaches a unique equilibrium where both parties adopt blockchain technology.

Tullock contest with reference‐dependent preferences

AbstractWe study the Tullock contest model with loss aversion and endogenously formed reference points. In a contest with n possibly heterogeneous players and convex effort costs, we establish sufficient conditions for a unique Nash equilibrium in pure strategies. Subsequently, we analyze the impact of loss aversion on players' spending behavior, probability of winning, and rent dissipation.

Equilibrium vertical structure with a common supplier

AbstractWe consider a vertically related market, in which each downstream firm produces a differentiated product by assembling a key input produced by a common supplier and another input produced by a dedicated upstream firm. On the one hand, vertical integration has the advantage of inducing the common supplier to set a lower input price, but the disadvantage of reducing downstream firms' competitiveness in the downstream market. On the other hand, vertical separation has the advantage of increasing downstream firms' competitiveness in the downstream market but the disadvantage of inducing the common supplier to set a higher input price. Contrary to results of previous studies, we find that the existence of a common supplier can lead to vertical integration under Cournot competition, which emerges as a unique equilibrium when a common supplier adopts input discrimination. Although vertical integration is better for the individual firms, it reduces the total welfare. Even when the common supplier uses uniform input pricing, vertical integration also emerges in equilibrium.