Stable Whenever Research Articles

Modelling financial crime population dynamics: optimal control and cost-effectiveness analysis

This work is designed to formulate and analyse a mathematical model for population dynamics of financial crime with optimal control measures. Necessary conditions for the existence and stability of financial crime steady states are derived. The financial crime reproduction number is determined. Based on construction of suitable Lyapunov functionals, crime-free equilibrium point of the formulated model is shown to be globally asymptotically stable when the crime reproduction number is below unity, while a unique crime-present equilibrium is proved to be globally asymptotically stable whenever the crime reproduction number exceeds unity. Sensitivity analysis is carried out to determine the relative importance of model parameters in financial crime spread. Furthermore, optimal control theory is employed to assess the impact of two time-dependent optimal control strategies, including public enlightenment campaign (preventive) and corrective measure, on the financial crime dynamics in a population. The cost-effectiveness analysis is carried out to determine the least costly and most effective strategy of the singular and combined implementations of the intervention strategies, when the available resources to combat the spread of financial crime are limited.

Assessing the Impact of Prophylactic Vaccines on HPV Prevalence

A mathematical model considering female and male individuals is proposed to evaluate vaccination strategies applied to control of HPV transmission in human population. The basic reproductive number of the disease, $R_0$, is given by the geometric mean of the basic reproductive number of female and male populations. The model has a globally asymptotically stable disease-free equilibrium whenever $R_0 <1$. Furthermore, it has an unique endemic state when $R_0$ exceeds unity which is globally asymptotically stable. Numerical simulations were done to compare several different vaccination schedules. The results showed that the vaccination strategies that do not include vaccination of men can only control the disease if more than 90\% of women are vaccinated. The sensitivity analysis indicated that the relevant parameters to control HPV transmission, in order of importance, are vaccine efficacy times the fraction of population that is vaccinated, disease recovery-rate, and disease transmission rate. Therefore, health politics that promoting the increase of vaccine coverage, and screening for the disease in both population can improve disease control.

A Maize Foliar Disease Mathematical Model with Standard Incidence Rate

In this paper, we present a mathematical model of maize foliar disease with standard incidence rate. The present model is an improvement model from Collins and Duffy, where Collins and Duffy consider a mathematical model with the bilinear incidence rate. The present model has two equilibria namely the disease-free equilibrium and endemic equilibrium. We find that the disease-free equilibrium is asymptotically stable whenever the basic reproductive ratio is less than one. On the other hand, the endemic equilibrium will exist and be asymptotically stable whenever the basic reproductive ratio is greater than one. Furthermore, we perform numerical simulations to confirm the analytical results.

Stabilization of Two-Port Networked Systems With Simultaneous Uncertainties in Plant, Controller, and Communication Channels

In this paper, we study robust stabilization for the networked control system (NCS) over the communication channels described by cascaded two-port networks. Such an NCS involves simultaneous uncertainties in the plant, controller, and two-port communication channels. The cascaded two-port connections arise when signals in the NCS are transmitted through bidirectional communication channels separated by a sequence of relays. Distortions and interferences occurring in communications are taken into account. We consider ${\mathcal H}_{\infty }$ -norm bounded uncertainties in the transmission matrices of the two-port channels, and the gap-type uncertainties in the plant and controller models. A necessary and sufficient condition for the robust stability of the NCS is presented in the form of an “arcsine” inequality, which states that the NCS is stable whenever the uncertainties quantified by the aforementioned metrics are well bounded and satisfy the “arcsine” inequality. A stability margin related to the Gang of Four transfer matrix is obtained, based on which the controller synthesis problem can be solved through a special ${\mathcal H}_{\infty }$ optimization with favorable properties. Furthermore, a generalized stability condition is studied in terms of frequency-wise bounded uncertainties, and the corresponding controller synthesis problem is proposed and solved.

A Two-Strain Deterministic Dengue Model With Seasonality Effect

A compartmental model is proposed to describe the dynamics of vector-host interations for dengue disease transmission with coexistence of two virus serotypes. The model is modified to incorporate seasonal-dependent mosquito birth rate in order to examine the influence of climatic factors such as rainfall and temperature on the dynamics of mosquito population and dengue disease transmission. The Next Generation Matrix method is used to obtain the basic reproduction number associated with the model without seasonality effect. The global dynamics of the model is analysed using the Comparison Theorem. The model is simulated in MATLAB with ode45 routine for two cases, namely: the less aggressive case (Case ) and the more aggressive case (Case ). Analysis of the model shows that the Disease-Free Equilibrium (DFE) is locally asymptotically stable whenever both the basic reproduction numbers R01 (associated with strain only) and R0j (associated with strain only) are below unity. It is shown that the DFE is globally asymptotically stable when the susceptibility indices for secondary infection in strain 1 ( sigma1) and strain j ( sigmaj), and are all less than 1.

Global Stability Analysis of Two-Stage Quarantine-Isolation Model with Holling Type II Incidence Function

A new two-stage model for assessing the effect of basic control measures, quarantine and isolation, on a general disease transmission dynamic in a population is designed and rigorously analyzed. The model uses the Holling II incidence function for the infection rate. First, the basic reproduction number (
 
 
 
 R
 0
 
 
 
 ) is determined. The model has both locally and globally asymptotically stable disease-free equilibrium whenever 
 
 
 
 
 R
 0
 
 <
 1
 
 
 
 . If 
 
 
 
 
 R
 0
 
 >
 1
 ,
 
 
 
 then the disease is shown to be uniformly persistent. The model has a unique endemic equilibrium when 
 
 
 
 
 R
 0
 
 >
 1
 .
 
 
 
 A nonlinear Lyapunov function is used in conjunction with LaSalle Invariance Principle to show that the endemic equilibrium is globally asymptotically stable for a special case.

Global analysis of multi-host and multi-vector epidemic models

We formulate a multi-group and multi-vector epidemic model in which hosts' dynamics is captured by staged-progression SEIR framework and the dynamics of vectors is captured by an SI framework. The proposed model describes the evolution of a class of zoonotic infections where the pathogen is shared by m host species and transmitted by p arthropod vector species. In each host, the infectious period is structured into n stages with a corresponding infectiousness parameter to each vector species. We determine the basic reproduction number R02(m,n,p) and investigate the dynamics of the systems when this threshold is less or greater than one. We show that the dynamics of the multi-host, multi-stage, and multi-vector system is completely determined by the basic reproduction number and the structure of the host-vector network configuration. Particularly, we prove that the disease-free equilibrium is globally asymptotically stable (GAS) whenever R02(m,n,p)<1, and a unique strongly endemic equilibrium exists and is GAS if R02(m,n,p)>1 and the host-vector configuration is irreducible. That is, either the disease dies out or persists in all hosts and all vector species.

Effects of quarantine on transmission dynamics of Lassa fever

In this paper, a mathematical model of Lassa fever is formulated. The model includes quarantine as a control strategy and allows re-infection. The model is shown to be well posed. The disease free equilibrium is shown to be locally asymptotically stable whenever the basic reproduction number is less than unity and unstable otherwise. Numerical simulations have been used to show the impact of the control measure. Keywords : Quarantine, re-infection, immunity, reproduction number, stability.

The role of houseflies in cholera transmission

In this paper, we propose and analyse a mathematical model that describes the dynamics of Cholera. The main aim of this model is to investigate the role of houseflies in the transmission of Cholera. Our analysis showed that the disease free equilibrium is globally asymptotically stable whenever the basic reproduction number is less than unity; and unstable otherwise; and our model posses only one endemic equilibrium which is locally asymptotically stable whenever basic reproduction number is greater than unity. Our sensitivity analysis showed that the basic reproduction number is very sensitive to ingesting vibrios rate from aquatic environment by vectors, the rate of contribution to V. cholera in the aquatic environment and the death rate of vector and death rate of vibrios in aquatic environment which indicates that the vector (i.e. the houseflies) play a very important role in the transmission procedure.Numerically, we shown that the rate of water contamination by infectious people shedding V. cholera into the environment has no impact in the infection because it depends on both bacteria shedding of the infected individuals and the level of sanitation in the environment and since the environment is safe, then it is obviously has no effects. In addition, if the contact rate of vectors with contaminated water is high in the presence of increased contribution of each infected vector to the aquatic environment then cholera will persist in the population. Therefore, to obtain a significant and effective control, the contribution of each infected vector to the aquatic environment and the rate of exposure to contaminated water must be reduced.

Global Exponential Nonlinear Stability for Double Diffusive Convection in Porous Medium

Nonlinear stability of the motionless double-diffusive solution of the problem of an infinite horizontal fluid layer saturated porous medium is studied. The layer is heated and salted from below. By introducing two balance fields and through defining new energy functionals it is proved that for CLe ≥ R, Le ≤ 1 the motionless double-diffusive solution is always stable and for CLe < R, Le < 1 the solution is globally exponentially and nonlinearly stable whenever R < 4π2+LeC, where Le, C and R are the Lewis number, Rayleigh number for solute and heat, respectively. Moreover, the nonlinear stability proved here is global and exponential, and the stabilizing effect of the concentration is also proved.

Global Dynamics of an SEIRS Compartmental Measles Model with Interrupted Vaccination

Measles is a reemerging disease that has a devastating impact, especially among children under 5. In this paper, an SEIRS model is developed to investigate a possible outbreak among the population of children under 5 in the Sunyani Municipality. We consider waning immunity or loss of immunity among those who were vaccinated, which leads to secondary attacks among some in the population. Using Routh-Hurwitz criterion, Matrix Theoretic and Goh-Volterra Lyapunov functions, the stability of the model was investigated around the equilibria. We have computed the threshold parameter, R0, using the Next Generation Matrix method. The disease-free equilibrium is globally stable whenever R0 ≤1 and unstable otherwise. The endemic equilibrium is globally stable when R0 >1.

Global stability of an age-structured epidemic model with general Lyapunov functional.

In this paper, we focus on the study of the dynamics of a certain age structured epidemic model. Our aim is to investigate the proposed model, which is based on the classical SIR epidemic model, with a general class of nonlinear incidence rate with some other generalization. We are interested to the asymptotic behavior of the system. For this, we have introduced the basic reproduction number R₀ of model and we prove that this threshold shows completely the stability of each steady state. Our approach is the use of general constructed Lyapunov functional with some results on the persistence theory. The conclusion is that the system has a trivial disease-free equilibrium which is globally asymptotically stable for R₀ < 1 and that the system has only a unique positive endemic equilibrium which is globally asymptotically stable whenever R₀ > 1. Several numerical simulations are given to illustrate our results.

Global Stability and Optimal Control Analysis of Malaria Dynamics in the Presence of Human Travelers

Introduction:The impact of unguarded human movement on the spread of infectious disease like malaria cannot be underestimated. Therefore, this study examines the significance of short term human travelers on malaria transmission dynamics.Methods:A non-autonomous system of ordinary differential equations incorporating four control strategies, namely personal protection, chemo-prophylaxis, chemotherapy and mosquito-reduction effort is presented to describe the dynamics of malaria transmission between two interacting populations. Suitable Lyapunov functions are constructed to analyze the global dynamics of the autonomous version. Moreover, the model which incorporates time-dependent vigilant controls is qualitatively analyzed with the overall goal of minimizing the spread of malaria and the associated costs of control implementation using the optimal control theory. An iterative method of forward-backward Runge-Kutta fourth order scheme is used to simulate the optimality system in order to investigate the effects of the control strategies on the magnitude of infected individuals in the population.Results:Analysis of the autonomous system shows that the disease-free equilibrium is globally asymptotically stable whenever the basic reproduction is less than unity and a uniquely determined endemic equilibrium is shown to be globally asymptotically stable whenever the associated basic reproduction number exceeds unity. In the case of non-autonomous system, necessary conditions for the optimal control of malaria are derived. It is shown that adherence to the combination of the control strategies by short term human travelers helps in curtailing the spread of malaria in the population.

Backward Bifurcation of an Epidemiological Model with Saturated Incidence, Isolation and Treatment Functions

This work considers an epidemic model with saturated incidence rate and saturated isolation and treatment functions. The incidence function (as well as the treatment and isolation functions) is of the Holling type II form, where, in the cases of treatment and isolation, these functions describes the effect of delayed treatment and isolation with large number of infected individuals in a population especially in situations where isolation facilities are available but fewer medical personnel. It is shown that the disease-free equilibrium (DFE) is locally-asymptotically stable whenever the effective reproduction number is less than unity. However, it is shown that the global-asymptotic stability of the disease-free equilibrium was largely dictated by the delay in treatment (of non-isolated infected individuals) and isolation of infected individuals; when such delay effect is weak, then the DFE is globally asymptotically stable when the effective reproduction number is less than unity. When the delay effect is strong, it is shown that there is the possibility of the existence of the backward bifurcation phenomenon whereby the DFE will co-exists with two endemic equilibria, when the effective reproduction number is less than unity. The backward bifurcation phenomenon existed when the delays occurred either singly or jointly. Mathematical analysis provides threshold conditions that allows for the global stability of the endemic equilibrium whenever the associated reproduction number is greater than unity. The results suggests that with timely isolation of infected individuals (for treatment under isolation) and treatment of non-isolated infectious cases, the ultimate goal of disease eradication is possible.

Mathematical assessment of the role of environmental factors on the dynamical transmission of cholera

In this paper, we investigate the impact of environmental factors on the dynamical transmission of cholera within a human community. We propose a mathematical model for the dynamical transmission of cholera that incorporates the virulence of bacteria and the commensalism relationship between bacteria and the aquatic reservoirs on the persistence of the disease. We provide a theoretical study of the model. We derive the basic reproduction number R0 which determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever R0≤1, while when R0>1, the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is locally asymptotically stable on a positively invariant region of the positive orthant. The sensitivity analysis of the model has been performed in order to determine the impact of related parameters on outbreak severity. Theoretical results are supported by numerical simulations, which further suggest the necessity to implement sanitation campaigns of aquatic environments by using suitable products against the bacteria during the periods of growth of aquatic reservoirs.

Stability criteria for complex microbial communities

Competition and mutualism are inevitable processes in microbial ecology, and a central question is which and how many taxa will persist in the face of these interactions. Ecological theory has demonstrated that when direct, pairwise interactions among a group of species are too numerous, or too strong, then the coexistence of these species will be unstable to any slight perturbation. Here, we refine and to some extent overturn that understanding, by considering explicitly the resources that microbes consume and produce. In contrast to more complex organisms, microbial cells consume primarily abiotic resources, and mutualistic interactions are often mediated through the mechanism of crossfeeding. We show that if microbes consume, but do not produce resources, then any positive equilibrium will always be stable to small perturbations. We go on to show that in the presence of crossfeeding, stability is no longer guaranteed. However, positive equilibria remain stable whenever mutualistic interactions are either sufficiently weak, or when all pairs of taxa reciprocate each other’s assistance.

Mathematical assessment of the role of Dengvaxia vaccine on the transmission dynamics of dengue serotypes

A new mathematical model is designed and used to assess the impact of the newly-released Dengvaxia vaccine on the transmission dynamics of two co-circulating dengue strains (where strain 1 consists of dengue serotypes 1, 3 and 4; and strain 2 consists of dengue serotype 2). It is shown that the model exhibits the phenomenon of backward bifurcation when the disease-induced mortality in the host population exceeds a certain threshold value or if the vaccine does not provide perfect protection against infection with the two strains. In the absence of backward bifurcation, the disease-free equilibrium of the model is shown to be globally-asymptotically stable whenever the associated reproduction number is less than unity. It is shown that the community-wide use of the vaccine could induce positive, negative or no population-level impact, depending on the sign of a certain epidemiological threshold quantity (known as the vaccine impact factor). Simulations of the model, using data from Oaxaca, Mexico, show that, although the community-wide use of the vaccine will significantly reduce dengue burden in the community, it is unable to lead to the elimination of the two dengue strains. It is further shown that the use of Dengvaxia vaccine in dengue-naive populations may induce increased risk of severe disease in these populations.

Mathematical Assessment of the Role of Early Latent Infections and Targeted Control Strategies on Syphilis Transmission Dynamics.

A new multi-stage deterministic model for the transmission dynamics of syphilis, which incorporates disease transmission by individuals in the early latent stage of syphilis infection and the reversions of early latent syphilis to the primary and secondary stages, is formulated and rigorously analysed. The model is used to assess the population-level impact of preventive (condom use) and therapeutic measures (treatment using antibiotics) against the spread of the disease in a community. It is shown that the disease-free equilibrium of the model is globally-asymptotically stable whenever the associated control reproduction number (denoted by [Formula: see text]) is less than unity. A special case of the model is shown to have a unique and globally-asymptotically stable endemic equilibrium whenever the associated reproduction number (denoted by [Formula: see text]) exceeds unity. Uncertainty and sensitivity analysis of the model, using parameter values and ranges relevant to syphilis transmission dynamics in Nigeria, show that the top three parameters that drive the syphilis infection (with respect to [Formula: see text]) are the disease transmission rate ([Formula: see text]), compliance in condom use (c) and efficacy of condom ([Formula: see text]). Numerical simulations of the model show that the targeted treatment of secondary syphilis cases is more effective than the targeted treatment of individuals in the primary or early latent stage of syphilis infection.

ON THE GLOBAL STABILITY OF CHOLERA MODEL WITH PREVENTION AND CONTROL

In this study, a system of first order ordinary differential equations is used to analyse the dynamics of cholera disease via a mathematical model extended from Fung (2014) cholera model. The global stability analysis is conducted for the extended model by suitable Lyapunov function and LaSalle’s invariance principle. It is shown that the disease free equilibrium (DFE) for the extended model is globally asymptotically stable if 𝑅0 𝑞 &lt; 1 and the disease eventually disappears in the population with time while there exists a unique endemic equilibrium that is globally asymptotically stable whenever 𝑅0 𝑞 &gt; 1 for the extended model or 𝑅0 &gt; 1 for the original model and the disease persists at a positive level though with mild waves (i.e few cases of cholera) in the case of𝑅0 𝑞 &gt; 1. Numerical simulations for strong, weak, and no prevention and control measures are carried out to verify the analytical results and Maple 18 is used to carry out the computations.

Study of the Population Dynamics of Busseola fusca, Maize Pest.

Busseola fusca is a maize and sorghum pest that can cause significant damage to both crops. Given that maize is one of the main cereals grown in the worldwide, this pest is a major challenge for maize production and therefore for the economies of several countries . In this paper , based on the life cycle of B. fusca, we propose a mathematical model to study the population dynamics of this insect pest . A sensitivity analysis using the eFast method was performed to show the most important parameters of the model. We present the theoretical analysis of the model. More precisely, we derive a threshold parameter [Formula: see text], called basic offspring number and show that the trivial equilibrium is globally asymptotically stable whenever [Formula: see text], while if [Formula: see text], the non trivial equilibrium is globally asymptotically stable. The theoretical results are supported by numerical simulations.