FIXED POINTS OF α-TYPE F-CONTRACTIVE MAPPINGS WITH AN APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we introduce new concepts of a-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in [21, 22] and different from a-GF-contractions given in [8]. Then, sufficient conditions for the existence and uniqueness of fixed point are established for these new types of contractive mappings, in the setting of complete metric space. Consequently, the obtained results encompass various generalizations of the Banach contraction principle. Moreover, some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.

Discrete Singular Convolution Method for Numerical Solutions of Fifth Order Korteweg-De Vries Equations

A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.

MATHEMATICAL ANALYSIS OF WEST NILE VIRUS MODEL WITH DISCRETE DELAYS

The paper presents the basic model for the transmission dynamics of West Nile virus （WNV）. The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease- free equilibrium whenever the associated reproduction number （R0） is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation （where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity）. It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever R0 〉 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by △b and △v are negative. On the other hand, increasing the length of the incubation period increases disease burden if △b 〉 0 and △v 〉 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence （considered in [2]） does not alter the qualitative dynamics of the autonomous ：system （with respect to the elimination or persistence of the disease）.

Mathematics of a single-locus model for assessing the impacts of pyrethroid resistance and temperature on population abundance of malaria mosquitoes

This study presents a genetic-ecology modeling framework for assessing the combined impacts of insecticide resistance,temperature variability,and insecticide-based interventions on the population abundance and control of malaria mosquitoes by genotype.Rigorous analyses of the model we developed reveal that the boundary equilibrium with only mosquitoes of homozygous sensitive(resistant)genotype is locally-asymptotically stable whenever a certain ecological threshold,denoted by R^(SS)_(0) eR^(RR)_(0) T,is less than one.Furthermore,genotype i drives genotype j to extinction whenever R j 0>1 and R i 0<1(where i,j SS or RR,with i s j).The model exhibits the phenomenon of bistability when both thresholds are less than one.In such a bistable situation,convergence to any of the two boundary equilibria depends on the initial allele distribution in the state variables of the model.Furthermore,in this bistable case,where maxfR^(SS)_(0);R^(RR)_(0)g<1,the basin of attraction of the boundary equilibrium of the mosquito genotype with lower value of the ecological threshold is larger.Specifically,the basin of attraction of the boundary equilibrium for genotype i is larger than that of genotype j if R^(i)_(0)<R^(j)_(0)<1.When both ecological thresholds exceed one eminfR^(SS)_(0);R^(RR)_(0)g>1T,the two boundary equilibria lose their stability,and a coexistence equilibrium(where all three mosquito genotypes coexist)becomes locally-asymptotically stable.Global sensitivity analysis shows that the key parameters that greatly influence the dynamics and population abundance of resistant mosquitoes include the proportion of new adult mosquitoes that are females,the insecticide-induced mortality rate of adult female mosquitoes,the coverage level and efficacy of adulticides used in the community,the oviposition rates for eggs of heterozygous and homozygous resistant genotypes,and the modification parameter accounting for the reduction in insecticide-induced mortality due to resistance.Numerical simulations show that the adult mosquito population increases with increasing temperature until a peak is reached at 31C,and declines thereafter.Simulating the model for moderate and high adulticide coverage,together with varying fitness costs of resistance,shows a switch in the dominant genotype at equilibrium as temperature is varied.In other words,this study shows that,for certain combinations of adulticide coverage and fitness costs of insecticide resistance,increases in temperature could result in effective management of resistance(by causing the switch from a stable resistant-only boundary equilibrium(at 18C)to a stable sensitive-only boundary equilibrium(at 25C)).Finally,this study shows that,for moderate fitness costs of resistance,density-dependent larval mortality suppresses the total population of adult mosquitoes with the resistant allele for all temperature values in the range[18Ce36C].

Dynamics of COVID-19 pandemic in India and Pakistan: A metapopulation modelling approach

India has been the latest global epicenter for COVID-19,a novel coronavirus disease that emerged in China in late 2019.We present a base mathematical model for the transmission dynamics of COVID-19 in India and its neighbor,Pakistan.The base model was rigorously analyzed and parameterized using cumulative COVID-19 mortality data from each of the two countries.The model was used to assess the population-level impact of the control and mitigation strategies implemented in the two countries(notably non-pharmaceutical interventions).Numerical simulations of the basic model indicate that,based on the current baseline levels of the control and mitigation strategies implemented,the pandemic trajectory in India is on a downward trend.This downward trend will be reversed,and India will be recording mild outbreaks,if the control and mitigation strategies are relaxed from their current levels.By early September 2021,our simulations suggest that India could record up to 460,000 cumulative deaths under baseline levels of the implemented control strategies,while Pakistan(where the pandemic is comparatively milder)could see over 24,000 cumulative deaths at current mitigation levels.The basic model was extended to assess the impact of back-and-forth mobility between the two countries.Simulations of the resulting metapopulation model show that the burden of the COVID-19 pandemic in Pakistan increases with increasing values of the average time residents of India spend in Pakistan,with daily mortality in Pakistan peaking in mid-August to mid-September of 2021.Under the respective baseline control scenarios,our simulations show that the backand-forth mobility between India and Pakistan could delay the time-to-elimination of the COVID-19 pandemic in India and Pakistan to November 2022 and July 2022,respectively.

A primer on using mathematics to understand COVID-19 dynamics: Modeling, analysis and simulations

The novel coronavirus(COVID-19)pandemic that emerged from Wuhan city in December 2019 overwhelmed health systems and paralyzed economies around the world.It became the most important public health challenge facing mankind since the 1918 Spanish flu pandemic.Various theoretical and empirical approaches have been designed and used to gain insight into the transmission dynamics and control of the pandemic.This study presents a primer for formulating,analysing and simulating mathematical models for understanding the dynamics of COVID-19.Specifically,we introduce simple compartmental,Kermack-McKendrick-type epidemic models with homogeneously-and heterogeneously-mixed populations,an endemic model for assessing the potential population-level impact of a hypothetical COVID-19 vaccine.We illustrate how some basic non-pharmaceutical interventions against COVID-19 can be incorporated into the epidemic model.A brief overview of other kinds of models that have been used to study the dynamics of COVID-19,such as agent-based,network and statistical models,is also presented.Possible extensions of the basic model,as well as open challenges associated with the formulation and theoretical analysis of models for COVID-19 dynamics,are suggested.

Modeling the impact of quarantine during an outbreak of Ebola virus disease

The quarantine of people suspected of being exposed to an infectious agent is one of the most basic public health measure that has historically been used to combat the spread of communicable diseases in human communities.This study presents a new deterministic model for assessing the population-level impact of the quarantine of individuals suspected of being exposed to disease on the spread of the 2014e2015 outbreaks of Ebola viral disease.It is assumed that quarantine is imperfect(i.e.,individuals can acquire infection during quarantine).In the absence of quarantine,the model is shown to exhibit global dynamics with respect to the disease-free and its unique endemic equilibrium when a certain epidemiological threshold(denoted byR 0)is either less than or greater than unity.Thus,unlike the full model with imperfect quarantine(which is known to exhibit the phenomenon of backward bifurcation),the version of the model with no quarantine does not undergo a backward bifurcation.Using data relevant to the 2014e2015 Ebola transmission dynamics in the three West African countries(Guinea,Liberia and Sierra Leone),uncertainty analysis of the model show that,although the current level and effectiveness of quarantine can lead to significant reduction in disease burden,they fail to bring the associated quarantine reproduction number(R Q0)to a value less than unity(which is needed to make effective disease control or elimination feasible).This reduction of R Q0 is,however,very possible with a modest increase in quarantine rate and effectiveness.It is further shown,via sensitivity analysis,that the parameters related to the effectiveness of quarantine(namely the parameter associated with the reduction in infectiousness of infected quarantined individuals and the contact rate during quarantine)are the main drivers of the disease transmission dynamics.Overall,this study shows that the singular implementation of a quarantine intervention strategy can lead to the effective control or elimination of Ebola viral disease in a community if its coverage and effectiveness levels are high enough.

Unique Factorization of Compositive Hereditary Graph Properties

A graph property is any class of graphs that is closed under isomorphisms, A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs

The Bulk-Surface Virtual Element Method for Reaction-Diffusion PDEs:Analysis and Applications

Bulk-surface partial differential equations(BS-PDEs)are prevalent in manyapplications such as cellular,developmental and plant biology as well as in engineeringand material sciences.Novel numerical methods for BS-PDEs in three space dimensions(3D)are sparse.In this work,we present a bulk-surface virtual elementmethod(BS-VEM)for bulk-surface reaction-diffusion systems,a form of semilinearparabolic BS-PDEs in 3D.Unlike previous studies in two space dimensions(2D),the3D bulk is approximated with general polyhedra,whose outer faces constitute a flatpolygonal approximation of the surface.For this reason,the method is restricted tothe lowest order case where the geometric error is not dominant.The BS-VEM guaranteesall the advantages of polyhedral methods such as easy mesh generation andfast matrix assembly on general geometries.Such advantages are much more relevantthan in 2D.Despite allowing for general polyhedra,general nonlinear reaction kineticsand general surface curvature,the method only relies on nodal values without needingadditional evaluations usually associated with the quadrature of general reactionkinetics.This latter is particularly costly in 3D.The BS-VEM as implemented in thisstudy retains optimal convergence of second order in space.

Fixed Points of Multivalued Quasi-nonexpansive Mappings Using a Faster Iterative Process

In this article, we prove some strong and weak convergence theorems for quasi-nonexpansive multivalued mappings in Banach spaces. The iterative process used is independent of Ishikawa iterative process and converges faster. Some examples are provided to validate our results. Our results extend and unify some results in the contemporary literature.

Will an imperfect vaccine curtail the COVID-19 pandemic in the U.S.?

The novel coronavirus(COVID-19)that emerged from Wuhan city of China in late December 2019 continue to pose devastating public health and economic challenges across the world.Although the community-wide implementation of basic non-pharmaceutical intervention measures,such as social distancing,quarantine of suspected COVID-19 cases,isolation of confirmed cases,use of face masks in public,contact tracing and testing,have been quite effective in curtailing and mitigating the burden of the pandemic,it is universally believed that the use of a vaccine may be necessary to effectively curtail and eliminating COVID-19 in human populations.This study is based on the use of a mathematical model for assessing the impact of a hypothetical imperfect anti-COVID-19 vaccine on the control of COVID-19 in the United States.An analytical expression for the minimum percentage of unvaccinated susceptible individuals needed to be vaccinated in order to achieve vaccine-induced community herd immunity is derived.The epidemiological consequence of the herd immunity threshold is that the disease can be effectively controlled or eliminated if the minimum herd immunity threshold is achieved in the community.Simulations of the model,using baseline parameter values obtained from fitting the model with COVID-19 mortality data for the U.S.,show that,for an anti-COVID-19 vaccine with an assumed protective efficacy of 80%,at least 82%of the susceptible US population need to be vaccinated to achieve the herd immunity threshold.The prospect of COVID-19 elimination in the US,using the hypothetical vaccine,is greatly enhanced if the vaccination program is combined with other interventions,such as face mask usage and/or social distancing.Such combination of strategies significantly reduces the level of the vaccine-induced herd immunity threshold needed to eliminate the pandemic in the US.For instance,the herd immunity threshold decreases to 72%if half of the US population regularly wears face masks in public(the threshold decreases to 46%if everyone wears a face mask).

Analytic Fragmentation Semigroups and Classical Solutions to Coagulation–fragmentation Equations——a Survey

In the paper we present a survey of recent results obtained by the author that concern discrete fragmentation–coagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical semigroup solutions to the fragmentation–coagulation equations.

Climate-dependent malaria disease transmission model and its analysis

Malaria infection continues to be a major problem in many parts of the world including Africa.Environmental variables are known to significantly affect the population dynamics and abundance of insects,major catalysts of vector-borne diseases,but the exact extent and consequences of this sensitivity are not yet well established.To assess the impact of the variability in temperature and rainfall on the transmission dynamics of malaria in a population,we propose a model consisting of a system of non-autonomous deterministic equations that incorporate the effect of both temperature and rainfall to the dispersion and mortality rate of adult mosquitoes.The model has been validated using epidemiological data collected from the western region of Ethiopia by considering the trends for the cases of malaria and the climate variation in the region.Further,a mathematical analysis is performed to assess the impact of temperature and rainfall change on the transmission dynamics of the model.The periodic variation of seasonal variables as well as the non-periodic variation due to the long-term climate variation have been incorporated and analyzed.In both periodic and non-periodic cases,it has been shown that the disease-free solution of the model is globally asymptotically stable when the basic reproduction ratio is less than unity in the periodic system and when the threshold function is less than unity in the non-periodic system.The disease is uniformly persistent when the basic reproduction ratio is greater than unity in the periodic system and when the threshold function is greater than unity in the non-periodic system.

On a diffusive bacteriophage dynamical model for bacterial infections

Bacteriophages or phages are viruses that infect bacteria and are increasingly used to control bacterial infections.We develop a reaction-diffusion model coupling the interactive dynamic of phages and bacteria with an epidemiological bacteria-borne disease model.For the submodel without phage absorption,the basic reproduction number Ro is computed.The disease-free equilibrium(DFE)is shown to be globally asymptotically stable whenever Ro is less than one,while a unique globally asymptotically endemic equilibrium is proven whenever Ro exceeds one.In the presence of phage absorption,the above stated classical condition based on Ro,as the average number of secondary human infections produced by susceptible/lysogen bacteria during their entire lifespan,is no longer suficient to guarantee the global stability of the DFE.We thus derive an additional threshold No,which is the average offspring number of lysogen bacteria produced by one infected human during the phage-bacteria interactions,and prove that the DFE is globally asymptotically stable whenever both Ro and No are under unity,and infections persist uniformly whenever Ro is greater than one.Finally,the discrete counterpart of the continuous partial differential equation model is derived by constructing a nonstandard finite difference scheme which is dynamically consistent.This consistency is shown by constructing suitable discrete Lyapunov functionals thanks to which the global stability results for the continuous model are replicated.This scheme is implemented in MatLab platform and used to assess the impact of spatial distribution of phages,on the dynamic of bacterial infections.

Will vaccine-derived protective immunity curtail COVID-19 variants in the US?

Multiple effective vaccines are currently being deployed to combat the COVID-19 pandemic,and are viewed as the major factor in marked reductions of disease burden in regions with moderate to high vaccination coverage.The effectiveness of COVID-19 vaccination programs is,however,significantly threatened by the emergence of new SARS-COV-2 variants that,in addition to being more transmissible than the wild-type(original)strain,may at least partially evade existing vaccines.A two-strain(one wildtype,one variant)and two-group(vaccinated or otherwise)mechanistic mathematical model is designed and used to assess the impact of the vaccine-induced cross-protective efficacy on the spread the COVID-19 pandemic in the United States.Rigorous analysis of the model shows that,in the absence of any co-circulating SARS-CoV-2 variant,the vaccine-derived herd immunity threshold needed to eliminate the wild-type strain can be achieved if 59%of the US population is fully-vaccinated with either the Pfizer or Moderna vaccine.This threshold increases to 76%if the wild-type strain is co-circulating with the Alpha variant(a SARS-CoV-2 variant that is 56%more transmissible than the wild-type strain).If the wild-type strain is co-circulating with the Delta variant(which is estimated to be 100%more transmissible than the wild-type strain),up to 82%of the US population needs to be vaccinated with either of the aforementioned vaccines to achieve the vaccine-derived herd immunity.Global sensitivity analysis of the model reveal the following four parameters as the most influential in driving the value of the reproduction number of the variant strain(hence,COVID-19 dynamics)in the US:(a)the infectiousness of the co-circulating SARS-CoV-2 variant,(b)the proportion of individuals fully vaccinated(using Pfizer or Moderna vaccine)against the wild-type strain,(c)the cross-protective efficacy the vaccines offer against the variant strain and(d)the modification parameter accounting for the reduced infectiousness of fully-vaccinated individuals experiencing breakthrough infection.Specifically,numerical simulations of the model show that future waves or surges of the COVID-19 pandemic can be prevented in the US if the two vaccines offer moderate level of cross-protection against the variant(at least 67%).This study further suggests that a new SARS-CoV-2 variant can cause a significant disease surge in the US if(i)the vaccine coverage against the wild-type strain is low(roughly<66%)(ii)the variant is much more transmissible(e.g.,100%more transmissible),than the wild-type strain,or(iii)the level of cross-protection offered by the vaccine is relatively low(e.g.,less than 50%).A new SARS-CoV-2 variant will not cause such surge in the US if it is only moderately more transmissible(e.g.,the Alpha variant,which is 56%more transmissible)than the wild-type strain,at least 66%of the population of the US is fully vaccinated,and the three vaccines being deployed in the US(Pfizer,Moderna,and Johnson&Johnson)offer a moderate level of cross-protection against the variant.