The ECHO (Extensions Community Healthcare Outcomes) model of healthcare delivery has grown rapidly since its establishment and increased in popularity in recent years. This expansion has developed alongside the growing incidence of chronic diseases and the need to better manage them. The increasing uptake in ECHO has presented a requirement to assess its true value as healthcare costs are increasing globally, resulting in a growing demand by governments and policy makers to ensure chronic disease management strategies provide true value. Therefore, the aim of this review is to examine the impact that ECHO has on clinical practice and how such impacts are measured or evaluated. A narrative literature review is carried out to examine the outcomes assessed in ECHO-related studies. Three key academic databases were utilised for the literature search: Web of Science, PubMed, and Medline. Keywords relating to the review were chosen and searched for. Papers were screened using specified inclusion and exclusion criteria relating to years of publication (2000–2020), type of publication (original research, review papers and meta-analyses) and language requirements (English language only). This review found that while the ECHO model is expanding, and improving the so-called “knowledge gap” between specialists and primary care physicians, there is also a gap in the ways value is examined within ECHO. Most studies on ECHO lack an examination of patient reported health outcomes and appropriate, comparative costing methods. Current ECHO-related studies lack vital components that demonstrate the value of the model. Such components include patient reported health outcomes and detailed costing comparisons between the ECHO model and the traditional care pathway it is replacing.
Citation: Christina Kenny, Anushree Priyadarshini. “Mind the Gap” - An overview of the role of the Extensions Community Healthcare Outcomes (ECHO) model in enhancing value in health care delivery[J]. AIMS Public Health, 2023, 10(1): 94-104. doi: 10.3934/publichealth.2023008
Related Papers:
[1]
Hamdy M. Youssef, Najat A. Alghamdi, Magdy A. Ezzat, Alaa A. El-Bary, Ahmed M. Shawky .
A new dynamical modeling SEIR with global analysis applied to the real data of spreading COVID-19 in Saudi Arabia. Mathematical Biosciences and Engineering, 2020, 17(6): 7018-7044.
doi: 10.3934/mbe.2020362
[2]
H. M. Srivastava, I. Area, J. J. Nieto .
Power-series solution of compartmental epidemiological models. Mathematical Biosciences and Engineering, 2021, 18(4): 3274-3290.
doi: 10.3934/mbe.2021163
[3]
Hongfan Lu, Yuting Ding, Silin Gong, Shishi Wang .
Mathematical modeling and dynamic analysis of SIQR model with delay for pandemic COVID-19. Mathematical Biosciences and Engineering, 2021, 18(4): 3197-3214.
doi: 10.3934/mbe.2021159
[4]
Salma M. Al-Tuwairqi, Sara K. Al-Harbi .
Modeling the effect of random diagnoses on the spread of COVID-19 in Saudi Arabia. Mathematical Biosciences and Engineering, 2022, 19(10): 9792-9824.
doi: 10.3934/mbe.2022456
[5]
Tao Chen, Zhiming Li, Ge Zhang .
Analysis of a COVID-19 model with media coverage and limited resources. Mathematical Biosciences and Engineering, 2024, 21(4): 5283-5307.
doi: 10.3934/mbe.2024233
[6]
Sarafa A. Iyaniwura, Musa Rabiu, Jummy F. David, Jude D. Kong .
Assessing the impact of adherence to Non-pharmaceutical interventions and indirect transmission on the dynamics of COVID-19: a mathematical modelling study. Mathematical Biosciences and Engineering, 2021, 18(6): 8905-8932.
doi: 10.3934/mbe.2021439
[7]
A. Q. Khan, M. Tasneem, M. B. Almatrafi .
Discrete-time COVID-19 epidemic model with bifurcation and control. Mathematical Biosciences and Engineering, 2022, 19(2): 1944-1969.
doi: 10.3934/mbe.2022092
[8]
Cheng-Cheng Zhu, Jiang Zhu .
Spread trend of COVID-19 epidemic outbreak in China: using exponential attractor method in a spatial heterogeneous SEIQR model. Mathematical Biosciences and Engineering, 2020, 17(4): 3062-3087.
doi: 10.3934/mbe.2020174
[9]
Sarita Bugalia, Vijay Pal Bajiya, Jai Prakash Tripathi, Ming-Tao Li, Gui-Quan Sun .
Mathematical modeling of COVID-19 transmission: the roles of intervention strategies and lockdown. Mathematical Biosciences and Engineering, 2020, 17(5): 5961-5986.
doi: 10.3934/mbe.2020318
[10]
Qinghua Liu, Siyu Yuan, Xinsheng Wang .
A SEIARQ model combine with Logistic to predict COVID-19 within small-world networks. Mathematical Biosciences and Engineering, 2023, 20(2): 4006-4017.
doi: 10.3934/mbe.2023187
Abstract
The ECHO (Extensions Community Healthcare Outcomes) model of healthcare delivery has grown rapidly since its establishment and increased in popularity in recent years. This expansion has developed alongside the growing incidence of chronic diseases and the need to better manage them. The increasing uptake in ECHO has presented a requirement to assess its true value as healthcare costs are increasing globally, resulting in a growing demand by governments and policy makers to ensure chronic disease management strategies provide true value. Therefore, the aim of this review is to examine the impact that ECHO has on clinical practice and how such impacts are measured or evaluated. A narrative literature review is carried out to examine the outcomes assessed in ECHO-related studies. Three key academic databases were utilised for the literature search: Web of Science, PubMed, and Medline. Keywords relating to the review were chosen and searched for. Papers were screened using specified inclusion and exclusion criteria relating to years of publication (2000–2020), type of publication (original research, review papers and meta-analyses) and language requirements (English language only). This review found that while the ECHO model is expanding, and improving the so-called “knowledge gap” between specialists and primary care physicians, there is also a gap in the ways value is examined within ECHO. Most studies on ECHO lack an examination of patient reported health outcomes and appropriate, comparative costing methods. Current ECHO-related studies lack vital components that demonstrate the value of the model. Such components include patient reported health outcomes and detailed costing comparisons between the ECHO model and the traditional care pathway it is replacing.
1.
Introduction
Mathematical models for the transmission dynamics of infectious diseases have been a powerful tool to understand and control epidemics. An important amount of these mathematical models are given by systems of ordinary differential equations (ODE's), considering a continuous-time framework. In the last decades, a great number of compartmental models has been proposed and many of them share some of the assumptions of the well known SIR model, first proposed by Kermack-McKendrick in 1927 [5] and also given by a system of ODE's.
The importance of compartmental models given by systems of ODE's has been even more highlighted since the beginning of the COVID-19 pandemic. In fact, SIR, SEIR, SEIRD-type models, among many others, have been used to analyze, predict and control the spread of SARS-CoV-2 virus worldwide, in what follows we refer to some of this models applied to COVID-19 given by systems of ODE's. The effect of the lockdown on the spread of COVID-19 was analyzed in [4] by considering a mathematical model that assesses the imposition of the lockdown in Nigeria. Besides lockdown measures, in [6] quarantine, and hospitalization of COVID-19 infected individuals is analyzed. A SIDARTHE compartmental model, is proposed in [10] which discriminates between infected individuals depending on whether they have been diagnosed and on the severity of their symptoms and is fitted to the COVID-19 epidemic in Italy. The authors claim that restrictive social-distancing measures will need to be combined with widespread testing and contact tracing to end the ongoing COVID-19 pandemic. The spread of COVID-19 in Portugal is modeled in [14], fitting the model to real data of simultaneously the number of active infected and hospitalized individuals, SEIR types models applied to COVID-19 are proposed in, for example, [12,16,21], were the models are fitted to real data. Quarantine and lockdown measures are considered in [24] with SEIQR type models. Fractional SEIR type models are also important in modeling and predicting the spread of infectious diseases. Their advantages in comparison to other type of models are highlighted in, for example, [1] were COVID-19 epidemic in Pakistan is analyzed through a fractional SEIR type model using the operator of Atangana-Baleanu. The Atangana-Baleanu derivative is also considered in [15]. A nabla discrete ABC-fractional order COVID-19 model is analyzed in [13]. In [2] a fractional model is proposed to study the first COVID-19 outbreak in Wuhan, China. The outbreak in Wuhan is also considered in [7,19]. A Caputo fractional model is proposed in [18] and fitted to COVID-19 data of Galicia, Spain and Portugal. A Bats-Hosts-Reservoir-People transmission fractional-order COVID-19 model for simulating the potential transmission with the thought of individual response and control measures by the government is analyzed in [23]. Stochastic epidemic models for COVID-19 spread and control are proposed in, e.g., [8,11]. A new class of distributions are applied to the generalized log-exponential transformation of Gumbel Type-II and implemented using real data of COVID-19, in [27]. The exponential transformation of Gumbel Type-II distribution for modeling COVID-19 data is used to analyze the number of deaths due to COVID-19 for Europe and China, in [28].
Among other important issues, in the mathematical analysis of compartmental models given by systems of ODE's, we may emphasize the stability analysis of the equilibrium points, the basic reproduction number and the model fitting to real data. Although the difficulty of this analysis depends on the complexity of the model under study, part of it is common to the majority of the models and may become simplified if we use adequate mathematical software. In this paper, we show how to use free and open-source software for the mathematical analysis of the models. We focus on the mathematical software SageMath (version 9.2) [22] and Python programming language [20], version 3.8, and the Python Libraries: Numpy (version 1.18.5), Pandas (version 0.25.3), Scipy (version 1.4.1) and Matplotlib (version 2.0.0).
In this note, we consider a SAIRP model, given by a system of five ODE's, for the transmission of SARS-CoV-2, first proposed in [26] and after generalized to piecewise constant parameters and complex networks model in [25]. We provide all the codes that allow us to compute the equilibrium points, basic reproduction number, visualize the global stability of the equilibrium points and by considering piecewise constant parameters. Moreover, we provide the Python code that allows to estimate some of the piecewise constant parameters and fit the model to the real data of COVID-19 transmission in Portugal, from March 2, 2020 until April 15, 2021. It is important to note, that all the codes are elementary and thus can be easily adapted to other compartmental models, see e.g., [3,6,9,17,24,30].
The paper is organized as follows. In Section 2, the SAIRP model, given by a system of ODE's, is described. The equilibrium points of the SAIRP model are computed in Section 3 and their global stability is illustrated using SageMath (version 9.2) [22] and Python (version 3.8) [20]. The generalized SAIRP model with piecewise constant parameters is presented and the Python code for the model fitting to COVID-19 Portuguese real data is given, in Section 4.
2.
Mathematical model
In this section, we consider the compartmental SAIRP mathematical model proposed in [25,26] for the transmission dynamics of COVID-19. We start by recalling the assumptions of the model. The total population N(t), with t∈[0,T] (in days) and T>0, is subdivided into five classes: susceptible individuals (S); asymptomatic infected individuals (A); active infected individuals (I); removed (including recovered individuals and COVID-19 induced deaths) (R); and protected individuals (P). Therefore, N(t)=S(t)+A(t)+I(t)+R(t)+P(t), considering a continuous time framework, with t∈[0,T]. The total population is homogeneous and has a variable size, with constant recruitment rate, Λ, and natural death rate, μ>0. The susceptible individuals S become infected by contact with active infected I and asymptomatic infected A individuals, at a rate of infection β(θA+I)N, where θ represents a modification parameter for the infectiousness of the asymptomatic infected individuals A and β represents the transmission rate. Only a fraction q of asymptomatic infected individuals A develop symptoms and are detected, at a rate v. Active infected individuals I are transferred to the recovered/removed individuals R, at a rate δ, by recovery from the disease or by COVID-19 induced death. A fraction p, with 0<p<1, is protected (without permanent immunity) from infection, and is transferred to the class of protected individuals P, at a rate ϕ. A fraction m of protected individuals P returns to the susceptible class S, at a rate w. Let ν=vq and ω=wm. The previous assumptions are described by the following system of ordinary differential equations:
The model (2.1) is biologically and mathematically well-posed, that is for any initial condition x0=(S0,A0,I0,R0,P0)T∈Ω, the system (2.1) admits a unique solution defined on [0,∞), whose components are non-negative. Furthermore, the region Ω defined by (2.2) is positively invariant [25].
3.
Equilibrium points and stability analysis
In this section, we use the free and open-source mathematical software SageMath (version 9.2) [22] to help us
compute the equilibrium points and the basic reproduction number of model (2.1). The global stability analysis of the equilibrium points is illustrated through numerical simulations developed using Python (version 3.8) [20].
3.1. Computing equilibrium points and basic reproduction number in SageMath
The model (2.1) has two equilibrium points:
● disease-free equilibrium, denoted by Σ0, given by
Theorem 1 (Local stability of the DFE, [25]). The disease-free equilibrium, Σ0, is locally asymptoticallystable whenever R0<1.
Theorem 2 (Global stability of the DFE, [25]). If R0<1, then the disease-free equilibrium, Σ0, is globally asymptotically stable in Ω.
Theorem 3 (Global stability of the EE, [25]). The compact region Γ defined by
Γ={x=(S,A,I,R,P)T∈(R+)5;S+A+I+R+P=Λμ}
is positively invariant under the flow induced by system (2.1).It contains the disease-free equilibrium, Σ0, and the endemic equilibrium, Σ+, if R0>1.Furthermore, if R0>1, then the endemic equilibrium Σ+is globally asymptotically stable in Γ.
The Python code to generate Figure 1 is given below. We first import several Python scientific libraries: numpy contains standard routines for numerical computations; scipy contains the function odeint, which implements the Runge-Kutta method for integrating ODE's systems; matplotlib contains useful functions for producing figures.
Next, we define the parameters for the SAIRP model (2.1).
Using expression (3.3), we easily compute the basic reproduction number R0.
Afterwards, we define the SAIRP model given by system (2.1).
Finally, we integrate the SAIRP model (2.1) with several randomly chosen initial conditions and we produce the 3D Figure 1.
4.
Mathematical model with piecewise constant parameters
In this section, we consider the SAIRP model with piecewise constant parameters, proposed in [25], which allows to model the impact of public health policies and the human behavior in the dynamics of the COVID-19 epidemic.
For the sake of simplicity, the equations of the SAIRP model (2.1) can be rewritten as
˙x(t)=f(x(t),α),t>0,
(4.1)
with x=(S,A,I,R,P)T∈R5 and α=(Λ,μ,β,p,θ,ϕ,ω,ν,δ)T∈R9, where the non-linear operator f is defined in R5×R9 by
The following theorem establishes the well-posedness of system (4.3).
Theorem 4.For any initial condition x0∈Ω, the sequence of Cauchyproblems given by model (4.3) admits a unique globalsolution, denoted again by x(t,x0), whose components are non-negative.Furthermore, the region Ω is positively invariant [25].
We recall that the solutions of problem (4.3) are continuous on the time interval [T0,Tend], but may not be of class C1 at t=Ti, 0≤i≤n−1. From the modeling point of view, each change of parameters occurring at time t=Ti (1≤i≤n−1) corresponds, for example, to a public announcement of confinement/lift of confinement or prohibition of displacement [25].
In what follows, we sub-divide the time interval [0,410] days into 9 sub-intervals and consider a set of piecewise parameters which are estimated in order to fit the real data of COVID-19 spread in Portugal, since the first confirmed case on March 2, 2020, until April 15, 2021.
Model fitting to real data with Python In order to fit the real data of active infected individuals by SARS-CoV-2 (detected by test) daily provided by the health authorities in Portugal [31], we use the model (4.3) with piecewise constant parameters and use the programming language Python (version 3.8) [20], and the Python Libraries: Numpy (version 1.18.5), Pandas (version 0.25.3), Scipy (version 1.4.1) and Matplotlib (version 2.0.0). The real data are available in [31] or, for example, in the data repository site, see e.g. [32]. In what follows we explain the goal of each code block.
First we need as previously to import several Python scientific libraries.
We consider the real data from [31,32], here denominated data-pt-covid19 with extension .xlsx. To read the real data one can use the following code.
Next, we subdivide the time window of the total 410 days into 9 subintervals. This specific subdivision is related to the increase/decrease of the transmission of the virus in the community, the public health measures implemented by the Portuguese authorities and the human behavior, that changes over time.
Next, we draw the curve of active infected individuals with COVID-19 in Portugal (real data), from the first confirmed case day, March 2, 2020, until April 15, 2021.
In order to solve the model (4.1) we need to define the initial conditions, for t=0, that correspond to the number of individuals in each class on March 2, 2020.
Then, we define the values of the parameters that take constant values for all time t∈[0,410].
The parameters β, p and m are assumed to be piecewise constant. Moreover, we estimate the parameter β for all t∈[0,410] days and the parameters m and p in some of the sub-intervals, using scipy.optimize.curve_fit from Python, see [33] for more details, that uses non-linear least squares method to fit a function to data.
Finally, we make the plot with the real data and the model solution I(t), for t∈[0,410].
The graphic that results from the previous Python code is given in Figure 2.
Figure 2.
Output of the Python code: fitting the SAIRP model (4.3) to COVID-19 active infected individuals in Portugal, between March 02, 2020 and April 15, 2021.
We recall, that the mathematical method and numerical code can be applied to other models given by systems of ODE's that describe the transmission dynamics of different virus or bacteria in human, animals or cells, for example.
5.
Conclusions
In this note, we provided a complete toolkit for performing both a symbolic and numerical analysis of a recent epidemic model, presented in [25], introduced in order to study the spreading of the COVID-19 pandemic.
This innovative compartmental model takes into account the possible transmission of the virus by asymptomatic individuals, as well as the possibility to protect a fraction of the affected population by public health strategies, such as confinement or quarantine. Since the public health policies have a strong influence on the spreading of the epidemic, we have also considered a piecewise constant parameters extension of the initial autonomous system, which have proved its ability to fit with real data. Once the mathematical analysis of such a complex compartmental model can be tedious, we have presented a computational approach, which is intended to support the theoretical analysis:
● using the free and open-source software Sagemath, we have first computed the symbolic expressions of the basic reproduction number R0 (Eq (3.3)), of the disease-free equilibrium Σ0 (Eq (3.1)) and of the endemic equilibrium Σ+ (Eq (3.2)), which highlight the role of each parameter of the model;
● using the scientific libraries numpy, scipy and matplotlib of the free and open-source language Python, we have integrated the epidemic model, in order to illustrate the theoretical stability statements (see Figure 1);
● finally, we fitted the piecewise constant parameters extension of the initial model with recent real-world data (see Figure 2).
Overall, the programs presented in this note have been written in a sufficiently general manner, so that they can easily be adapted to a great number of other epidemic models.
In a near future, we aim to apply our computational approach to an improved version of our complex compartmental model, so as to consider the effects of the mutations of the virus and the benefits of vaccination.
Acknowledgments
This research is partially supported by the Portuguese Foundation for Science and Technology (FCT) by the project UIDB/04106/2020 (CIDMA). Cristiana J. Silva is also supported by FCT via the FCT Researcher Program CEEC Individual 2018 with reference CEECIND/00564/2018.
Conflict of interest
The authors declare that they have no conflict of interest.
Bardhan I, Chen H, Karahanna E (2020) Connecting systems, data, and people: A multidisciplinary research roadmap for chronic disease management. MIS Quart 44: 185-200. https://doi.org/10.25300/MISQ/2020/14644
[3]
Nicholson K, Makovski TT, Griffith LE, et al. (2019) Multimorbidity and comorbidity revisited: refining the concepts for international health research. J Clin Epidemiol 105: 142-146. https://doi.org/10.1016/j.jclinepi.2018.09.008
[4]
Violan C, Foguet-Boreu Q, Flores-Mateo G, et al. (2014) Prevalence, determinants, and patterns of multimorbidity in primary care: A systematic review of observational studies. PLoS One 9: e102149. https://doi.org/10.1371/journal.pone.0102149
McPhail SM (2016) Multimorbidity in chronic disease: impact on health care resources and costs. Risk Manag Healthc Policy 9: 143-156. https://doi.org/10.2147/RMHP.S97248
[8]
Rapoport J, Jacobs P, Bell NR, et al. (2004) Refining the measurement of the economic burden of chronic diseases in Canada. Age 20: 1-643.
Rosenblatt RA (2004) A view from the periphery—health care in rural America. N Engl J Med 351: 1049-1051. https://doi.org/10.1056/NEJMp048073
[13]
Arora S, Geppert CM, Kalishman S, et al. (2007) Academic health center management of chronic diseases through knowledge networks: Project ECHO. Acad Med 82: 154-160. https://doi.org/10.1097/ACM.0b013e31802d8f68
Arora S, Thornton K, Murata G, et al. (2011) Outcomes of treatment for hepatitis C virus infection by primary care providers. N Engl J Med 364: 2199-2207. https://doi.org/10.1056/NEJMoa1009370
[16]
Harkins M, Raissy H, Moseley K, et al. (2011) Project ECHO: improving asthma care in New Mexico with telehealth technology. Chest 140: 861A. https://doi.org/10.1378/chest.1107313
[17]
Lewiecki EM, Rochelle R (2019) Project ECHO: telehealth to expand capacity to deliver best practice medical care. Rheum Dis Clin North Am 45: 303-314. https://doi.org/10.1016/j.rdc.2019.01.003
Bennett MI, Kaasa S, Barke A, et al. (2019) The IASP classification of chronic pain for ICD-11: chronic cancer-related pain. Pain 160: 38-44. https://doi.org/10.1097/j.pain.0000000000001363
[20]
Furlan AD, Pajer KA, Gardner W, et al. (2019) Project ECHO: building capacity to manage complex conditions in rural, remote, and underserved areas. Can J Rural Med 24: 115-120. https://doi.org/10.4103/CJRM.CJRM_20_18
[21]
Dowell D (2022) Draft updated CDC guideline for prescribing opioids: background, overview, and progress. CDC Stacks .
[22]
Hassan S, Carlin L, Zhao J, et al. (2021) Promoting an interprofessional approach to chronic pain management in primary care using Project ECHO. J Interprof Care 35: 464-467. https://doi.org/10.1080/13561820.2020.1733502
[23]
Chaple MJ, Freese TE, Rutkowski BA, et al. (2018) Using ECHO clinics to promote capacity building in clinical supervision. Am J Prev Med 54: S275-S280. https://doi.org/10.1016/j.amepre.2018.01.015
[24]
Bikinesi L, O'Bryan G, Roscoe C, et al. (2020) Implementation and evaluation of a Project ECHO telementoring program for the Namibian HIV workforce. Hum Resour Health 18: 61. https://doi.org/10.1186/s12960-020-00503-w
[25]
Wood BR, Mann MS, Martinez-Paz N, et al. (2018) Project ECHO: telementoring to educate and support prescribing of HIV pre-exposure prophylaxis by community medical providers. Sex health 15: 601-605. https://doi.org/10.1071/SH18062
Hariprasad R, Arora S, Babu R, et al. (2018) Retention of knowledge levels of health care providers in cancer screening through telementoring. J Glob Oncol 4: 1-7. https://doi.org/10.1200/JGO.18.00048
[29]
Davis DA, Mazmanian PE, Fordis M, et al. (2006) Accuracy of physician self-assessment compared with observed measures of competence: A systematic review. JAMA 296: 1094-1102. https://doi.org/10.1001/jama.296.9.1094
[30]
Lalloo C, Osei-Twum JA, Rapoport A, et al. (2021) Pediatric project ECHO®: A virtual Community of Practice to improve palliative care knowledge and self-efficacy among Interprofessional health care providers. J Palliat Med 24: 1036-1044. https://doi.org/10.1089/jpm.2020.0496
[31]
Newcomb D, Moss PW (2018) Empowering General Practitioners to manage children with ADHD using the ECHO® model. Int J Integr Care 2: 18. https://doi.org/10.5334/ijic.s1080
[32]
Zhou C, Crawford A, Serhal E, et al. (2016) The impact of project ECHO on participant and patient outcomes: A systematic review. Acad Med 91: 1439-1461. https://doi.org/10.1097/ACM.0000000000001328
[33]
Katzman JG, Qualls CR, Satterfield WA, et al. (2019) Army and navy ECHO pain telementoring improves clinician opioid prescribing for military patients: an observational cohort study. J Gen Intern Med 34: 387-395. https://doi.org/10.1007/s11606-018-4710-5
[34]
Rattay T, Dumont IP, Heinzow HS, et al. (2017) Cost-effectiveness of access expansion to treatment of hepatitis C virus infection through primary care providers. Gastroenterology 153: 1531-1543.e2. https://doi.org/10.1053/j.gastro.2017.10.016
Katzman JG, Galloway K, Olivas C, et al. (2016) Expanding health care access through education: dissemination and implementation of the ECHO model. Mil Med 181: 227-235. https://doi.org/10.7205/MILMED-D-15-00044
[37]
Xie Y, Lu L, Gao F, et al. (2021) Integration of artificial intelligence, blockchain, and wearable technology for chronic disease management: A new paradigm in smart healthcare. Curr Med Sci 41: 1123-1133. https://doi.org/10.1007/s11596-021-2485-0
This article has been cited by:
1.
Azhar Iqbal Kashif Butt, Muhammad Rafiq, Waheed Ahmad, Naeed Ahmad,
Implementation of computationally efficient numerical approach to analyze a Covid-19 pandemic model,
2023,
69,
11100168,
341,
10.1016/j.aej.2023.01.052
2.
R. Prem Kumar, P.K. Santra, G.S. Mahapatra,
Global stability and analysing the sensitivity of parameters of a multiple-susceptible population model of SARS-CoV-2 emphasising vaccination drive,
2023,
203,
03784754,
741,
10.1016/j.matcom.2022.07.012
3.
Harald Øverby, Jan A. Audestad, Gabriel Andy Szalkowski,
Compartmental market models in the digital economy—extension of the Bass model to complex economic systems,
2023,
47,
03085961,
102441,
10.1016/j.telpol.2022.102441
4.
Manoj Kumar Singh, Brajesh K. Singh, Carlo Cattani,
Impact of general incidence function on three-strain SEIAR model,
2023,
20,
1551-0018,
19710,
10.3934/mbe.2023873
5.
Zenebe Shiferaw Kifle, Legesse Lemecha Obsu,
Mathematical modeling and analysis of COVID-19 and TB co-dynamics,
2023,
9,
24058440,
e18726,
10.1016/j.heliyon.2023.e18726
6.
Sofia Tedim, Vera Afreixo, Miguel Felgueiras, Rui Pedro Leitão, Sofia J. Pinheiro, Cristiana J. Silva,
Evaluating COVID-19 in Portugal: Bootstrap confidence interval,
2023,
9,
2473-6988,
2756,
10.3934/math.2024136
7.
Azhar Iqbal Kashif Butt, Waheed Ahmad, Muhammad Rafiq, Naeed Ahmad, Muhammad Imran,
Computationally efficient optimal control analysis for the mathematical model of Coronavirus pandemic,
2023,
234,
09574174,
121094,
10.1016/j.eswa.2023.121094
Christina Kenny, Anushree Priyadarshini. “Mind the Gap” - An overview of the role of the Extensions Community Healthcare Outcomes (ECHO) model in enhancing value in health care delivery[J]. AIMS Public Health, 2023, 10(1): 94-104. doi: 10.3934/publichealth.2023008
Christina Kenny, Anushree Priyadarshini. “Mind the Gap” - An overview of the role of the Extensions Community Healthcare Outcomes (ECHO) model in enhancing value in health care delivery[J]. AIMS Public Health, 2023, 10(1): 94-104. doi: 10.3934/publichealth.2023008