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1.
Bull Math Biol ; 85(7): 66, 2023 Jun 09.
Article in English | MEDLINE | ID: covidwho-20240982

ABSTRACT

Diagnostic testing may represent a key component in response to an ongoing epidemic, especially if coupled with containment measures, such as mandatory self-isolation, aimed to prevent infectious individuals from furthering onward transmission while allowing non-infected individuals to go about their lives. However, by its own nature as an imperfect binary classifier, testing can produce false negative or false positive results. Both types of misclassification are problematic: while the former may exacerbate the spread of disease, the latter may result in unnecessary isolation mandates and socioeconomic burden. As clearly shown by the COVID-19 pandemic, achieving adequate protection for both people and society is a crucial, yet highly challenging task that needs to be addressed in managing large-scale epidemic transmission. To explore the trade-offs imposed by diagnostic testing and mandatory isolation as tools for epidemic containment, here we present an extension of the classical Susceptible-Infected-Recovered model that accounts for an additional stratification of the population based on the results of diagnostic testing. We show that, under suitable epidemiological conditions, a careful assessment of testing and isolation protocols can contribute to epidemic containment, even in the presence of false negative/positive results. Also, using a multi-criterial framework, we identify simple, yet Pareto-efficient testing and isolation scenarios that can minimize case count, isolation time, or seek a trade-off solution for these often contrasting epidemic management objectives.


Subject(s)
COVID-19 , Humans , COVID-19/diagnosis , COVID-19/epidemiology , COVID-19/prevention & control , SARS-CoV-2 , Pandemics/prevention & control , Models, Biological , Mathematical Concepts
2.
PLoS Comput Biol ; 19(5): e1011088, 2023 05.
Article in English | MEDLINE | ID: covidwho-2323248

ABSTRACT

Modelling the transmission dynamics of an infectious disease is a complex task. Not only it is difficult to accurately model the inherent non-stationarity and heterogeneity of transmission, but it is nearly impossible to describe, mechanistically, changes in extrinsic environmental factors including public behaviour and seasonal fluctuations. An elegant approach to capturing environmental stochasticity is to model the force of infection as a stochastic process. However, inference in this context requires solving a computationally expensive "missing data" problem, using data-augmentation techniques. We propose to model the time-varying transmission-potential as an approximate diffusion process using a path-wise series expansion of Brownian motion. This approximation replaces the "missing data" imputation step with the inference of the expansion coefficients: a simpler and computationally cheaper task. We illustrate the merit of this approach through three examples: modelling influenza using a canonical SIR model, capturing seasonality using a SIRS model, and the modelling of COVID-19 pandemic using a multi-type SEIR model.


Subject(s)
COVID-19 , Influenza, Human , Humans , Pandemics , Stochastic Processes , Influenza, Human/epidemiology , Models, Biological
3.
Bull Math Biol ; 85(6): 54, 2023 05 11.
Article in English | MEDLINE | ID: covidwho-2318476

ABSTRACT

Metapopulation models have been a popular tool for the study of epidemic spread over a network of highly populated nodes (cities, provinces, countries) and have been extensively used in the context of the ongoing COVID-19 pandemic. In the present work, we revisit such a model, bearing a particular case example in mind, namely that of the region of Andalusia in Spain during the period of the summer-fall of 2020 (i.e., between the first and second pandemic waves). Our aim is to consider the possibility of incorporation of mobility across the province nodes focusing on mobile-phone time-dependent data, but also discussing the comparison for our case example with a gravity model, as well as with the dynamics in the absence of mobility. Our main finding is that mobility is key toward a quantitative understanding of the emergence of the second wave of the pandemic and that the most accurate way to capture it involves dynamic (rather than static) inclusion of time-dependent mobility matrices based on cell-phone data. Alternatives bearing no mobility are unable to capture the trends revealed by the data in the context of the metapopulation model considered herein.


Subject(s)
COVID-19 , Humans , COVID-19/epidemiology , Pandemics , Models, Biological , Mathematical Concepts , Time
4.
J Math Biol ; 86(5): 82, 2023 04 25.
Article in English | MEDLINE | ID: covidwho-2312809

ABSTRACT

We formulate a general age-of-infection epidemic model with two pathways: the symptomatic infections and the asymptomatic infections. We then calculate the basic reproduction number [Formula: see text] and establish the final size relation. It is shown that the ratio of accumulated counts of symptomatic patients and asymptomatic patients is determined by the symptomatic ratio f which is defined as the probability of eventually becoming symptomatic after being infected. We also formulate and study a general age-of-infection model with disease deaths and with two infection pathways. The final size relation is investigated, and the upper and lower bounds for final epidemic size are given. Several numerical simulations are performed to verify the analytical results.


Subject(s)
Asymptomatic Infections , Epidemics , Humans , Asymptomatic Infections/epidemiology , Basic Reproduction Number , Probability , Models, Biological
5.
J Math Biol ; 86(5): 65, 2023 03 30.
Article in English | MEDLINE | ID: covidwho-2311810

ABSTRACT

The perception of susceptible individuals naturally lowers the transmission probability of an infectious disease but has been often ignored. In this paper, we formulate and analyze a diffusive SIS epidemic model with memory-based perceptive movement, where the perceptive movement describes a strategy for susceptible individuals to escape from infections. We prove the global existence and boundedness of a classical solution in an n-dimensional bounded smooth domain. We show the threshold-type dynamics in terms of the basic reproduction number [Formula: see text]: when [Formula: see text], the unique disease-free equilibrium is globally asymptotically stable; when [Formula: see text], there is a unique constant endemic equilibrium, and the model is uniformly persistent. Numerical analysis exhibits that when [Formula: see text], solutions converge to the endemic equilibrium for slow memory-based movement and they converge to a stable periodic solution when memory-based movement is fast. Our results imply that the memory-based movement cannot determine the extinction or persistence of infectious disease, but it can change the persistence manner.


Subject(s)
Communicable Diseases , Epidemics , Humans , Computer Simulation , Models, Biological , Communicable Diseases/epidemiology , Basic Reproduction Number , Disease Susceptibility/epidemiology
6.
Clin Transl Sci ; 16(4): 618-630, 2023 04.
Article in English | MEDLINE | ID: covidwho-2299386

ABSTRACT

This study aimed to determine the effects of pregnancy and ontogeny on risperidone and paliperidone pharmacokinetics by assessing their serum concentrations in two subjects and constructing a customized physiologically-based pharmacokinetic (PBPK) model. Risperidone and paliperidone serum concentrations were determined in a pregnant woman and her newborn. PBPK models for risperidone and paliperidone in adults, pediatric, and pregnant populations were developed and verified using the Simcyp simulator. These models were then applied to our two subjects, generating their "virtual twins." Effects of pregnancy on both drugs were examined using models with fixed pharmacokinetic parameters. In the neonatal PBPK simulation, 10 different models for estimating the renal function of neonates were evaluated. Risperidone was not detected in the serum of both pregnant woman and her newborn. Maternal and neonatal serum paliperidone concentrations were between 2.05-3.80 and 0.82-1.03 ng/ml, respectively. Developed PBPK models accurately predicted paliperidone's pharmacokinetics, as shown by minimal bias and acceptable precision across populations. The individualized maternal model predicted all observed paliperidone concentrations within the 90% prediction interval. Fixed-parameter simulations showed that CYP2D6 activity largely affects risperidone and paliperidone pharmacokinetics during pregnancy. The Flanders metadata equation showed the lowest absolute bias (mean error: 22.3% ± 6.0%) and the greatest precision (root mean square error: 23.8%) in predicting paliperidone plasma concentration in the neonatal population. Our constructed PBPK model can predict risperidone and paliperidone pharmacokinetics in pregnant and neonatal populations, which could help with precision dosing using the PBPK model-informed approach in special populations.


Subject(s)
Paliperidone Palmitate , Risperidone , Humans , Adult , Female , Pregnancy , Child , Infant, Newborn , Pregnant Women , Cytochrome P-450 CYP2D6 , Models, Biological
7.
J Biol Dyn ; 17(1): 2182373, 2023 12.
Article in English | MEDLINE | ID: covidwho-2284511

ABSTRACT

In this paper, we developed a mathematical model to simulate virus transport through a viscous background flow driven by the natural pumping mechanism. Two types of respiratory pathogens viruses (SARS-Cov-2 and Influenza-A) are considered in this model. The Eulerian-Lagrangian approach is adopted to examine the virus spread in axial and transverse directions. The Basset-Boussinesq-Oseen equation is considered to study the effects of gravity, virtual mass, Basset force, and drag forces on the viruses transport velocity. The results indicate that forces acting on the spherical and non-spherical particles during the motion play a significant role in the transmission process of the viruses. It is observed that high viscosity is responsible for slowing the virus transport dynamics. Small sizes of viruses are found to be highly dangerous and propagate rapidly through the blood vessels. Furthermore, the present mathematical model can help to better understand the viruses spread dynamics in a blood flow.


Subject(s)
COVID-19 , Humans , SARS-CoV-2 , Viscosity , Models, Biological , Biological Transport
8.
J Biol Dyn ; 17(1): 2189001, 2023 12.
Article in English | MEDLINE | ID: covidwho-2261032

ABSTRACT

We derive a stochastic epidemic model for the evolving density of infective individuals in a large population. Data shows main features of a typical epidemic consist of low periods interspersed with outbreaks of various intensities and duration. In our stochastic differential model, a novel reproductive term combines a factor expressing the recent notion of 'attenuated Allee effect' and a capacity factor is controlling the size of the process. Simulation of this model produces sample paths of the stochastic density of infectives, which behave much like long-time Covid-19 case data of recent years. Writing the process as a stochastic diffusion allows us to derive its stationary distribution, showing the relative time spent in low levels and in outbursts. Much of the behaviour of the density of infectives can be understood in terms of the interacting drift and diffusion coefficient processes, or, alternatively, in terms of the balance between noise level and the attenuation parameter of the Allee effect. Unexpected results involve the effect of increasing overall noise variance on the density of infectives, in particular on its level-crossing function.


Subject(s)
COVID-19 , Epidemics , Humans , Stochastic Processes , Models, Biological , COVID-19/epidemiology , Computer Simulation
9.
J Math Biol ; 86(3): 47, 2023 02 16.
Article in English | MEDLINE | ID: covidwho-2259140

ABSTRACT

A continuous time multivariate stochastic model is proposed for assessing the damage of a multi-type epidemic cause to a population as it unfolds. The instants when cases occur and the magnitude of their injure are random. Thus, we define a cumulative damage based on counting processes and a multivariate mark process. For a large population we approximate the behavior of this damage process by its asymptotic distribution. Also, we analyze the distribution of the stopping times when the numbers of cases caused by the epidemic attain levels beyond certain thresholds. We focus on introducing some tools for statistical inference on the parameters related with the epidemic. In this regard, we present a general hypothesis test for homogeneity in epidemics and apply it to data of Covid-19 in Chile.


Subject(s)
COVID-19 , Communicable Diseases , Epidemics , Humans , Stochastic Processes , Models, Biological , COVID-19/epidemiology , Communicable Diseases/epidemiology
10.
Bull Math Biol ; 85(5): 32, 2023 03 17.
Article in English | MEDLINE | ID: covidwho-2252039

ABSTRACT

One of the driving concerns during any epidemic is the strain on the healthcare system. As we have seen many times over the globe with the COVID-19 pandemic, hospitals and ICUs can quickly become overwhelmed by cases. While strict periods of public health mitigation have certainly helped decrease incidence and thus healthcare demand, vaccination is the only clear long-term solution. In this paper, we develop a two-module model to forecast the effects of relaxation of non-pharmaceutical intervention and vaccine uptake on daily incidence, and the cascade effects on healthcare demand. The first module is a simple epidemiological model which incorporates non-pharmaceutical intervention, the relaxation of such measures and vaccination campaigns to predict caseloads into the Fall of 2021. This module is then fed into a healthcare module which can forecast the number of doctor visits, the number of occupied hospital beds, number of occupied ICU beds and any excess demand of these. From this module, we can also estimate the length of stay of individuals in ICU. For model verification and forecasting, we use the four most populous Canadian provinces as a case study.


Subject(s)
COVID-19 , Humans , COVID-19/epidemiology , COVID-19/prevention & control , SARS-CoV-2 , COVID-19 Vaccines , Pandemics/prevention & control , Canada , Mathematical Concepts , Models, Biological , Health Services Needs and Demand , Vaccination
11.
Int J Environ Res Public Health ; 20(5)2023 03 04.
Article in English | MEDLINE | ID: covidwho-2275180

ABSTRACT

The health effects of particles are directly related to their deposition patterns (deposition site and amount) in human airways. However, estimating the particle trajectory in a large-scale human lung airway model is still a challenge. In this work, a truncated single-path, large-scale human airway model (G3-G10) with a stochastically coupled boundary method were employed to investigate the particle trajectory and the roles of their deposition mechanisms. The deposition patterns of particles with diameters (dp) of 1-10 µm are investigated under various inlet Reynolds numbers (Re = 100-2000). Inertial impaction, gravitational sedimentation, and combined mechanism were considered. With the increasing airway generations, the deposition of smaller particles (dp < 4 µm) increased due to gravitational sedimentation, while that of larger particles decreased due to inertial impaction. The obtained formulas of Stokes number and Re can predict the deposition efficiency due to the combined mechanism in the present model, and the prediction can be used to assess the dose-effect of atmospheric aerosols on the human body. Diseases in deeper generations are mainly attributed to the deposition of smaller particles under lower inhalation rates, while diseases at the proximal generations mainly result from the deposition of larger particles under higher inhalation rates.


Subject(s)
Lung , Models, Biological , Humans , Particle Size , Computer Simulation , Aerosols , Administration, Inhalation
12.
CPT Pharmacometrics Syst Pharmacol ; 12(2): 148-153, 2023 02.
Article in English | MEDLINE | ID: covidwho-2269400

ABSTRACT

Pregnant individuals are at high risk for severe illness from COVID-19, and there is an urgent need to identify safe and effective therapeutics for this population. Remdesivir (RDV) is a SARS-CoV-2 nucleotide analog RNA polymerase inhibitor. Limited RDV pharmacokinetic (PK) and safety data are available for pregnant women receiving RDV. The aims of this study were to translate a previously published nonpregnant adult physiologically based PK (PBPK) model for RDV to pregnancy and evaluate model performance with emerging clinical PK data in pregnant women with COVID-19. The pregnancy model was built in the Open Systems Pharmacology software suite (Version 10) including PK-Sim® and MoBi® with pregnancy-related changes of relevant enzymes applied. PK were predicted in a virtual population of 1000 pregnant subjects, and prediction results were compared with in vivo PK data from the International Maternal, Pediatric, Adolescent AIDS Clinical Trials (IMPAACT) Network  2032 study. The developed PBPK model successfully captured RDV and its metabolites' plasma concentrations during pregnancy. The ratios of prediction versus observation for RDV area under the curve from time 0 to infinity (AUC0-∞ ) and maximum concentration (Cmax ) were 1.61 and 1.17, respectively. For GS-704277, the ratios of predicted versus observed were 0.94 for AUC0-∞ and 1.20 for Cmax . For GS-441524, the ratios of predicted versus observed were 1.03 for AUC0-24 , 1.05 for Cmax , and 1.07 for concentrations at 24 h. All predictions of AUC and Cmax for RDV and its metabolites were within a twofold error range, and about 60% of predictions were within a 10% error range. These findings demonstrate the feasibility of translating PBPK models to pregnant women to potentially guide trial design, clinical decision making, and drug development.


Subject(s)
COVID-19 , Pregnant Women , Adult , Adolescent , Pregnancy , Female , Child , Humans , SARS-CoV-2 , COVID-19 Drug Treatment , Models, Biological
13.
Bull Math Biol ; 85(1): 6, 2022 12 19.
Article in English | MEDLINE | ID: covidwho-2246486

ABSTRACT

Most models of COVID-19 are implemented at a single micro or macro scale, ignoring the interplay between immune response, viral dynamics, individual infectiousness and epidemiological contact networks. Here we develop a data-driven model linking the within-host viral dynamics to the between-host transmission dynamics on a multilayer contact network to investigate the potential factors driving transmission dynamics and to inform how school closures and antiviral treatment can influence the epidemic. Using multi-source data, we initially determine the viral dynamics and estimate the relationship between viral load and infectiousness. Then, we embed the viral dynamics model into a four-layer contact network and formulate an agent-based model to simulate between-host transmission. The results illustrate that the heterogeneity of immune response between children and adults and between vaccinated and unvaccinated infections can produce different transmission patterns. We find that school closures play a significant effect on mitigating the pandemic as more adults get vaccinated and the virus mutates. If enough infected individuals are diagnosed by testing before symptom onset and then treated quickly, the transmission can be effectively curbed. Our multiscale model reveals the critical role played by younger individuals and antiviral treatment with testing in controlling the epidemic.


Subject(s)
COVID-19 , Child , Humans , Mathematical Concepts , Models, Biological , Pandemics/prevention & control , Schools , Vaccination
14.
J Biol Dyn ; 17(1): 2175920, 2023 12.
Article in English | MEDLINE | ID: covidwho-2245703

ABSTRACT

HIV/AIDS-COVID-19 co-infection is a major public health concern especially in developing countries of the world. This paper presents HIV/AIDS-COVID-19 co-infection to investigate the impact of interventions on its transmission using ordinary differential equation. In the analysis of the model, the solutions are shown to be non-negative and bounded, using next-generation matrix approach the basic reproduction numbers are computed, sufficient conditions for stabilities of equilibrium points are established. The sensitivity analysis showed that transmission rates are the most sensitive parameters that have direct impact on the basic reproduction numbers and protection and treatment rates are more sensitive and have indirect impact to the basic reproduction numbers. Numerical simulations shown that some parameter effects on the transmission of single infections as well as co-infection, and applying the protection rates and treatment rates have effective roles to minimize and also to eradicate the HIV/AIDS-COVID-19 co-infection spreading in the community.


Subject(s)
Acquired Immunodeficiency Syndrome , COVID-19 , Coinfection , Humans , Acquired Immunodeficiency Syndrome/epidemiology , Coinfection/epidemiology , Models, Biological , Computer Simulation
15.
Bull Math Biol ; 85(1): 9, 2022 12 24.
Article in English | MEDLINE | ID: covidwho-2238820

ABSTRACT

Predicting infectious disease outbreak impacts on population, healthcare resources and economics and has received a special academic focus during coronavirus (COVID-19) pandemic. Focus on human disease outbreak prediction techniques in current literature, Marques et al. (Predictive models for decision support in the COVID-19 crisis. Springer, Switzerland, 2021) state that there are four main methods to address forecasting problem: compartmental models, classic statistical models, space-state models and machine learning models. We adopt their framework to compare our research with previous works. Besides being divided by methods, forecasting problems can also be divided by the number of variables that are considered to make predictions. Considering this number of variables, forecasting problems can be classified as univariate, causal and multivariate models. Multivariate approaches have been applied in less than 10% of research found. This research is the first attempt to evaluate, over real time-series data of 3 different countries with univariate and multivariate methods to provide a short-term prediction. In literature we found no research with that scope and aim. A comparison of univariate and multivariate methods has been conducted and we concluded that besides the strong potential of multivariate methods, in our research univariate models presented best results in almost all regions' predictions.


Subject(s)
COVID-19 , Humans , COVID-19/epidemiology , Models, Biological , Mathematical Concepts , Disease Outbreaks , Models, Statistical
16.
Bull Math Biol ; 84(9): 99, 2022 08 09.
Article in English | MEDLINE | ID: covidwho-2220201

ABSTRACT

COVID-19, caused by the infection of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has been a global pandemic and created unprecedented public health challenges throughout the world. Despite significant progresses in understanding the disease pathogenesis and progression, the epidemiological triad of pathogen, host, and environment remains unclear. In this paper, we develop a multiscale model to study the coupled within-host and between-host dynamics of COVID-19. The model includes multiple transmission routes (both human-to-human and environment-to-human) and connects multiple scales (both the population and individual levels). A detailed analysis on the local and global dynamics of the fast system, slow system and full system shows that rich dynamics, including both forward and backward bifurcations, emerge with the coupling of viral infection and epidemiological models. Model fitting to both virological and epidemiological data facilitates the evaluation of the influence of a few infection characteristics and antiviral treatment on the spread of the disease. Our work underlines the potential role that the environment can play in the transmission of COVID-19. Antiviral treatment of infected individuals can delay but cannot prevent the emergence of disease outbreaks. These results highlight the implementation of comprehensive intervention measures such as social distancing and wearing masks that aim to stop airborne transmission, combined with surface disinfection and hand hygiene that can prevent environmental transmission. The model also provides a multiscale modeling framework to study other infectious diseases when the environment can serve as a reservoir of pathogens.


Subject(s)
COVID-19 , Antiviral Agents , COVID-19/epidemiology , COVID-19/prevention & control , Humans , Mathematical Concepts , Models, Biological , SARS-CoV-2
17.
Viruses ; 14(12)2022 12 15.
Article in English | MEDLINE | ID: covidwho-2216897

ABSTRACT

Influenza epidemics cause considerable morbidity and mortality every year worldwide. Climate-driven epidemiological models are mainstream tools to understand seasonal transmission dynamics and predict future trends of influenza activity, especially in temperate regions. Testing the structural identifiability of these models is a fundamental prerequisite for the model to be applied in practice, by assessing whether the unknown model parameters can be uniquely determined from epidemic data. In this study, we applied a scaling method to analyse the structural identifiability of four types of commonly used humidity-driven epidemiological models. Specifically, we investigated whether the key epidemiological parameters (i.e., infectious period, the average duration of immunity, the average latency period, and the maximum and minimum daily basic reproductive number) can be uniquely determined simultaneously when prevalence data is observable. We found that each model is identifiable when the prevalence of infection is observable. The structural identifiability of these models will lay the foundation for testing practical identifiability in the future using synthetic prevalence data when considering observation noise. In practice, epidemiological models should be examined with caution before using them to estimate model parameters from epidemic data.


Subject(s)
Epidemics , Influenza, Human , Humans , Humidity , Influenza, Human/epidemiology , Epidemiological Models , Climate , Models, Biological
18.
J Biol Dyn ; 16(1): 619-639, 2022 12.
Article in English | MEDLINE | ID: covidwho-2187649

ABSTRACT

In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number R0 and show that if R0≤1, then the disease-free equilibrium is globally asymptotically stable, whereas if R0>1, then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves.


Subject(s)
Epidemics , Quarantine , Basic Reproduction Number , Computer Simulation , Models, Biological
19.
Bull Math Biol ; 85(2): 13, 2023 01 13.
Article in English | MEDLINE | ID: covidwho-2174876

ABSTRACT

In response to the COVID-19 pandemic, many higher educational institutions moved their courses on-line in hopes of slowing disease spread. The advent of multiple highly-effective vaccines offers the promise of a return to "normal" in-person operations, but it is not clear if-or for how long-campuses should employ non-pharmaceutical interventions such as requiring masks or capping the size of in-person courses. In this study, we develop and fine-tune a model of COVID-19 spread to UC Merced's student and faculty population. We perform a global sensitivity analysis to consider how both pharmaceutical and non-pharmaceutical interventions impact disease spread. Our work reveals that vaccines alone may not be sufficient to eradicate disease dynamics and that significant contact with an infectious surrounding community will maintain infections on-campus. Our work provides a foundation for higher-education planning allowing campuses to balance the benefits of in-person instruction with the ability to quarantine/isolate infectious individuals.


Subject(s)
COVID-19 , Humans , COVID-19/epidemiology , COVID-19/prevention & control , Pandemics/prevention & control , SARS-CoV-2 , Mathematical Concepts , Models, Biological
20.
J Biol Dyn ; 16(1): 859-879, 2022 12.
Article in English | MEDLINE | ID: covidwho-2187651

ABSTRACT

Contact tracing is an important intervention measure to control infectious diseases. We present a new approach that borrows the edge dynamics idea from network models to track contacts included in a compartmental SIR model for an epidemic spreading in a randomly mixed population. Unlike network models, our approach does not require statistical information of the contact network, data that are usually not readily available. The model resulting from this new approach allows us to study the effect of contact tracing and isolation of diagnosed patients on the control reproduction number and number of infected individuals. We estimate the effects of tracing coverage and capacity on the effectiveness of contact tracing. Our approach can be extended to more realistic models that incorporate latent and asymptomatic compartments.


Subject(s)
Communicable Diseases , Epidemics , Humans , Contact Tracing/methods , Epidemiological Models , Models, Biological , Communicable Diseases/epidemiology
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