Research

19 pages, 1416 KiB

Open AccessArticle

Mathematical Model of Antiviral Immune Response against the COVID-19 Virus

by Juan Carlos Chimal-Eguia

Mathematics 2021, 9(12), 1356; https://doi.org/10.3390/math9121356 - 11 Jun 2021

Cited by 12 | Viewed by 3405

This work presents a mathematical model to investigate the current outbreak of the coronavirus disease 2019 (COVID-19) worldwide. The model presents the infection dynamics and emphasizes the role of the immune system: both the humoral response as well as the adaptive immune response. [...] Read more.

This work presents a mathematical model to investigate the current outbreak of the coronavirus disease 2019 (COVID-19) worldwide. The model presents the infection dynamics and emphasizes the role of the immune system: both the humoral response as well as the adaptive immune response. We built a mathematical model of delay differential equations describing a simplified view of the mechanism between the COVID-19 virus infection and the immune system. We conduct an analysis of the model exploring different scenarios, and our numerical results indicate that some theoretical immunotherapies are successful in eradicating the COVID-19 virus. Full article

(This article belongs to the Special Issue Mathematical Biology: Developments in Epidemic and Endemic Models)

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16 pages, 2270 KiB

Open AccessArticle

Mathematical Modelling of the Impact of Non-Pharmacological Strategies to Control the COVID-19 Epidemic in Portugal

by Constantino Caetano, Maria Luísa Morgado, Paula Patrício, João F. Pereira and Baltazar Nunes

Mathematics 2021, 9(10), 1084; https://doi.org/10.3390/math9101084 - 12 May 2021

Cited by 9 | Viewed by 2919

Abstract

In this paper, we present an age-structured SEIR model that uses contact patterns to reflect the physical distance measures implemented in Portugal to control the COVID-19 pandemic. By using these matrices and proper estimates for the parameters in the model, we were able [...] Read more.

In this paper, we present an age-structured SEIR model that uses contact patterns to reflect the physical distance measures implemented in Portugal to control the COVID-19 pandemic. By using these matrices and proper estimates for the parameters in the model, we were able to ascertain the impact of mitigation strategies employed in the past. Results show that the March 2020 lockdown had an impact on disease transmission, bringing the effective reproduction number (

R (t)

) below 1. We estimate that there was an increase in the transmission after the initial lift of the measures on 6 May 2020 that resulted in a second wave that was curbed by the October and November measures. December 2020 saw an increase in the transmission reaching an

R (t)

= 1.45 in early January 2021. Simulations indicate that the lockdown imposed on the 15 January 2021 might reduce the intensive care unit (ICU) demand to below 200 cases in early April if it lasts at least 2 months. As it stands, the model was capable of projecting the number of individuals in each infection phase for each age group and moment in time. Full article

(This article belongs to the Special Issue Mathematical Biology: Developments in Epidemic and Endemic Models)

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18 pages, 1163 KiB

Open AccessArticle

A Viral Disease That Damages the Immunity Conferred by Different Viral Diseases or Vaccination

by James P. Braselton and Martha L. Abell

Mathematics 2021, 9(8), 808; https://doi.org/10.3390/math9080808 - 08 Apr 2021

Viewed by 1161

Abstract

In this paper we modify a standard SIR model used to study the spread of some diseases by incorporating a disease that destroys the immunity that is conferred by having one of the other diseases or being vaccinated against the disease. A specific [...] Read more.

In this paper we modify a standard SIR model used to study the spread of some diseases by incorporating a disease that destroys the immunity that is conferred by having one of the other diseases or being vaccinated against the disease. A specific biological example of this occurs with measles. Studies of recent measles’ patients has shown that many patients have lost some (or all) of their immunity to other diseases from which they were previously protected. In the future, models like those developed here might be helpful in understanding how viruses that affect multiple organ systems can impact the effect the disease has on at risk populations. Full article

(This article belongs to the Special Issue Mathematical Biology: Developments in Epidemic and Endemic Models)

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10 pages, 371 KiB

Open AccessArticle

Mathematical and Statistical Analysis of Doubling Times to Investigate the Early Spread of Epidemics: Application to the COVID-19 Pandemic

by Alexandra Smirnova, Linda DeCamp and Gerardo Chowell

Mathematics 2021, 9(6), 625; https://doi.org/10.3390/math9060625 - 16 Mar 2021

Cited by 7 | Viewed by 2444

Abstract

Simple mathematical tools are needed to quantify the threat posed by emerging and re-emerging infectious disease outbreaks using minimal data capturing the outbreak trajectory. Here we use mathematical analysis, simulation and COVID-19 epidemic data to demonstrate a novel approach to numerically and mathematically [...] Read more.

Simple mathematical tools are needed to quantify the threat posed by emerging and re-emerging infectious disease outbreaks using minimal data capturing the outbreak trajectory. Here we use mathematical analysis, simulation and COVID-19 epidemic data to demonstrate a novel approach to numerically and mathematically characterize the rate at which the doubling time of an epidemic is changing over time. For this purpose, we analyze the dynamics of epidemic doubling times during the initial epidemic stage, defined as the sequence of times at which the cumulative incidence doubles. We introduce new methodology to characterize epidemic threats by analyzing the evolution of epidemics as a function of (1) the number of times the epidemic doubles until the epidemic peak is reached and (2) the rate at which the doubling times increase. In our doubling-time approach, the most dangerous epidemic threats double in size many times and the doubling times change at a relatively low rate (e.g., doubling times remain nearly invariant) whereas the least transmissible threats double in size only a few times and the doubling times rapidly increases in the period of emergence. We derive analytical formulas and test and illustrate our methodology using synthetic and COVID-19 epidemic data. Our mathematical analysis demonstrates that the series of epidemic doubling times increase approximately according to an exponential function with a rate that quantifies the rate of change of the doubling times. Our analytic results are in excellent agreement with numerical results. Our methodology offers a simple and intuitive approach that relies on minimal outbreak trajectory data to characterize the threat posed by emerging and re-emerging infectious diseases. Full article

(This article belongs to the Special Issue Mathematical Biology: Developments in Epidemic and Endemic Models)

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21 pages, 907 KiB

Open AccessArticle

A Mathematical Model of Contact Tracing during the 2014–2016 West African Ebola Outbreak

by Danielle Burton, Suzanne Lenhart, Christina J. Edholm, Benjamin Levy, Michael L. Washington, Bradford R. Greening, Jr., K. A. Jane White, Edward Lungu, Obias Chimbola, Moatlhodi Kgosimore, Faraimunashe Chirove, Marilyn Ronoh and M. Helen Machingauta

Mathematics 2021, 9(6), 608; https://doi.org/10.3390/math9060608 - 12 Mar 2021

Cited by 8 | Viewed by 3452

Abstract

The 2014–2016 West African outbreak of Ebola Virus Disease (EVD) was the largest and most deadly to date. Contact tracing, following up those who may have been infected through contact with an infected individual to prevent secondary spread, plays a vital role in [...] Read more.

The 2014–2016 West African outbreak of Ebola Virus Disease (EVD) was the largest and most deadly to date. Contact tracing, following up those who may have been infected through contact with an infected individual to prevent secondary spread, plays a vital role in controlling such outbreaks. Our aim in this work was to mechanistically represent the contact tracing process to illustrate potential areas of improvement in managing contact tracing efforts. We also explored the role contact tracing played in eventually ending the outbreak. We present a system of ordinary differential equations to model contact tracing in Sierra Leonne during the outbreak. Using data on cumulative cases and deaths, we estimate most of the parameters in our model. We include the novel features of counting the total number of people being traced and tying this directly to the number of tracers doing this work. Our work highlights the importance of incorporating changing behavior into one’s model as needed when indicated by the data and reported trends. Our results show that a larger contact tracing program would have reduced the death toll of the outbreak. Counting the total number of people being traced and including changes in behavior in our model led to better understanding of disease management. Full article

(This article belongs to the Special Issue Mathematical Biology: Developments in Epidemic and Endemic Models)

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32 pages, 4983 KiB

Open AccessArticle

On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility

by Manuel De la Sen, Santiago Alonso-Quesada and Asier Ibeas

Mathematics 2021, 9(5), 520; https://doi.org/10.3390/math9050520 - 02 Mar 2021

Cited by 11 | Viewed by 1932

Abstract

A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic [...] Read more.

A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic equilibrium points is also performed. The model has the following main characteristics: (a) the exposed subpopulation is infective, as it is the infectious one, but their respective transmission rates may be distinct; (b) a feedback vaccination control law on the Susceptible is incorporated; and (c) the model is subject to delayed partial re-susceptibility in the sense that a partial immunity loss in the recovered individuals happens after a certain delay. In this way, a portion of formerly recovered individuals along a range of previous samples is incorporated again to the susceptible subpopulation. The rate of loss of partial immunity of the considered range of previous samples may be, in general, distinct for the various samples. It is found that the endemic equilibrium point is not reachable in the transmission rate range of values, which makes the disease-free one to be globally asymptotically stable. The critical transmission rate which confers to only one of the equilibrium points the property of being asymptotically stable (respectively below or beyond its value) is linked to the unity basic reproduction number and makes both equilibrium points to be coincident. In parallel, the endemic equilibrium point is reachable and globally asymptotically stable in the range for which the disease-free equilibrium point is unstable. It is also discussed the relevance of both the vaccination effort and the re-susceptibility level in the modification of the disease-free equilibrium point compared to its reached component values in their absence. The influences of the limit control gain and equilibrium re-susceptibility level in the reached endemic state are also explicitly made viewable for their interpretation from the endemic equilibrium components. Some simulation examples are tested and discussed by using disease parameterizations of COVID-19. Full article

(This article belongs to the Special Issue Mathematical Biology: Developments in Epidemic and Endemic Models)

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16 pages, 458 KiB

Open AccessArticle

Estimating Equations for Density Dependent Markov Jump Processes

by Oluseyi Odubote and Daniel F. Linder

Mathematics 2021, 9(4), 391; https://doi.org/10.3390/math9040391 - 16 Feb 2021

Viewed by 1285

Abstract

Reaction networks are important tools for modeling a variety of biological phenomena across a wide range of scales, for example as models of gene regulation within a cell or infectious disease outbreaks in a population. Hence, calibrating these models to observed data is [...] Read more.

Reaction networks are important tools for modeling a variety of biological phenomena across a wide range of scales, for example as models of gene regulation within a cell or infectious disease outbreaks in a population. Hence, calibrating these models to observed data is useful for predicting future system behavior. However, the statistical estimation of the parameters of reaction networks is often challenging due to intractable likelihoods. Here we explore estimating equations to estimate the reaction rate parameters of density dependent Markov jump processes (DDMJP). The variance–covariance weights we propose to use in the estimating equations are obtained from an approximating process, derived from the Fokker–Planck approximation of the chemical master equation for stochastic reaction networks. We investigate the performance of the proposed methodology in a simulation study of the Lotka–Volterra predator–prey model and by fitting a susceptible, infectious, removed (SIR) model to real data from the historical plague outbreak in Eyam, England. Full article

(This article belongs to the Special Issue Mathematical Biology: Developments in Epidemic and Endemic Models)

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19 pages, 602 KiB

Open AccessArticle

COVID-19 Transmission: Bangladesh Perspective

by Masud M A, Md Hamidul Islam, Khondaker A. Mamun, Byul Nim Kim and Sangil Kim

Mathematics 2020, 8(10), 1793; https://doi.org/10.3390/math8101793 - 15 Oct 2020

Cited by 19 | Viewed by 3238

Abstract

The sudden emergence of the COVID-19 pandemic has tested the strength of the public health system of the most developed nations and created a “new normal”. Many nations are struggling to curb the epidemic in spite of expanding testing facilities. In this study, [...] Read more.

The sudden emergence of the COVID-19 pandemic has tested the strength of the public health system of the most developed nations and created a “new normal”. Many nations are struggling to curb the epidemic in spite of expanding testing facilities. In this study, we consider the case of Bangladesh, and fit a simple compartmental model holding a feature to distinguish between identified infected and infectious with time series data using least square fitting as well as the likelihood approach; prior to which, dynamics of the model were analyzed mathematically and the identifiability of the parameters has also been confirmed. The performance of the likelihood approach was found to be more promising and was used for further analysis. We performed fitting for different lengths of time intervals starting from the beginning of the outbreak, and examined the evolution of the key parameters from Bangladesh’s perspective. In addition, we deduced profile likelihood and

95 %

confidence interval for each of the estimated parameters. Our study demonstrates that the parameters defining the infectious and quarantine rates change with time as a consequence of the change in lock-down strategies and expansion of testing facilities. As a result, the value of the basic reproduction number

R_{0}

was shown to be between

1.5

and 12. The analysis reveals that the projected time and amplitude of the peak vary following the change in infectious and quarantine rates obtained through different lock-down strategies and expansion of testing facilities. The identification rate determines whether the observed peak shows the true prevalence. We find that by restricting the spread through quick identification and quarantine, or by implementing lock-down to reduce overall contact rate, the peak could be delayed, and the amplitude of the peak could be reduced. Another novelty of this study is that the model presented here can infer the unidentified COVID cases besides estimating the officially confirmed COVID cases. Full article

(This article belongs to the Special Issue Mathematical Biology: Developments in Epidemic and Endemic Models)

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