A Mathematical Model of COVID-19 with Vaccination and Treatment

We formulate and theoretically analyze a mathematical model of COVID-19 transmission mechanism incorporating vital dynamics of the disease and two key therapeutic measures—vaccination of susceptible individuals and recovery/treatment of infected individuals. Both the disease-free and endemic equilibrium are globally asymptotically stable when the effective reproduction number R 0 v is, respectively, less or greater than unity. The derived critical vaccination threshold is dependent on the vaccine efficacy for disease eradication whenever R 0 v > 1 , even if vaccine coverage is high. Pontryagin’s maximum principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions to optimally mitigate the spread of the disease. The model is fitted with cumulative daily Senegal data, with a basic reproduction number R 0 = 1.31 at the onset of the epidemic. Simulation results suggest that despite the effectiveness of COVID-19 vaccination and treatment to mitigate the spread of COVID-19, when R 0 v > 1 , additional efforts such as nonpharmaceutical public health interventions should continue to be implemented. Using partial rank correlation coefficients and Latin hypercube sampling, sensitivity analysis is carried out to determine the relative importance of model parameters to disease transmission. Results shown graphically could help to inform the process of prioritizing public health intervention measures to be implemented and which model parameter to focus on in order to mitigate the spread of the disease. The effective contact rate b , the vaccine efficacy ε , the vaccination rate v , the fraction of exposed individuals who develop symptoms, and, respectively, the exit rates from the exposed and the asymptomatic classes σ and ϕ are the most impactful parameters.

Download Full-text

Investigating the Role of Environmental Transmission in COVID-19 Dynamics: A Mathematical Model Based Study

10.21203/rs.3.rs-32476/v1 ◽

2020 ◽

Author(s):

Ibrahim M. ELmojtaba ◽

Fatma Al-Musalhi ◽

Asma Al-Ghassani ◽

Nasser Al-Salti

Keyword(s):

Mathematical Model ◽

Disease Transmission ◽

Reproduction Number ◽

Equilibrium Points ◽

Transmission Dynamics ◽

Model Parameters ◽

Environmental Transmission ◽

Estimated Parameters ◽

Existence And Stability

Abstract A mathematical model with environmental transmission has been proposed and analyzed to investigate its role in the transmission dynamics of the ongoing COVID-19 outbreak. Two expressions for the basic reproduction number R0 have been analytically derived using the next generation matrix method. The two expressions composed of a combination of two terms related to human to human and environment to human transmissions. The value of R0 has been calculated using estimated parameters corresponding to two datasets. Sensitivity analysis of the reproduction number to the corresponding model parameters has been carried out. Existence and stability analysis of disease free and endemic equilibrium points have been presented in relation with the obtained expressions of R0. Numerical simulations to demonstrate the effect of some model parameters related to environmental transmission on the disease transmission dynamics have been carried out and the results have been demonstrated graphically.

Download Full-text

Mathematical Modeling to Estimate the Reproductive Number and the Outbreak Size of COVID-19: The case of India and the World

10.21203/rs.3.rs-26261/v1 ◽

2020 ◽

Author(s):

Durgesh Nandini Sinha

Keyword(s):

Mathematical Model ◽

Reproduction Number ◽

Model Fitting ◽

Rank Correlation ◽

Latin Hypercube Sampling ◽

Reproductive Number ◽

Global Pandemic ◽

The World ◽

Partial Rank Correlation ◽

Coefficient Method

Abstract Coronavirus disease (COVID-19) has become a global pandemic with more than 218,000 deaths in 211 different countries around the world. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the virus responsible for this deadliest disease. This paper describes a mathematical model for India, a country with the second highest population in the world with an extremely high population density of about 464 people per km2. This disease has multiphasic actions and reaction mode and our model SEIAQIm is based on six compartmental groups in the form of susceptible, exposed, infectious, asymptomatic, quarantine, and recovered immune factions. Latin Hypercube Sampling Partial Rank Correlation Coefficient method was used for the data analysis and model fitting. According to our model, India would reach its basic reproduction number R0=0.97 on May 14, 2020 with a total number of 73,800 estimated cases. Further, this study also equates the world's situation using the same model system and predicts by May 7, 2020 with a total number of 3,772,000 estimated confirmed cases. Moreover, the current mathematical model highlights the importance of social distancing as an effective method of containing spread of COVID-19.

Download Full-text

The minimal COVID-19 vaccination coverage and efficacy to compensate for potential increase of transmission contacts, and increased transmission probability of the emerging strains

10.21203/rs.3.rs-140717/v1 ◽

2021 ◽

Author(s):

Biao Tang ◽

Xue Zhang ◽

Qian Li ◽

Nicola Bragazzi ◽

Dasantila Golemi-Kotra ◽

...

Keyword(s):

Public Health ◽

Vaccine Efficacy ◽

Attack Rate ◽

Vaccine Coverage ◽

Public Health Intervention ◽

Contact Tracing ◽

Efficacy Rate ◽

Case Scenario ◽

Worst Case ◽

The Impact

Abstract Background: Mass immunization is a potentially effective approach to finally control the local outbreak and global spread of COVID-19 pandemic. However, it can also lead to undesirable outcomes if mass vaccination results in increased transmission effective contacts and relaxation of other public health interventions due to the perceived immunity from the vaccine. Methods: We designed a mathematical model of COVID-19 transmission dynamics that takes into consideration the epidemiological status, public health intervention status (quarantined/isolated), immunity status of the population, and the strain variations. Comparing the control reproduction numbers and the final epidemic sizes (attack rate) in the cases with and without vaccination, we quantified some key factors determining when vaccination in the population is beneficial for preventing and controlling future outbreaks. Results: Our analyses predicted that there is a critical (minimal) vaccine efficacy rate (or a critical quarantine rate) below which the control reproduction number with vaccination is higher than that without vaccination, and the final attack rate in the population is also higher with the vaccination. We also predicted the worst case scenario occurs when a high vaccine coverage is achieved for a vaccine with lower efficacy rate and when the vaccines increase the transmission efficient contacts.Conclusions: The analyses show that an immunization program with a vaccine efficacy rate below the predicted critical values will not be as effective as simply investing in the contact tracing/quarantine/isolation implementation. We reached similar conclusions by considering the final epidemic size (or attack rates). This research then highlights the importance of monitoring the impact on transmissibility and vaccine efficacy of emerging strains.

Download Full-text

The impact of prophylaxis of healthcare workers on influenza pandemic burden

Journal of The Royal Society Interface ◽

10.1098/rsif.2006.0204 ◽

2007 ◽

Vol 4 (15) ◽

pp. 727-734 ◽

Cited By ~ 9

Author(s):

Michael Gardam ◽

Dong Liang ◽

Seyed M Moghadas ◽

Jianhong Wu ◽

Qingling Zeng ◽

...

Keyword(s):

Public Health ◽

Mathematical Model ◽

General Population ◽

Disease Transmission ◽

Healthcare Workers ◽

Reproduction Number ◽

Control Measure ◽

Influenza Pandemic ◽

High Level ◽

The Impact

Several models have rationalized the use of antiviral drugs as an early control measure for delaying the progression and limiting the size of outbreaks during an influenza pandemic. However, the strategy for use of these drugs is still under debate. We evaluated the impact of prophylaxis of healthcare workers (HCWs) through a mathematical model that considers attack rates in a range of 25–35% in the general population and 25–50% among HCWs. Simulations and uncertainty analysis using the demographics of the province of Ontario, Canada show that increasing prophylaxis coverage of HCWs has little impact on reducing the reproduction number of disease transmission and may not prevent the occurrence of an outbreak if expected. However, it does enable a high level of treatment, which substantially reduces morbidity and mortality in the population as a whole. Therefore, prophylaxis of HCWs should be considered an important part of public health efforts for minimizing influenza pandemic burden and its socio-economic disruption.

Download Full-text

Evolution of disease transmission during the COVID-19 pandemic: patterns and determinants

Scientific Reports ◽

10.1038/s41598-021-90347-8 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Jie Zhu ◽

Blanca Gallego

Keyword(s):

Public Health ◽

Disease Transmission ◽

Clustering Algorithm ◽

Health Interventions ◽

Transmission Model ◽

Model Parameters ◽

Public Health Interventions ◽

Forecasting Accuracy ◽

Time Lagged ◽

The Impact

AbstractEpidemic models are being used by governments to inform public health strategies to reduce the spread of SARS-CoV-2. They simulate potential scenarios by manipulating model parameters that control processes of disease transmission and recovery. However, the validity of these parameters is challenged by the uncertainty of the impact of public health interventions on disease transmission, and the forecasting accuracy of these models is rarely investigated during an outbreak. We fitted a stochastic transmission model on reported cases, recoveries and deaths associated with SARS-CoV-2 infection across 101 countries. The dynamics of disease transmission was represented in terms of the daily effective reproduction number ($$R_t$$ R t ). The relationship between public health interventions and $$R_t$$ R t was explored, firstly using a hierarchical clustering algorithm on initial $$R_t$$ R t patterns, and secondly computing the time-lagged cross correlation among the daily number of policies implemented, $$R_t$$ R t , and daily incidence counts in subsequent months. The impact of updating $$R_t$$ R t every time a prediction is made on the forecasting accuracy of the model was investigated. We identified 5 groups of countries with distinct transmission patterns during the first 6 months of the pandemic. Early adoption of social distancing measures and a shorter gap between interventions were associated with a reduction on the duration of outbreaks. The lagged correlation analysis revealed that increased policy volume was associated with lower future $$R_t$$ R t (75 days lag), while a lower $$R_t$$ R t was associated with lower future policy volume (102 days lag). Lastly, the outbreak prediction accuracy of the model using dynamically updated $$R_t$$ R t produced an average AUROC of 0.72 (0.708, 0.723) compared to 0.56 (0.555, 0.568) when $$R_t$$ R t was kept constant. Monitoring the evolution of $$R_t$$ R t during an epidemic is an important complementary piece of information to reported daily counts, recoveries and deaths, since it provides an early signal of the efficacy of containment measures. Using updated $$R_t$$ R t values produces significantly better predictions of future outbreaks. Our results found variation in the effect of early public health interventions on the evolution of $$R_t$$ R t over time and across countries, which could not be explained solely by the timing and number of the adopted interventions.

Download Full-text

Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination

Scientific Reports ◽

10.1038/s41598-021-95913-8 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Md Abdul Kuddus ◽

M. Mohiuddin ◽

Azizur Rahman

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Equilibrium Points ◽

Rank Correlation ◽

Dynamical Analysis ◽

Model Parameters ◽

Progression Rate ◽

Double Dose ◽

The Basic Reproduction Number

AbstractAlthough the availability of the measles vaccine, it is still epidemic in many countries globally, including Bangladesh. Eradication of measles needs to keep the basic reproduction number less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}<1)$$ ( i . e . R 0 < 1 ) . This paper investigates a modified (SVEIR) measles compartmental model with double dose vaccination in Bangladesh to simulate the measles prevalence. We perform a dynamical analysis of the resulting system and find that the model contains two equilibrium points: a disease-free equilibrium and an endemic equilibrium. The disease will be died out if the basic reproduction number is less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{ R}}_{0}<1)$$ ( i . e . R 0 < 1 ) , and if greater than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}>1)$$ ( i . e . R 0 > 1 ) epidemic occurs. While using the Routh-Hurwitz criteria, the equilibria are found to be locally asymptotically stable under the former condition on $${\mathrm{R}}_{0}$$ R 0 . The partial rank correlation coefficients (PRCCs), a global sensitivity analysis method is used to compute $${\mathrm{R}}_{0}$$ R 0 and measles prevalence $$\left({\mathrm{I}}^{*}\right)$$ I ∗ with respect to the estimated and fitted model parameters. We found that the transmission rate $$(\upbeta )$$ ( β ) had the most significant influence on measles prevalence. Numerical simulations were carried out to commissions our analytical outcomes. These findings show that how progression rate, transmission rate and double dose vaccination rate affect the dynamics of measles prevalence. The information that we generate from this study may help government and public health professionals in making strategies to deal with the omissions of a measles outbreak and thus control and prevent an epidemic in Bangladesh.

Download Full-text

Estimasi Reproduction Number Model Matematika Penyebaran Malaria di Sumba Tengah, Indonesia

Jambura Journal of Biomathematics (JJBM) ◽

10.34312/jjbm.v2i1.9971 ◽

2021 ◽

Vol 2 (1) ◽

pp. 13-19

Author(s):

Ervin Mawo Banni ◽

Maria A Kleden ◽

Maria Lobo ◽

Meksianis Zadrak Ndii

Keyword(s):

Mathematical Modeling ◽

Mathematical Model ◽

Health Center ◽

Disease Transmission ◽

Reproduction Number ◽

Transmission Dynamics ◽

Awareness Program ◽

Awareness Programs ◽

Malaria Eradication

Malaria is transmitted via a bite of mosquitoes and it is dangerous if it is not properly treated. Mathematical modeling can be formulated to understand the disease transmission dynamics. In this paper, a mathematical model with an awareness program has been formulated and the reproduction number has been estimated against the data from Weeluri Health Center, Mamboro District, Central Sumba. The calculation showed that the reproduction number is R0 = 1.2562. Results showed that if the efficacy of the awareness program is lower than 20%, the reproduction number is still above unity. If the efficacy of the awareness program is higher than 20%, the reproduction number is lower than unity. This implies that the efficacy of awareness programs is the key to the success of Malaria eradication.

Download Full-text

Melatonin in Epilepsy: A New Mathematical Model of Diurnal Secretion

International Journal of Endocrinology ◽

10.1155/2016/3861461 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8 ◽

Cited By ~ 7

Author(s):

Justyna Paprocka ◽

Marek Kijonka ◽

Marcin Pęcka ◽

Maria Sokół

Keyword(s):

Mathematical Model ◽

Least Squares Method ◽

Rank Correlation ◽

Nonlinear Least Squares ◽

Wilcoxon Test ◽

Melatonin Level ◽

Model Parameters ◽

Melatonin Secretion ◽

Spearman’S Rank Correlation ◽

Spearman’S Rank Correlation Coefficient

Purpose. The main objective of the study was to create a mathematical model that describes the melatonin circadian secretion and, then the functionality of the model was tested by a comparison of the melatonin secretions in children with and without epilepsy.Material and Methods. The patients were divided into the epilepsy group (EG,n=52) and the comparison group (CG,n=30). The melatonin level was assessed by a radioimmunoassay method. The diurnal melatonin secretion was described using a nonlinear least squares method. Spearman’s rank correlation coefficient was chosen to estimate the dependence of the acquired data. The model reproduces blood concentration profiles and its parameters were statistically analyzed using the Mann-Whitney-Wilcoxon test and logistic regression.Results. The correlation analysis performed for the EG and CG groups showed moderate correlations between age and the melatonin secretion model parameters. Patients with epilepsy are characterized by an increased phase shift of melatonin release.

Download Full-text

A prototype vaccination model for endemic Covid-19 under waning immunity and imperfect vaccine take-up

10.1101/2021.11.06.21266002 ◽

2021 ◽

Author(s):

John S Dagpunar ◽

ChenChen Wu

Keyword(s):

Basic Reproduction Number ◽

Vaccine Efficacy ◽

Reproduction Number ◽

Vaccine Coverage ◽

Infection Prevalence ◽

State Variables ◽

Susceptible Population ◽

Waning Immunity ◽

The Impact ◽

The Basic Reproduction Number

In this paper, for an infectious disease such as Covid-19, we present a SIR model which examines the impact of waning immunity, vaccination rates, vaccine efficacy, and the proportion of the susceptible population who aspire to be vaccinated. Under an assumed constant control reproduction number, we provide simple conditions for the disease to be eliminated, and conversely for it to exhibit the more likely endemic behaviour. With regard to Covid-19, it is shown that if the control reproduction number is set to the basic reproduction number (say 6) of the dominant delta (B1.617.2) variant, vaccination alone, even under the most optimistic of assumptions about vaccine efficacy and high vaccine coverage, is very unlikely to lead to elimination of the disease. The model is not intended to be predictive but more an aid to understanding the relative importance of various biological and control parameters. For example, from a long-term perspective, it may be found that in the UK, through changes in societal behaviour (such as mask use, ventilation, and level of homeworking), without formal government interventions such as on-off lockdowns, the control reproduction number can still be maintained at a level significantly below the basic reproduction number. Even so, our simulations show that endemic behaviour ensues. The model obtains equilibrium values of the state variables such as the infection prevalence and mortality rate under various scenarios.

Download Full-text

The effects of non-pharmaceutical interventions on SARS-CoV-2 transmission in different socioeconomic populations in Kuwait: A modelling study

University of Toronto Journal of Public Health ◽

10.33137/utjph.v2i2.36995 ◽

2021 ◽

Vol 2 (2) ◽

Author(s):

Fatima Khadadah ◽

Abdullah A. Al-Shammari ◽

Ahmad Alhashemi ◽

Dari Alhuwail ◽

Bader Al-Saif ◽

...

Keyword(s):

Disease Transmission ◽

Reproduction Number ◽

Public Health Intervention ◽

Population Level ◽

Socioeconomic Groups ◽

Before And After ◽

Cross Transmission ◽

Community Transmission ◽

The Impact ◽

Pharmaceutical Interventions

Background: Aggressive non-pharmaceutical interventions (NPIs) may reduce transmission of SARS-CoV2. The extent to which these interventions are successful in stopping the spread have not been characterized in countries with distinct socioeconomic groups. We compared the effects of a partial lockdown on disease transmission among Kuwaitis (P1) and non-Kuwaitis (P2) living in Kuwait. Methods: We fit a metapopulation Susceptible-Exposed-Infectious-Recovered (SEIR) model to reported cases stratified by two groups to estimate the impact of a lockdown on the effective reproduction number (Re). We estimated the basic reproduction number (R0) for the transmission in each group and simulated the potential trajectories of an outbreak from the first recorded case of community transmission until 12 days after the lockdown. We estimated Re values of both groups before and after the lockdown, simulated the effect of these values on epidemic curves and explored a range of cross-transmission scenarios. Results: We estimate R0 at 1·06 (95% CI: 1·05-1·28) for P1 and 1·83 (1·58-2·33) for P2. On March 22nd, Re for P1 and P2 are estimated at 1·13 (1·07-1·17) and 1·38 (1·25-1·63) respectively. After the curfew had taken effect, Re for P1 dropped modestly to 1·04 (1·02-1·06) but almost doubled for P2 to 2·47 (1·98-3·45). Our simulated epidemic trajectories show that the partial curfew measure modestly reduced and delayed the height of the peak in P1, yet significantly elevated and hastened the peak in P2. Modest cross-transmission from P2 to P1 elevated the height of the peak in P1 and brought it forward in time closer to the peak of P2. Conclusion: Our results demonstrate that a lockdown can reduce SARS-CoV2 transmission in one subpopulation but accelerate it in another. At the population level, the consequences of lockdowns may vary across the socioeconomic spectrum. Any public health intervention needs to be sensitive to disparities within populations.

Download Full-text