Qualitative analysis of a mathematical model with presymptomatic individuals and two SARS-CoV-2 variants

Modeling the dynamics of COVID-19 in the presence of Delta and Omicron variants with vaccination and non-pharmaceutical interventions

Since its inception in December 2019, many safe and effective vaccines have been invented and approved for use against COVID-19 along with various non-pharmaceutical interventions. But the emergence of numerous SARS-CoV-2 variants has put the effectiveness of these vaccines, and other intervention measures under threat. So it is important to understand the dynamics of COVID-19 in the presence of its variants of concern (VOC) in controlling the spread of the disease. To address these situations and to find a way out of this problem, a new mathematical model consisting of a system of non-linear differential equations considering the original COVID-19 strain with its two variants of concern (Delta and Omicron) has been proposed and formulated in this paper. We then analyzed the proposed model to study the transmission dynamics of this multi-strain model and to investigate the consequences of the emergence of multiple new SARS-CoV-2 variants which are more transmissible than the previous ones. The control reproduction number, an important threshold parameter, is then calculated using the next-generation matrix method. Further, we presented the discussion about the stability of the model equilibrium. It is shown that the disease-free equilibrium (DFE) of the model is locally asymptotic stable when the control reproduction is less than unity. It is also shown that the model has a unique endemic equilibrium (EEP) which is locally asymptotic stable when the control reproduction number is greater than unity. Using the Center Manifold theory it is shown that the model also exhibits the backward bifurcation phenomenon when the control reproduction number is less than unity. Again without considering the re-infection of the recovered individuals, it is proved that the disease-free equilibrium is globally asymptotically stable when the reproduction threshold is less than unity. Finally, numerical simulations are performed to verify the analytic results and to show the impact of multiple new SARS-CoV-2 variants in the population which are more contagious than the previous variants. Global uncertainty and sensitivity analysis has been done to identify which parameters have a greater impact on disease dynamics and control disease transmission. Numerical simulation suggests that the emergence of new variants of concern increases COVID-19 infection and related deaths. It also reveals that a combination of non-pharmaceutical interventions with vaccination programs of new more effective vaccines should be continued to control the disease outbreak. This study also suggests that more doses of vaccine should provide to combat new and deadly variants like Delta and Omicron.

COVID-19 dynamics and immune response: Linking within-host and between-host dynamics

The global impact of COVID-19 has led to the development of numerous mathematical models to understand and control the pandemic. However, these models have not fully captured how the disease's dynamics are influenced by both within-host and between-host factors. To address this, a new mathematical model is proposed that links these dynamics and incorporates immune response. The model is compartmentalized with a fractional derivative in the sense of Caputo-Fabrizio, and its properties are studied to show a unique solution. Parameter estimation is carried out by fitting real-life data, and sensitivity analysis is conducted using various methods. The model is then numerically implemented to demonstrate how the dynamics within infected hosts drive human-to-human transmission, and various intervention strategies are compared based on the percentage of averted deaths. The simulations suggest that a combination of medication to boost the immune system, prevent infected cells from producing the virus, and adherence to COVID-19 protocols is necessary to control the spread of the virus since no single intervention strategy is sufficient.

A survey on Lyapunov functions for epidemic compartmental models

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey-predator or rumor spreading.

Study of optimal vaccination strategies for early COVID-19 pandemic using an age-structured mathematical model: A case study of the USA

In this paper we study different vaccination strategies that could have been implemented for the early COVID-19 pandemic. We use a demographic epidemiological mathematical model based on differential equations in order to investigate the efficacy of a variety of vaccination strategies under limited vaccine supply. We use the number of deaths as the metric to measure the efficacy of each of these strategies. Finding the optimal strategy for the vaccination programs is a complex problem due to the large number of variables that affect the outcomes. The constructed mathematical model takes into account demographic risk factors such as age, comorbidity status and social contacts of the population. We perform simulations to assess the performance of more than three million vaccination strategies which vary depending on the vaccine priority of each group. This study focuses on the scenario corresponding to the early vaccination period in the USA, but can be extended to other countries. The results of this study show the importance of designing an optimal vaccination strategy in order to save human lives. The problem is extremely complex due to the large amount of factors, high dimensionality and nonlinearities. We found that for low/moderate transmission rates the optimal strategy prioritizes high transmission groups, but for high transmission rates, the optimal strategy focuses on groups with high CFRs. The results provide valuable information for the design of optimal vaccination programs. Moreover, the results help to design scientific vaccination guidelines for future pandemics.

Implementation of computationally efficient numerical approach to analyze a Covid-19 pandemic model

Corona virus disease (Covid-19) which has caused frustration in the human community remains the concern of the globe as every government struggles to defeat the pandemic. To deal with the situation, we have extensively studied a deadly Covid-19 model to provide a deep insight into the disease dynamics. A mathematical analysis of the model utilizing preventive measures is performed with the aim to reduce the disease burden. Some comprehensive mathematical techniques are employed to demonstrate several essential properties of solutions. To start with, we proved the existence and uniqueness of solutions. Equilibrium points are stated both in the absence and presence of the pandemic. Biologically important quantity known as threshold parameter is computed to handle the future disease dynamics and analyzed for its sensitivity. We proved the stability of the proposed model at equilibrium points by employing necessary conditions on threshold parameter. A reliable and competitive numerical analysis is conducted to observe the effectiveness of implemented strategies and to verify obtained analytical results. The most sensitive parameters are determined through sensitivity analysis. An important feature of this study is to employ Non-Standard Finite Difference (NSFD) numerical scheme to solve the system instead of other standard methods like Runge–Kutta method of order 4 (RK4). Finally, several numerical simulations are performed to validate our former theoretical analysis. Numerical results exhibiting dynamical behavior of Covid-19 system under the influence of involved parameters suggest that both the implemented strategies, especially quarantine of exposed individuals, are effective for the substantial reduction in the diseased population and to achieve the herd immunity.

Global stability and analysing the sensitivity of parameters of a multiple-susceptible population model of SARS-CoV-2 emphasising vaccination drive

The study explores the dynamics of a COVID-19 epidemic in multiple susceptible populations, including the various stages of vaccination administration. In the model, there are eight human compartments: completely susceptible; susceptible with dose-1 vaccination; susceptible with dose-2 vaccination; susceptible with booster dose vaccination; exposed; infected with and without symptoms, and recovered compartments. The biological feasibility of the model is analysed. The threshold value, R 0 , is derived using the next-generation matrix. The stability analysis of the equilibrium points was performed locally and globally using the threshold parameter of the model. The conditions determining disease persistence is obtained. The model is subjected to sensitivity analysis, and the most sensitive parameters are identified. Also, MATLAB is used to verify the mathematical outcomes of the system's dynamic behaviour and suggests that necessary steps should be taken to keep the spread of the omicron variant infectious disease under control. The findings of this study could aid health officials in their efforts to combat the spread of COVID-19.

Parameter identifiability and optimal control of an SARS-CoV-2 model early in the pandemic

We fit an SARS-CoV-2 model to US data of COVID-19 cases and deaths. We conclude that the model is not structurally identifiable. We make the model identifiable by prefixing some of the parameters from external information. Practical identifiability of the model through Monte Carlo simulations reveals that two of the parameters may not be practically identifiable. With thus identified parameters, we set up an optimal control problem with social distancing and isolation as control variables. We investigate two scenarios: the controls are applied for the entire duration and the controls are applied only for the period of time. Our results show that if the controls are applied early in the epidemic, the reduction in the infected classes is at least an order of magnitude higher compared to when controls are applied with 2-week delay. Further, removing the controls before the pandemic ends leads to rebound of the infected classes.

Mathematical Analysis of Two Waves of COVID-19 Disease with Impact of Vaccination as Optimal Control

This paper is devoted to answering some questions using a mathematical model by analyzing India’s first and second phases of the COVID-19 pandemic. A new mathematical model is introduced with a nonmonotonic incidence rate to incorporate the psychological effect of COVID-19 in society. The paper also discusses the local stability and global stability of an endemic equilibrium and a disease-free equilibrium. The basic reproduction number is evaluated using the proposed COVID-19 model for disease spread in India based on the actual data sets. The study of nonperiodic solutions at a positive equilibrium point is also analyzed. The model is rigorously studied using MATLAB to alert the decision-making bodies to hinder the emergence of any other pandemic outbreaks or the arrival of subsequent pandemic waves. This paper shows the excellent prediction of the first wave and very commanding for the second wave. The exciting results of the paper are as follows: (i) psychological effect on the human population has an impact on propagation; (ii) lockdown is a suitable technique mathematically to control the COVID spread; (iii) different variants produce different waves; (iv) the peak value always crosses its past value.

Impact of Infective Immigrants on COVID-19 Dynamics

The COVID-19 epidemic is an unprecedented and major social and economic challenge worldwide due to the various restrictions. Inflow of infective immigrants have not been given prominence in several mathematical and epidemiological models. To investigate the impact of imported infection on the number of deaths, cumulative infected and cumulative asymptomatic, we formulate a mathematical model with infective immigrants and considering vaccination. The basic reproduction number of the special case of the model without immigration of infective people is derived. We varied two key factors that affect the transmission of COVID-19, namely the immigration and vaccination rates. In addition, we considered two different SARS-CoV-2 transmissibilities in order to account for new more contagious variants such as Omicron. Numerical simulations using initial conditions approximating the situation in the US when the vaccination program was starting show that increasing the vaccination rate significantly improves the outcomes regarding the number of deaths, cumulative infected and cumulative asymptomatic. Other factors are the natural recovery rates of infected and asymptomatic individuals, the waning rate of the vaccine and the vaccination rate. When the immigration rate is increased significantly, the number of deaths, cumulative infected and cumulative asymptomatic increase. Consequently, accounting for the level of inflow of infective immigrants may help health policy/decision-makers to formulate policies for public health prevention programs, especially with respect to the implementation of the stringent preventive lock down measure.