How Large Fraction of A Population Must be Vaccinated before A Disease is Controlled?

The ongoing Covid-19 pandemic has already caused more than 5 million casualties despite hard restrictions and relatively high vaccine coverage in many countries. The crucial question is therefore, how large vaccination rate and how severe restrictions are required to terminate the spread of the decease, assuming that the vaccine efficiency and the basic reproduction ratio (R0) are known? To answer this question, a mathematical equation was applied to visualize the required vaccination level as function of vaccine efficiency, restriction efficiency and basic reproduction ratio (R0). In addition to the modelling study, Covid-19 data from Europe was collected during 19/11-26/11 (2021) to assess the relation between vaccination rate and incidence. The analysis indicates that a vaccination rate of ~92% (2 doses) is required to stop Delta (B.1.617.2) without severe restrictions, under conditions like those in Europe late November 2021. A third vaccine dose, improved vaccines, higher vaccination rates and/or stronger restrictions will be required to force Omicron (B.1.1.529) to expire without infecting a large fraction of the population.

How Large Fraction of A Population Must be Vaccinated before A Disease is Controlled?

The ongoing Covid-19 pandemic has already caused more than 5 million casualties despite hard restrictions and relatively high vaccine coverage in many countries. The crucial question is therefore, how large vaccination rate and how severe restrictions are required to terminate the spread of the decease, assuming that the vaccine efficiency and the basic reproduction ratio (R0) are known? To answer this question, a simple mathematical equation was developed to visualize the required vaccination level as function of vaccine efficiency, restriction efficiency and basic reproduction ratio (R0). In addition to the modelling study, Covid-19 data from Europe was collected during 19/11-26/11 (2021) to assess the relation between vaccination rate and incidence. The analysis indicates that a vaccination rate of ~92% (2 doses) is currently required to stop Delta (B.1.617.2) without severe restrictions, using the vaccines that are most common in Europe today. A third vaccine dose, improved vaccines, higher vaccination rates and/or stronger restrictions will be required to force Omicron (B.1.1.529) to expire without infecting a large fraction of the population.

Minimal wave speed in an HIV-1 virus integrodifference system

This paper is concerned with the minimal wave speed of nonconstant traveling wave solutions in an HIV-1 virus integrodifference system. Here, the traveling wave solution models the spatial spreading process of infected cells and virus. When the basic reproduction ratio of the corresponding ordinary differential system or difference system is larger than one, we establish the existence of nonconstant traveling wave solutions if the wave speed is not less than a threshold, and if the speed is smaller than the threshold, we prove the nonexistence of nonconstant traveling wave solutions. Moreover, when the basic reproduction ratio of the corresponding ordinary differential system or difference system is not larger than one, we also confirm the nonexistence of nonconstant traveling wave solutions.

Mathematical analysis of a within-host model of SARS-CoV-2

In this paper, we have mathematically analyzed a within-host model of SARS-CoV-2 which is used by Li et al. in the paper “The within-host viral kinetics of SARS-CoV-2” published in (Math. Biosci. Eng. 17(4):2853–2861, 2020). Important properties of the model, like nonnegativity of solutions and their boundedness, are established. Also, we have calculated the basic reproduction number which is an important parameter in the infection models. From stability analysis of the model, it is found that stability of the biologically feasible steady states are determined by the basic reproduction number $(\chi _{0})$ ( χ 0 ) . Numerical simulations are done in order to substantiate analytical results. A biological implication from this study is that a COVID-19 patient with less than one basic reproduction ratio can automatically recover from the infection.

The correlation between antiviral drug, immune response and HIV viral load

Developing antiviral drugs is an exigent task since viruses mutate to overcome the effect of antiviral drugs. As a result, the efficacy of most antiviral drugs is short-lived. To include this effect, we modify the Neumann and Dahari model. Considering the fact that the efficacy of the antiviral drug varies in time, the differential equations introduced in the previous model systems are rewritten to study the correlation between the viral load and antiviral drug. The effect of antiviral drug that either prevents infection or stops the production of a virus is explored. First, the efficacy of the drug is considered to decreases monotonously as time progresses. In this case, our result depicts that when the efficacy of the drug is low, the viral load decreases and increases back in time revealing the effect of the antiviral drugs is short-lived. On the other hand, for the antiviral drug with high efficacy, the viral load as well as the number of infected cells monotonously decreases while the number of uninfected cells increases. The dependence of the critical drug efficacy on time is also explored. Moreover, the correlation between viral load, the antiviral drug, and CTL response is also explored. In this case, not only the dependence for the basic reproduction ratio on the model parameters is explored but also we analyze the critical drug efficacy as a function of time. We show that the term related to the basic reproduction ratio increases when the CTL response step up. A simple analytically solvable mathematical model is also presented to analyze the correlation between viral load and antiviral drugs.PACS numbersValid PACS appear here

Mathematical Modelling of Transmission Dynamics of COVID-19: A Case Study of Nepal

In this study, the SIR compartmental mathematical model has been proposed to predict the transmission dynamics of COVID-19 in Nepal. The model is analysed by deriving some important expressions such as the basic reproduction ratio and possible maximum number of infectives in the future. This study examines the applicability of the SIR model for the study of the COVID-19 pandemic and other similar infectious diseases. The prime objective of the study is to analyse and forecast the COVID-19 pandemic in Nepal for the upcoming time. The estimation of the parameters of the model is based upon data from January 20, 2020 to July 14, 2020. The model presented in the paper fitted to the time-series data well for the whole Nepal and its neighbouring countries such as India and China. The findings suggest that there is a potential for this model to contribute to better public health policy in combating COVID-19.