Reproduction Ratio Research Articles

A Mathematical Model for the Control of Cholera Epidemic without Natural Recovery

In this research work, we present a mathematical model for the control of cholera outbreak without natural recovery. This follows a slight modification as compared to previous cholera models for the Nigerian case. Our model incorporates treatment, water hygiene as well as environmental sanitation. The model employs a system of nonlinear ordinary differential equations, which is analyzed in detail for its stability properties. We compute the basic reproduction ratio R0 for the various control parameters and discover that with proper combination of control measures, the spread of cholera could be minimized. Numerical simulation of the cholera model is done using MathCAD14, and the graphical profiles of the main variables are depicted. We conclude that improvement in treatment, water hygiene and the environmental sanitation is indeed effective in eradicating the cholera epidemic.

On threshold dynamics for periodic and time-delayed impulsive systems and application to a periodic disease model.

The basic reproduction ratio R0 of more general periodic and time-delayed impulsive model which the period of model coefficients is different from that of fixed impulsive moments, is developed. That R0 is the threshold parameter for the stability of the zero solution of the associated linear system is also shown. The developed theory is further applied to a swine parasitic disease model with pulse therapy. Threshold results on its global dynamics in terms of R0 are obtained. Some numerical simulation results are also given to support our main results.

Insights of How Lung Microbiome can Contribute to COVID-19 Severity in Intensive Care Unit Patients

Objective: We present a country-specific, modified Susceptible, Exposed, Infectious, and Removed (SEIR) model of SARS-CoV-2 transmission aiming to provide accurate prediction of COVID-19 cases to optimize clinical trial recruitment, inform mitigation strategies, and facilitate rapid medication development. Methods: Epidemiological data from more than 170 countries were obtained from the Johns Hopkins University COVID-19 Dashboard. Intercountry differences in initial exposure, cultural/environmental factors, and stringency of mitigation strategies were incorporated. Asymptomatic patients and “super-spreaders” were also factored into our model. Simulations were limited to a period of 2 months considering that the effects of certain parameters (e.g. seasonality of virus transmission, wearing of face masks, and deployment of vaccines) were sufficiently ambiguous to limit confidence in a longer simulation period. Results: Using these data, our model estimated 71.5% of cases as asymptomatic. Without mitigation, a mean maximum infection rate of 1.08 cases/day (Intercountry range, 0.68–1.65) was estimated in symptomatic cases. From here, symptomatic and asymptomatic people were estimated to infect 3.39 and 7.71 other people, respectively, suggesting that asymptomatic persons could be responsible for 85% of new infections. An estimated 10.6% of cases were super-spreaders with a 2.86-fold higher transmission rate than average. Mitigation strategies with a stringency index value of ≥ 45% were estimated to be required to reduce the reproduction ratio below 1 for symptomatic cases. Simulated cases over the next 2 months differed between countries, with certain countries (eg, Argentina and Japan) likely to experience an accelerated accumulation of cases. Conclusion: Together, results from our model can guide the distribution of diagnostic tests, impact clinical trial development, support medication development and distribution, and inform mitigation strategies to reduce COVID-19 spread. The large contribution of asymptomatic cases in the transmission also suggests that measures such as wearing masks, social distancing, testing, and vaccination deployment are foundational to slowing the spread of COVID-19.

The dynamics of rubella virus with two-dose vaccination strategy

An age-structured SEIR model were constructed to study the transmission of rubella virus. The study investigated the impact of 2-dose vaccination strategy to eradicate the virus transmission. The population were focused on women population and due to the implementation of 2-dose vaccination, the population were divided into two age groups, namely age group 1 of girls who received MMR1 at age 12 – 15 months old and age group 2 of girls who received MMR2 at age 4 years old or older. The analysis was started with determining the equilibria of the system and then used it to calculate the basic reproduction ratio (R0) of the system. The basic reproduction ratio was used to identify the stability of the equilibria. Some numerical simulations were undertaken to confirm the analytical calculations. From the results of the study it can be concluded that the model is well fitted to the facts that 2-dose vaccination strategy for elimination rubella virus transmission really makes sense to be applied by countries all over the world.

Impacts of tidal restriction caused by embankments on the plastic growth responses of Bolboschoenus planiculmis in Korea

Tidal restriction caused by embankments can have large impacts on estuarial ecosystems. It changes the tidal regimes, which influence the growth and development of marsh plants. Although large-scale tidal freshwater marshes are negatively or positively affected by the embankments especially around the Yellow Sea in Korea, little is known regarding how the tidal restriction changes the plastic growth responses of marsh plants, which may cause a population decline in the field. Therefore, we evaluated the impacts of tidal restriction on the growth, plasticity in morphology, reproduction, and biomass allocation of the important but vulnerable marsh plant Bolboschoenus planiculmis in Korea. These responses of B. planiculmis in tidal marshes and tide-restricted marshes were compared. The tide-restricted marshes experienced deeper water (9 cm) and greater dominance of other species (51%) than the tidal marshes (respectively 0 cm, 20%). Populations in the tide-restricted marshes revealed a lower density (128 ramets m−2), decreased sexual reproduction ratio (6%), and higher short rhizome density ratio (17%) with a rapid increase in allocation to below ground biomass compared to the tidal marsh populations (respectively 1046 m−2, 35%, and 0.3%). A redundancy analysis indicated that B. planiculmis with a low density in the tide-restricted marshes developed more short rhizomes and allocated more energy to below ground biomass in order to persist in the original habitats where neighbor species dominated. On the other hand, the tidal marsh populations with higher densities invested more energy to sexual reproduction and developed more long rhizomes to expand into new areas. In this paper, tidal restriction caused the dominance of neighbor species such as Phragmites australis in standing water and negative impacts on B. planiculmis. These results indicate that the impacts of tidal restriction caused by artificial structures should be minimized by maintaining natural tides for sustainable management of B. planiculmis.

Parameter estimation from ICC curves

Incidence vs. Cumulative Cases (ICC) curves are introduced and shown to provide a simple framework for parameter identification in the case of the most elementary epidemiological model, consisting of susceptible, infected, and removed compartments. This novel methodology is used to estimate the basic reproduction ratio of recent outbreaks, including those associated with the ongoing COVID-19 pandemic.

Mathematical analysis and adaptive control spreading of coronavirus disease 2019 (COVID-19)

Coronavirus disease 2019 (COVID-19) is an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which is spreading all over the world and the main causes of worldwide death. For this reason, the control analysis of this coronavirus disease 2019 has a significant importance prevent the spread of it. In this paper, we present the modeling, mathematical analysis, and adaptive control of spreading coronavirus disease 2019. The mathematical analysis shows that the two fixed points of the coronavirus disease 2019 are globally asymptotically stable and the basic reproduction ratio R 0 is obtained, which characterizes the disease transmission. Moreover, an adaptive control is designed to control and treat coronavirus outbreak. The sufficient control conditions are derived for the existence of stable coronavirus disease 2019 free is presented.

Mathematical analysis of an HIV model with latent reservoir, delayed CTL immune response and immune impairment.

In this paper, an in-host HIV infection model with latent reservoir, delayed CTL immune response and immune impairment is investigated. By using suitable Lyapunov functions and LaSalle's invariance principle, it is shown that when time delay is equal to zero, the immunity-inactivated reproduction ratio is a threshold determining the global dynamics of the model. By means of the persistence theory for infinite dimensional systems, it is proven that if the immunity-inactivated reproduction ratio is greater than unity, the model is permanent. Choosing time delay as the bifurcation parameter and analyzing the corresponding characteristic equation of the linearized system, the existence of a Hopf bifurcation at the immunity-activated equilibrium is established. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics.

A SIQ mathematical model on COVID-19 investigating the lockdown effect.

This research paper aims at studying the impact of lockdown on the dynamics of novel Corona Virus Disease (COVID-19) emerged in Wuhan city of China in December 2019. Perceiving the pandemic situation throughout the world, Government of India restricted international passenger traffic through land check post (Liang, 2020) and imposed complete lockdown in the country on 24 March 2020. To study the impact of lockdown on disease dynamics we consider a three-dimensional mathematical model using nonlinear ordinary differential equations. The proposed model has been studied using stability theory of nonlinear ordinary differential equations. Basic reproduction ratio is computed and significant parameters responsible to keep basic reproduction ratio less than one are identified. The study reveals that disease vanishes from the system only if complete lockdown is imposed otherwise disease will always persist in the population. However, disease can be kept under control by implementing contact tracing and quarantine measures as well along with lockdown if lockdown is imposed partially.

Mathematical model to assess the imposition of lockdown during COVID-19 pandemic

Nigeria, like most other countries in the world, imposes lockdown as a measure to curtail the spread of COVID-19. But, it is known fact that in some countries the lockdown strategy could bring the desired results while in some the situation could worsen the spread of the virus due to poor management and lack of facilities, palliatives and incentives. To this regard, we feel motivated to develop a new mathematical model that assesses the imposition of the lockdown in Nigeria. The model comprises of a system of five ODE. Mathematical analysis of the model were carried out, where boundedness, computation of equilibria, calculation of the basic reproduction ratio and stability analysis of the equilibria were carried out. We finally study the numerical outcomes of the governing model in respect of the approximate solutions. To this aim, we employed the effective ODE45, Euler, RK-2 and RK-4 schemes and compare the results.

R0 and Re of COVID-19: Can We Predict When the Pandemic Outbreak will be Contained?

R0 (R naught) is the basic reproduction number, also known as basic reproduction ratio or rate which is an epidemiological metric used to measure the transmissibility of infectious agents. R0 is a derivative of the following variables—the duration of infectivity after the patient gets infected, the likelihood of transmission of infection per contact between a susceptible person and an infectious individual, and the contact rate. R0 is usually estimated retrospectively from serial epidemiological data or using theoretical mathematical models. Epidemiologists can calculate R0 using contact-tracing data, the most common method is to use cumulative incidence data. When mathematical models are used, R0 values are estimated using ordinary differential equations. R0 of COVID-19 as initially estimated by the World Health Organization (WHO) was between 1.4 and 2.4. The forecast is of critical importance as it will help the governments to have an estimate as well as strategize quickly to avoid any unfavorable condition.How to cite this articleAchaiah NC, Subbarajasetty SB, Shetty RM. R0 and Re of COVID-19: Can We Predict When the Pandemic Outbreak will be Contained? Indian J Crit Care Med 2020;24(11):1125–1127.

Extinction time for the weaker of two competing SIS epidemics

We consider a simple Markov model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size $N$. The spread of each strain in the absence of the other one is described by the stochastic SIS logistic epidemic process, and we assume that there is perfect cross-immunity between the two strains, that is, individuals infected by one are temporarily immune to re-infections and infections by the other. For the case where one strain is strictly stronger than the other, and the stronger strain on its own is supercritical, we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population. We further extend our results to certain parameter values where the difference between the basic reproductive ratios of the two strains may tend to 0 as $N\to\infty$. In our proofs, we illustrate a new approach to a fluid limit approximation for a sequence of Markov chains in the vicinity of a stable fixed point of the limit differential equation, valid over long time intervals.

Assessing the Risk of Occurrence of Bluetongue in Senegal.

Bluetongue is a non-contagious viral disease affecting small ruminants and cattle that can cause severe economic losses in the livestock sector. The virus is transmitted by certain species of the genus Culicoides and consequently, understanding their distribution is essential to enable the identification of high-risk transmission areas. In this work we use bioclimatic and environmental variables to predict vector abundance, and estimate spatial variations in the basic reproductive ratio . The resulting estimates were combined with livestock mobility and serological data to assess the risk of Bluetongue outbreaks in Senegal. The results show an increasing abundance of C. imicola, C. oxystoma, C. enderleini, and C. miombo from north to south. < 1 for most areas of Senegal, whilst southern (Casamance) and southeastern (Kedougou and part of Tambacounda) agro-pastoral areas have the highest risk of outbreak ( = 2.7 and 2.9, respectively). The next higher risk areas are in the Senegal River Valley ( = 1.07), and the Atlantic coast zones. Seroprevalence rates, shown by cELISA, weren’t positively correlated with outbreak probability. Future works should include follow-up studies of competent vector abundancies and serological surveys based on the results of the risk analysis conducted here to optimize the national epidemiological surveillance system.

The Introduction of a Mandatory Mask Policy Was Associated with Significantly Reduced COVID-19 Cases in a Major Metropolitan City

Background: Whilst evidence of face masks’ effectiveness in reducing COVID-19 cases is increasing, the impact of mandatory use across a large population has been difficult to assess. Introduction of mandatory mask use on July 22, 2020 during a resurgence of COVID-19 in Melbourne, Australia created a natural experiment to assess the impact of the policy on the epidemic growth rate as its introduction occurred in the absence of other changes to restrictions. Methods: Exponential epidemic growth or decay rates in daily COVID-19 diagnoses were estimated using a non-weighted linear regression of the natural logarithm of the daily cases against time, using a linear spline model with one knot (lspline package in R v 3.6.3). The model’s two linear segments straddle the hinge day, determined from the model fit, on which the mask policy began to take effect. We used two forms of data to assess change in mask usage: population-based survey data and images of people wearing masks in public places obtained from a major media outlet. Potential confounding factors (including daily COVID-19 tests, number of COVID-19 cases among population subsets affected differentially by the mask policy – e.g. healthcare workers) were examined for their impact on the results.Findings: Introduction of a mandatory mask policy was associated with an estimated 23% reduction in the effective reproduction ratio, from 1.18 to 0.91. The hinge day in the linear spline model occurred eight days after the policy was introduced on July 23, 2020. Daily cases grew exponentially in the first log-linear segment (k = +0.042, s.e. = 0.007), and decayed exponentially in the second (k = -0.023, s.e. = 0.017). Analysis of images of people in public spaces showed mask usage rose from approximately 43% to 97%. Analysis of survey data found that on the third day before policy introduction, 44% of participants reported “often” or “always” wearing a mask; on the fourth day after, 100% reported “always” doing so. No potentially confounding factors were associated with the observed change in growth rates.Interpretation: The mandatory mask use policy substantially increased public use of masks and was associated with a significant decline in COVID-19 transmission after introduction of the policy. The hinge date (where the shift in epidemic growth was observed) occurred eight days after policy introduction, consistent with the incubation time of COVID-19 and the time needed to test and report new cases. Although studies emphasise the importance of masks in a clinical setting, to our knowledge no previous peer reviewed publications measured the role of masks for controlling epidemics in the broader community.Funding Statement: This work was funded from Burnet Institute resources.Declaration of Interests: BC serves on the National COVID-19 Health Research Advisory Committee that provides guidance to the Federal Government’s COVID-19 response. NS, MH, A Saeri, EG and LS provide guidance to the Victorian Government’s COVID-19 response. MH and NS receive funding from the Victorian Government for the Optimise Study and modelling work. A Saeri and EG receive funding from the Victorian Government for the SCRUB project. CRM serves on the National COVID-19 Health Research Advisory Committee that provides guidance to the Federal Government’s COVID-19 response. She currently receives funding from the National Health and Medical Research Council and Medical Research Futures Fund, Office of Health Protection, Department of Health, Sanofi and from the Defence Intelligence Organisation. All authors have nothing to declare.

Ambient PM2.5 exposure and rapid spread of COVID-19 in the United States

It has been posited that populations being exposed to long-term air pollution are more susceptible to COVID-19. Evidence is emerging that long-term exposure to ambient PM2.5 (particulate matter with aerodynamic diameter 2.5 μm or less) associates with higher COVID-19 mortality rates, but whether it also associates with the speed at which the disease is capable of spreading in a population is unknown. Here, we establish the association between long-term exposure to ambient PM2.5 in the United States (US) and COVID-19 basic reproduction ratio R0– a dimensionless epidemic measure of the rapidity of disease spread through a population. We inferred state-level R0 values using a state-of-the-art susceptible, exposed, infected, and recovered (SEIR) model initialized with COVID-19 epidemiological data corresponding to the period March 2–April 30. This period was characterized by a rapid surge in COVID-19 cases across the US states, implementation of strict social distancing measures, and a significant drop in outdoor air pollution. We find that an increase of 1 μg/m3 in PM2.5 levels below current national ambient air quality standards associates with an increase of 0.25 in R0 (95% CI: 0.048–0.447). A 10% increase in secondary inorganic composition, sulfate-nitrate-ammonium, in PM2.5 associates with ≈10% increase in R0 by 0.22 (95% CI: 0.083–0.352), and presence of black carbon (soot) in the ambient environment moderates this relationship. We considered several potential confounding factors in our analysis, including gaseous air pollutants and socio-economical and meteorological conditions. Our results underscore two policy implications – first, regulatory standards need to be better guided by exploring the concentration-response relationships near the lower end of the PM2.5 air quality distribution; and second, pollution regulations need to be continually enforced for combustion emissions that largely determine secondary inorganic aerosol formation.

Estimation and Monitoring of COVID-19's Transmissibility From Publicly Available Data

The COVID-19 pandemic began in the city of Wuhan, China, at the end of 2019 and quickly spread worldwide. The disease is caused by contact with the SARS-CoV-2 virus, which probably jumped from an animal host to humans. SARS-CoV-2 infects various tissues in the body, notably the lungs, and patients usually die from respiratory complications. Mathematical models of the disease have been instrumental to guide the implementation of mitigation strategies aimed at slowing the spread of the disease. One of the key parameters of mathematical models is the basic reproduction ratio R0, which measures the degree of infectivity of affected individuals. The goal of mitigation is to reduce R0 as close or below 1 as possible, as it means that new infections are in decline. In the present work, we use the recursive least-squares algorithm to establish the stochastic variability of a time-varying R0(t) from eight different countries: Argentina, Belgium, Brazil, Germany, Italy, New Zealand, Spain, and the United States of America. The proposed system can be implemented as an online tracking application providing information about the dynamics of the pandemic to health officials and the public at large.

Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media

<p style='text-indent:20px;'>This paper is concerned with propagation phenomena for an epidemic model describing the circulation of a disease within two populations or two subgroups in periodic media, where the susceptible individuals are assumed to be motionless. The spatial dynamics for the cooperative system obtained by a classical transformation are investigated, including spatially periodic steady state, spreading speeds and pulsating travelling fronts. It is proved that the minimal wave speed is linearly determined and given by a variational formula involving linear eigenvalue problem. Further, we prove that the existence and non-existence of travelling wave solutions of the model are entirely determined by the basic reproduction ratio <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_{0} $\end{document}</tex-math></inline-formula>. As an application, we prove that if the localized amount of infectious individuals are introduced at the beginning, then the solution of such a system has an asymptotic spreading speed in large time and that is exactly coincident with the minimal wave speed.

Hopf and backward bifurcations induced by immune effectors in a cancer oncolytic virotherapy dynamics

Oncolytic virotherapy is emerging as a promising new method for cancer treatment. In this paper, we consider a mathematical model for treatment of cancer by using oncolytic virotherapy in the presence of immune effectors. We provide a theoretical analysis of the model and derive the basic reproduction number $$\mathcal {R}_0$$ which determines the extinction and the presence of oncolytic viruses during therapy. The existence of equilibria and their stability are investigated. More precisely, we show that, depending of the values of the parameters, there exits a quantity $$\varepsilon $$ so that, when $$\varepsilon <1$$ , the tumor can be eliminated in the body. However, we also show that, if $$\mathcal {R}_0<1$$ and $$\varepsilon >1$$ , the infection-free equilibrium is stable and the model is shown to exhibit the phenomenon of backward bifurcation (where a stable infection-free equilibrium coexist with one or more stable endemic equilibria when the associated basic reproduction ratio is less than unity). Furthermore the model presents a Hopf bifurcation which is supercritical, from which birth of oscillation occurs. Numerical simulations support our theoretical results.

A Herpesvirus of Turkey-Based Vector Vaccine Reduces Transmission of Newcastle Disease Virus in Commercial Broiler Chickens with Maternally Derived Antibodies

Newcastle Disease is one of the most important infectious poultry diseases worldwide and is associated with high morbidity, mortality, and economic loss. In several countries, vaccination is applied to prevent and control outbreaks; however, information on the ability of vaccines to reduce transmission of ND virus (NDV) is sparse. Here we quantified the transmission of velogenic NDV among 42-day-old broilers. Chickens were either vaccinated with a single dose of a vector vaccine expressing the F protein (rHVT-ND) at day-old in the presence of maternally derived antibodies or kept unvaccinated. Seeders were challenged 8 h before the co-mingling with the corresponding contacts from the same group. Infection was monitored by daily testing of cloacal and oro-nasal swabs with reverse transcription-real-time PCR and by serology. Vaccinated birds were completely protected against clinical disease and virus excretion was significantly reduced compared to the unvaccinated controls that all died during the experiment. The reproduction ratio, which is the average number of secondary infections caused by an infectious bird, was significantly lower in the vaccinated group (0.82 (95% CI 0.38–1.75)) than in the unvaccinated group (3.2 (95% CI 2.06–4.96)). Results of this study demonstrate the potential of rHVT-ND vaccine in prevention and control of ND outbreaks.

The Epidemiological Framework for Biological Invasions (EFBI): an interdisciplinary foundation for the assessment of biosecurity threats

Emerging microparasite (e.g. viruses, bacteria, protozoa and fungi) epidemics and the introduction of non-native pests and weeds are major biosecurity threats worldwide. The likelihood of these threats is often estimated from probabilities of their entry, establishment, spread and ease of prevention. If ecosystems are considered equivalent to hosts, then compartment disease models should provide a useful framework for understanding the processes that underpin non-native species invasions. To enable greater cross-fertilisation between these two disciplines, the Epidemiological Framework for Biological Invasions (EFBI) is developed that classifies ecosystems in relation to their invasion status: Susceptible, Exposed, Infectious and Resistant. These states are linked by transitions relating to transmission, latency and recovery. This viewpoint differs markedly from the species-centric approaches often applied to non-native species. It allows generalisations from epidemiology, such as the force of infection, the basic reproductive ratio R0, super-spreaders, herd immunity, cordon sanitaire and ring vaccination, to be discussed in the novel context of non-native species and helps identify important gaps in the study of biological invasions. The EFBI approach highlights several limitations inherent in current approaches to the study of biological invasions including: (i) the variance in non-native abundance across ecosystems is rarely reported; (ii) field data rarely (if ever) distinguish source from sink ecosystems; (iii) estimates of the susceptibility of ecosystems to invasion seldom account for differences in exposure to non-native species; and (iv) assessments of ecosystem susceptibility often confuse the processes that underpin patterns of spread within -and between- ecosystems. Using the invasion of lakes as a model, the EFBI approach is shown to present a new biosecurity perspective that takes account of ecosystem status and complements demographic models to deliver clearer insights into the dynamics of biological invasions at the landscape scale. It will help to identify whether management of the susceptibility of ecosystems, of the number of vectors, or of the diversity of pathways (for movement between ecosystems) is the best way of limiting or reversing the population growth of a non-native species. The framework can be adapted to incorporate increasing levels of complexity and realism and to provide insights into how to monitor, map and manage biological invasions more effectively.