Reproduction Number R0 Research Articles

Mathematical Analysis of Drugs and Substance Abuse in Kenya among the Adolescents

It is incontestable that the mortality rate among drugs and substance abusers is higher than that in thegeneral population. The National Authority for the campaign against alcohol and substance abuse (NACADA)has painted a grim picture of the incessant rise in the number of youth becoming addicted. In this research, adeterministic model for drugs and substance abuse (DSA) driven by light drug abusers (LDA) and heavy drugabusers (HDA) was proposed. The basic reproduction number R0; , the foundation upon which the model’sstability analysis is established, was determined by utilizing the next-generation matrix (NGM) approach.The analysis showed that drug-free equilibrium (DFE) is locally asymptotically stable for R0 < 1 and unstableif R0 > 1. The global stability of both DFE and drugs endemic equilibrium (DEE) are explored by utilizingLyapunov functions. The bifurcation analysis was carried out using the center manifold theorem, where themethod utilized by Castilo-Chavez and Song was implemented and revealed that the rate of drug reinitiationdrove backward bifurcation. The contribution of the important parameters to DSA are investigated, andresults are presented graphically. Results from the simulation revealed that delayed exposure of the youth todrugs increased identification and treatment of the LDA and HDA, which would curtail DSA menace in Kenya.

Stability and Hopf bifurcation for age-structured SVIR epidemic model with different compartment ages and two delays effects

An age-structured SVIR epidemic model with different compartment ages and two delays is investigated. The model is transformed into a non-densely defined abstract Cauchy problem. The basic properties, including the positivity and boundedness of solutions, basic reproduction number R0 and the existence of equilibria, are first derived. The linearized system and characteristic equation at an equilibrium from the corresponding abstract Cauchy problem are also obtained. When R0<1, the local stability of the disease-free equilibrium is proved, and when R0>1 and the two delays vanish, the local stability of the endemic equilibrium is proved. When R0>1, and Assumptions 1 and 2 are satisfied, the existence of Hopf bifurcation are established respectively in the general case with two delays as bifurcation parameters and in three special cases with only one delay as the bifurcation parameter. The results show that disease with the incubation period and the infection period of infection age has very complex dynamical behavior at the endemic equilibrium. Finally, numerical examples validate the theoretical results.

Analyzing the Asymptotic Behavior of an Extended SEIR Model with Vaccination for COVID-19

Several research papers have attempted to describe the dynamics of COVID-19 based on systems of differential equations. These systems have taken into account quarantined or isolated cases, vaccinations, control measures, and demographic parameters, presenting propositions regarding theoretical results that often investigate the asymptotic behavior of the system. In this paper, we discuss issues that concern the theoretical results proposed in the paper “An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter”. We propose detailed explanations regarding the resolution of these issues. Additionally, this paper focuses on extending the local stability analysis of the disease-free equilibrium, as presented in the aforementioned paper, while emphasizing the derivation of theorems that validate the global stability of both epidemic equilibria. Emphasis is placed on the basic reproduction number R0, which determines the asymptotic behavior of the system. This index represents the expected number of secondary infections that are generated from an already infected case in a population where almost all individuals are susceptible. The derived propositions can inform health authorities about the long-term behavior of the phenomenon, potentially leading to more precise and efficient public measures. Finally, it is worth noting that the examined paper still presents an interesting epidemiological scheme, and the utilization of the Kalman filtering approach remains one of the state-of-the-art methods for modeling epidemic phenomena.

Threshold dynamics of stochastic H7N9 model with Markov switching and hybrid strategy

This article aims to study the impact of different environmental states on the transmission dynamics of avian influenza A (H7N9) for poultry by using an epidemic model with general incidence rate and Markov switching. The model couples the essential roles of quarantine and hybrid strategy of vaccination and elimination. By establishing the basic reproduction number R0 as a sharp threshold for the spread of H7N9. When R0<1, the disease almost surely dies out exponentially. We use the compactness of space to prove that when R0>1, the disease almost surely persistent, and this method can be applied to other infectious disease models considering vaccination. In addition, under a mild extra condition, the ergodicity of the Markov process is proved. Our research indicates that: (i) theoretical results and sensitivity analysis show that implementing hybrid strategy is an effective method to control H7N9 propagation; (ii) under the influence of environmental switching, numerical simulations find that the transmission state of H7N9 will neutralize multiple regimes, leading to the possibility of multi wave transmission. This can be used to explain the multiple outbreaks of H7N9 in China.

Algorithmic Approach for a Unique Definition of the Next-Generation Matrix

The basic reproduction number R0 is a concept which originated in population dynamics, mathematical epidemiology, and ecology and is closely related to the mean number of children in branching processes (reflecting the fact that the phenomena of interest are well approximated via branching processes, at their inception). Despite the very extensive literature around R0 for deterministic epidemic models, we believe there are still aspects which are not fully understood. Foremost is the fact that R0 is not a function of the original ODE model, unless we also include in it a certain (F,V) gradient decomposition, which is not unique. This is related to the specification of the “infected compartments”, which is also not unique. A second interesting question is whether the extinction probabilities of the natural continuous time Markovian chain approximation of an ODE model around boundary points (disease-free equilibrium and invasion points) are also related to the (F,V) gradient decomposition. We offer below several new contributions to the literature: (1) A universal algorithmic definition of a (F,V) gradient decomposition (and hence of the resulting R0). (2) A fixed point equation for the extinction probabilities of a stochastic model associated to a deterministic ODE model, which may be expressed in terms of the (F,V) decomposition. Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe offers always reasonable results, but that sometimes other reasonable (F,V) decompositions are available as well.

A cell-level dynamical model for malaria parasite infection with antimalarial drug treatment

Malaria is an infectious disease caused by intracellular parasites of the genus Plasmodium. It is a major health problem around the world. In this study, a cell-level mathematical model of malaria parasites with antimalarial drug treatments is formulated and analyzed. The model consists of seven compartments for cell populations. We analyzed the qualitative behavior of the model using various techniques. The stability analysis of the parasite-free equilibrium is obtained, whereas it is locally and globally stable if the basic reproduction number R0&lt;1. The parasite persistence equilibrium point exists, and it is locally asymptotically stable if R0&gt;1. The sensitivity analysis of the basic reproduction number is computed, and the results show that the infection rate of the erythrocyte by merozoites, the average number of merozoites per ruptured infected erythrocyte cells, the natural death rate of merozoites, and the requirement rate of the uninfected erythrocyte are the most influential parameters within-host dynamics of malaria infection. Different numerical simulations are performed to supplement our analytical findings. The effect of primary tissue schizontocides, blood schizontocides, and gametocytocides on infected hepatocytes, infected erythrocytes, and gametocytes have been investigated, respectively. Finally, some counterplots are presented in order to investigate the impact of parameters on the basic reproduction number. The in-host basic reproduction number decreases as the antimalarial treatment administration increases. Therefore, increasing antimalarial treatment administration is the best way to mitigate the in-host malaria infection.

The analysis of a new fractional model to the Zika virus infection with mutant

We present a new mathematical model to analyze the dynamics of the Zika virus (ZV) disease with the mutant under the real confirmed cases in Colombia. We give the formulation of the model initially in integer order derivative and then extend it to a fractional order system in the sense of the Mittag-Leffler kernel. We study the properties of the model in the Mittag-Leffler kernel and establish the result. The basic reproduction of the fractional system is computed. The equilibrium points of the Zika virus model are obtained and found that the endemic equilibria exist when the threshold is greater than unity. Further, we show that the model does not possess the backward bifurcation phenomenon. The numerical procedure to solve the problem using the Atangana-Baleanu derivative is shown using the newly established numerical scheme. We consider the real cases of the Zika virus in Colombia outbreak are considered and simulate the model using the nonlinear least square curve fit and computed the basic reproduction number R0=0.4942, whereas in previous work (Alzahrani et al., 2021) [1], the authors computed the basic reproduction number R0=0.5447. This is due to the fact that our work in the present paper provides better fitting to the data when using the fractional order model, and indeed the result regarding the data fitting using the fractional model is better than integer order model. We give a sensitivity analysis of the parameters involved in the basic reproduction number and show them graphically. The results obtained through the present numerical method converge to its equilibrium for the fractional order, indicating the proposed scheme's reliability.

Analysis of a delayed spatiotemporal model of HBV infection with logistic growth

In this paper, following previous works of ours, we deal with a mathematical model of hepatitis B virus (HBV) infection. We assume spatial diffusion of free HBV particles, logistic growth for both healthy and infected hepatocytes, and use the standard incidence function for viral infection. Moreover, one time delay is introduced to account for actual virus production. Another time delay is used to account for virus maturation. The existence, uniqueness, positivity and boundedness of solutions are established. Analyzing the model qualitatively and using a Lyapunov functional, we establish the existence of a threshold T 0 such that, if the basic reproduction number R 0 verifies R 0 ≤ T 0 < 1 , the infection-free equilibrium is globally asymptotically stable. When R 0 is greater than one, we discuss the local asymptotic stability of the unique endemic equilibrium and the occurrence of a Hopf bifurcation. Also, when R 0 is greater than one, the system is uniformly persistent, which means that the HBV infection is endemic. Finally, we carry out some relevant numerical simulations to clarify and interpret the theoretical results.

Extinction in host–vector infection models and the role of heterogeneity

For infections that become endemic in a population, the process may appear stable over a long time scale, but stochastic fluctuations can lead to eventual disease extinction. We consider the effects of model parameters and of population heterogeneities upon the expected time to extinction for host–vector disease systems. We find that non-homogeneous host selection by vectors increases persistence times relative to the homogeneous case, and that the effect becomes even more marked when there are strong associations between particular groups of vectors and hosts. Heterogeneity in vector lifespans, in contrast, is found to decrease persistence times relative to the homogeneous case. Neither the basic reproduction number R0, nor the endemic prevalence level in the corresponding deterministic model, is found to be sufficient to predict (for a given population size) time to extinction. The endemic level, in particular, proves a very unreliable guide to the duration of long-term persistence.

Global Dynamics in an Alcoholism Epidemic Model with Saturation Incidence Rate and Two Distributed Delays

In this study, considering the delays for a susceptible individual becoming an alcoholic and the relapse of a recovered individual back into being an alcoholic, we formulate an epidemic model for alcoholism with distributed delays and relapse. The basic reproduction number R0 is calculated, and the threshold property of R0 is established. By analyzing the characteristic equation, we demonstrate the local asymptotic stability of the different equilibria under various conditions: when R0&lt;1, the alcoholism-free equilibrium is locally asymptotically stable; when R0&gt;1, the alcoholism equilibrium exists and is locally asymptotically stable. Furthermore, we demonstrate the global asymptotic stability at each equilibrium using a suitable Lyapunov function. Specifically, when R0&lt;1, the alcoholism-free equilibrium is globally asymptotically stable; when R0&gt;1, the alcoholism equilibrium is globally asymptotically stable. The sensitivity analysis of R0 shows that reducing exposure is more effective than treatment in controlling alcoholism. Interestingly, we found that extending the latency delay h1 and relapse delay h2 also effectively contribute to the control of the spread of alcoholism. Numerical simulations are also provided to support our theoretical results.

Dynamically adjustable SVEIR(MH) model of multiwave epidemics: Estimating the effects of public health measures against COVID-19.

The COVID-19 pandemic was characterized by multiple subsequent, overlapping outbreaks, as well as extremely rapid changes in viral genomes. The information about local epidemics spread and the epidemic control measures was shared on a daily basis (number of cases and deaths) via centralized repositories. The vaccines were developed within the first year of the pandemic. New modes of monitoring and sharing of epidemic data were implemented using Internet resources. We modified the basic SEIR compartmental model to include public health measures, multiwave scenarios, and the variation of viral infectivity and transmissibility reflected by the basic reproduction number R0 of emerging viral variants. SVEIR(MH) model considers the capacity of the medical system, lockdowns, vaccination, and changes in viral reproduction rate on the epidemic spread. The developed model uses daily infection reports for assessing the epidemic dynamics, and daily changes of mobility data from mobile phone networks to assess the lockdown effectiveness. This model was deployed to six European regions Baden-Württemberg (Germany), Belgium, Czechia, Lombardy (Italy), Sweden, and Switzerland for the first 2 years of the pandemic. The correlation coefficients between observed and reported infection data showed good concordance for both years of the pandemic (ρ = 0.84-0.94 for the raw data and ρ = 0.91-0.98 for smoothed 7-day averages). The results show stability across the regions and the different epidemic waves. Optimal control of epidemic waves can be achieved by dynamically adjusting epidemic control measures in real-time. SVEIR(MH) model can simulate different scenarios and inform adjustments to the public health policies to achieve the target outcomes. Because this model is highly representative of actual epidemic situations, it can be used to assess both the public health and socioeconomic effects of the public health measures within the first 7 days of the outbreak.

Mathematical and Numerical Investigations for a Cholera Dynamics With a Seasonal Environment

We propose a mathematical model for the vibrio cholerae spread under the influence of a seasonal environment with two routes of infection. We proved the existence of a unique bounded positive solution, and that the system admits a global attractor set. The basic reproduction number R0 was calculated for both cases, the fixed and seasonal environment which permits to characterise both, the extinction and the persistence of the disease. We proved that the phage-free equilibrium point is globally asymptotically stable if R0≤1, while the disease will be persist if R0&gt;1. Finally, extensive numerical simulations are given to confirm the theoretical findings.

Periodic traveling waves for a diffusive influenza model with treatment and seasonality

This paper is concerned with the existence and nonexistence of time-periodic traveling waves for a diffusive influenza model with treatment and seasonality. By using the next generation operator theory, we first get basic reproduction number R0 for the corresponding periodic ODEs. Then, by constructing sub-and super-solutions and using Schauder’s fixed point theorem, we obtain the existence of time-periodic traveling wave solutions for the system with wave speed c>c∗ and R0>1. We further prove the existence of time-periodic traveling waves with wave speed c=c∗ by a delicate limitation argument. For du=dh, the nonexistence of traveling waves is proved by a contradiction argument for two cases involved with c∗ and R0, while the exponential decay of traveling waves with the critical speed is obtained by a dynamical system approach combined with Laplace transform.

Evaluating the Interplay between Transmissibility and Virulence of SARS-CoV-2 by Mathematical Modeling

During the first months of 2020 SARS-CoV-2 spread to all continents, virtually reaching all countries. In the subsequent months, new variants emerged in different regions of the world. A mathematical model based on the Covid-19 natural history encompassing the age-dependent fatality was applied to evaluate SARS-CoV-2 transmissibility and virulence. Transmissibility was assessed by calculating the basic reproduction number R0 and virulence by counting the proportion of severe Covid-19 cases and deaths. The model parameters were adjusted against the data observed in the state of São Paulo, Brazil, considering two different levels of virulence. The severe Covid-19 cases and deaths were three times higher and R0 was 25% lower when the more virulent SARS-CoV-2 variant was compared to the less virulent variant. However, under the high-virulence scenario the number of transmitting individuals is 25% lower, mainly due to the isolation of symptomatic individuals. The corollary that transmission increases in the low virulence scenario is also true. The estimated parameters, using data from São Paulo up to May 13, 2020, showed that the Covid-19 epidemic predicted with low virulence SARS-CoV-2 transmission matched the observed data just before the beginning of the relaxation, which occurred by mid-June 2020. The assessment of the interplay between transmissibility can be applied to explain in somehow the appearance of gamma and omicron variants of concern in São Paulo.

Investigations of transmission dynamics of Nipah virus in Bangladesh

This study has focused on the transmission dynamics of the Nipah Virus (NiV) and its controlling system. In this respect, we have developed a virus transmission model based on several compartments and discussed the model's basic reproduction number and stability analysis at disease-free equilibrium (DFE) and endemic equilibrium (EE) points. It is observed that the model is locally and globally asymptotically stable at DFE and EE for the basic reproduction number R0<1 and R0>1, respectively. We have also solved the model equations numerically using the Runge-Kutta (RK-45) method to investigate the effect of disease parameters. In the calculation, some parameter values are taken from several references. Lastly, the calculated results show that the infection rate of the exposed individuals highly influences the disease compartments. Furthermore, drug resistance is observed to be less effective than the treatment rate. In addition, it is found that the number of infected people is reducing with the increasing treatment rate.

Contact-dependent infection and mobility in the metapopulation SIR model from a birth–death process perspective

Given the widespread impact of COVID-19, modeling and analysis of epidemic propagation has been critical to epidemic prevention and control. However, previous studies have overlooked the significant influence of individual heterogeneity in behavior and physiology, including contact-dependent infection and migration on epidemic propagation. In this paper, we propose two metapopulation SIR models from individual and population perspectives. The first individual model introduces individual contact-dependent infection considering activity potential and infection rate, which leads to the derivation of the basic reproduction number R0 of our model. The birth–death process, used in the second population model, is represented by a compound Poisson process flow and Poisson process decomposition, respectively, to depict population mobility among subpopulations. In simulations, the number of individuals in each state and the converged number are illustrated to demonstrate the impact of various parameters. The relationship between the basic reproduction number R0 and various parameters is also demonstrated. Furthermore, the validity of our model is also confirmed on a real clinical report dataset of COVID-19 disease.

Global Dynamics of a Within-Host Model for Usutu Virus

We propose a within-host mathematical model for the dynamics of Usutu virus infection, incorporating Crowley–Martin functional response. The basic reproduction number R0 is found by applying the next-generation matrix approach. Depending on this threshold, parameter, global asymptotic stability of one of the two possible equilibria is also established via constructing appropriate Lyapunov functions and using LaSalle’s invariance principle. We present numerical simulations to illustrate the results and a sensitivity analysis of R0 was also completed. Finally, we fit the model to actual data on Usutu virus titers. Our study provides new insights into the dynamics of Usutu virus infection.

Mathematical analysis of an epidemic model with direct and indirect transmission modes and two delays

Abstract In this article, we consider an epidemiological model in which we take into account the effects of direct and indirect transmissions. The first mode occurs through direct contact between infectious and susceptible individuals, and the second one will take place through the shedding of virus particles by infectious individuals and their acquisition by susceptible ones. We also study the effect of latency period and time needed for a susceptible person to become infected by indirect transmission mode. By considering the direct and indirect basic reproduction numbers, we define the basic reproduction number R 0 {R}_{0} of the model, which helps us to analyze the stability of equilibria and bifurcation and determine the most sensitive parameters. In conclusion, some numerical simulations are given to confirm the analytical analysis.

A deterministic mathematical model for optimal control of diphtheria disease with booster vaccination

Diphtheria is an infectious disease caused by a strain of Corynebacterium diphtheria and forms part of the childhood vaccine-preventable diseases. The Diphtheria vaccine is a component of one of the routine vaccines given to children thrice before their first birthday. The protection against diphtheria derived from the diphtheria vaccine in infancy wanes in later childhood, necessitating a booster dose to protect the child as they grow older. To determine the impact of a booster dose of the diphtheria vaccine amidst a contaminated environment, a diphtheria model that incorporates a vaccine booster and a contaminated environment is formulated. The reproduction number R0 is computed and used to prove the local and global stability of the disease-free equilibrium. Global sensitivity analysis is conducted via the application of Latin Hypercube Sampling (LHS) with a Partial Rank Correlation coefficient on the infected humans and the contaminated environment to deduce the most sensitive parameters of the dynamics of diphtheria disease. Then, the model is further extended based on the result of the global sensitivity analysis by introducing four time-dependent controls, disinfection, screening/treatment, booster vaccination, and hygiene practice, to form an optimal control model. The control model is analyzed using Pontryagin’s maximum principle. The numerical simulation shows that diphtheria disease will reduce drastically in the community if any control combination involves booster vaccination since the diphtheria vaccine in infancy wanes after ten years. In a situation where there are limited resources to implement all the controls simultaneously, it is recommended to implement any two of the combined controls: disinfection of the environment and administration of booster vaccination or screening/treatment of the asymptomatic infected and administration of booster vaccination. The study shows that the best combination is to disinfect the environment, screen/treat the asymptomatic infected humans, and administer booster vaccination to the community.

Analysis of global stability and asymptotic properties of a Cholera model with multiple transmission routes, spatial diffusion and incomplete immunity

A novel reaction diffusion model is proposed, in this paper, to consider the impacts on the spatially heterogeneity, horizontal and environmental transmission, incomplete immunity and exposed individuals on the spread and control of Cholera. Firstly, the existence of the solution of this model, the boundedness and the existence of the global attractor are investigated. The basic reproduction number R0 of model is further defined, and then the threshold results of its global dynamics are established based on R0. In particular, we analyze the asymptotic behavior of the steady state when the diffusion rate is small or large in susceptible and exposed individuals. With the help of numerical simulations, we found that the spread of susceptible individuals does not change the spatial distribution of the disease and the local epidemic risk level during the spread of the disease in different regions, while the spread of exposed individuals has an important effect on the spatial distribution of the infected population.