Next Generation Matrix Method Research Articles

Atangana-Baleanu fractional dynamics of dengue fever with optimal control strategies

<abstract><p>Dengue fever, a vector-borne disease, has affected the whole world in general and the Indian subcontinent in particular for the last three decades. Dengue fever has a significant economic and health impact worldwide; it is essential to develop new mathematical models to study not only the dynamics of the disease but also to suggest cost-effective mechanisms to control disease. In this paper, we design modified facts about the dynamics of this disease more realistically by formulating a new basic $ S_hE_hI_hR_h $ host population and $ S_vI_v $ vector population integer order model, later converting it into a fractional-order model with the help of the well-known Atangana-Baleanu derivative. In this design, we introduce two more compartments, such as the treatment compartment $ T_h $, and the protected traveler compartment $ P_h $ in the host population to produce $ S_hE_hI_hT_hR_hP_h $. We present some observational results by investigating the model for the existence of a unique solution as well as by proving the positivity and boundedness of the solution. We compute reproduction number $ \mathcal{R}_{0} $ by using a next-generation matrix method to estimate the contagious behavior of the infected humans by the disease. In addition, we prove that disease free and endemic equilibrium points are locally and globally stable with restriction to reproduction number $ \mathcal{R}_{0} $. The second goal of this article is to formulate an optimal control problem to study the effect of the control strategy. We implement the Toufik-Atangana scheme for the first time to solve both of the state and adjoint fractional differential equations with the ABC derivative operator. The numerical results show that the fractional order and the different constant treatment rates affect the dynamics of the disease. With an increase in the fractional order and the treatment rate, exposed and infected humans, as well as the infected mosquitoes, decrease. However, the optimal control analysis reveals that the implemented optimal control strategy is very effective for disease control.</p></abstract>

A soil water indicator for a dynamic model of crop and soil water interaction.

Water scarcity is a critical issue in agriculture, and the development of reliable methods for determining soil water content is crucial for effective water management. This study proposes a novel, theoretical, non-physiological indicator of soil water content obtained by applying the next-generation matrix method, which reflects the water-soil-crop dynamics and identifies the minimum viable value of soil water content for crop growth. The development of this indicator is based on a two-dimensional, nonlinear dynamic that considers two different irrigation scenarios: the first scenario involves constant irrigation, and the second scenario irrigates in regular periods by assuming each irrigation as an impulse in the system. The analysis considers the study of the local stability of the system by incorporating parameters involved in the water-soil-crop dynamics. We established a criterion for identifying the minimum viable value of soil water content for crop growth over time. Finally, the model was calibrated and validated using data from an independent field study on apple orchards and a tomato crop obtained from a previous field study. Our results suggest the advantages of using this theoretical approach in modeling the plants' conditions under water scarcity as the first step before an empirical model. The proposed indicator has some limitations, suggesting the need for future studies that consider other factors that affect soil water content.

Analysis of a stochastic model of corruption transmission dynamics with temporary immunity

In this investigation, we developed a deterministic SCHRS mathematical model of corruption transmission dynamics and its extended stochastic model after incorporating stochastic factors and Brownian motion. We presented both model analyses in their respective order. The positivity, invariant region, corruption-free equilibrium point, basic reproduction numbers, both local and global stabilities of corruption-free equilibrium, and endemic equilibrium with their stabilities were discussed. The basic reproduction numbers of both deterministic and stochastic approaches are obtained using the next-generation matrix method and twice Itô’s differentiable formula. Sensitivity analyses of the impact of relevant parameters are performed. Furthermore, the outcomes of numerical simulations were presented by comparing the deterministic and stochastic approaches to the model's relevant parameters. We observed that as the rate at which the corrupted population interacts with the susceptible population increases, so does the number of corrupted individuals in the community. Moreover, when individuals recover more through education or punishment, the number of corrupted individuals decreases.

Investigating the Role of Mobility between Rural Areas and Forests on the Spread of Zika

A mathematical model of Zika virus transmission, incorporating human movement between rural areas and nearby forests, is presented to investigate the role of human movement in the spread of Zika virus infections in human and mosquito populations. Proportions of both susceptible and infected humans living in rural areas are assumed to move to nearby forest areas. Direct, indirect, and vertical transmission routes are incorporated for all populations. A mathematical analysis of the proposed model is presented. The analysis starts with normalizing the proposed model. The positivity and boundedness of solutions to the normalized model are then addressed. The basic reproduction number is calculated using the next-generation matrix method and its relation to the three routes of disease transmission has been presented. The sensitivity analysis of the basic reproduction number to all model parameters is investigated. The analysis also includes the existence and stability of disease-free and endemic equilibrium points. Bifurcation analysis is also carried out. Finally, numerical solutions to the normalized model are obtained to confirm the theoretical results and demonstrate human movement's role in disease transmission in human and mosquito populations.

Chewing Khat Transmission Dynamics: A Mathematical Model and Stability Analysis

In this study, the authors proposed a nonlinear deterministic model and stability analysis for the transmission dynamics of chewing khat. The model’s solution is proved to be positive and bounded, and the basic reproduction number ( R 0 ) is calculated using the next-generation matrix method. Following that, the authors have looked at the local and global stability of the model’s khat-free and endemic equilibrium points. When R 0 < 1 , the chewing khat-free equilibrium point is locally and globally asymptotically stable, whereas when R 0 > 1 , the endemic equilibrium point is locally and globally asymptotically stable. The simulation results demonstrate the analytical results.

Mathematical modeling of vaccination as a control measure of stress to fight COVID-19 infections

The world experienced the life-threatening COVID-19 disease worldwide since its inversion. The whole world experienced difficult moments during the COVID-19 period, whereby most individual lives were affected by the disease socially and economically. The disease caused millions of illnesses and hundreds of thousands of deaths worldwide. To fight and control the COVID-19 disease intensity, mathematical modeling was an essential tool used to determine the potentiality and seriousness of the disease. Due to the effects of the COVID-19 disease, scientists observed that vaccination was the main option to fight against the disease for the betterment of human lives and the world economy. Unvaccinated individuals are more stressed with the disease, hence their body’s immune system are affected by the disease. In this study, the SVEIHR deterministic model of COVID-19 with six compartments was proposed and analyzed. Analytically, the next-generation matrix method was used to determine the basic reproduction number (R0). Detailed stability analysis of the no-disease equilibrium (E0) of the proposed model to observe the dynamics of the system was carried out and the results showed that E0 is stable if R0<1 and unstable when R0>1. The Bayesian Markov Chain Monte Carlo (MCMC) method for the parameter identifiability was discussed. Moreover, the sensitivity analysis of R0 showed that vaccination was an essential method to control the disease. With the presence of a vaccine in our SVEIHR model, the results showed that R0=0.208, which means COVID-19 is fading out of the community and hence minimizes the transmission. Moreover, in the absence of a vaccine in our model, R0=1.7214, which means the disease is in the community and spread very fast. The numerical simulations demonstrated the importance of the proposed model because the numerical results agree with the sensitivity results of the system. The numerical simulations also focused on preventing the disease to spread in the community.

A SEIAR mathematical model to analyse the effect of COVID-19 pandemic over the crowded and dense populated regions

In this study, an attempt has been made to formulate a SEIR-type mathematical model for the dynamics of SARS-CoV-2 (COVID-19) pandemic by incorporating an additional asymptomatic compartment. The formulated model contains a system of ordinary differential equations. The basic reproduction number R 0 is obtained by applying the next-generation matrix method. For India, daily death cases and cumulative COVID-19 death cases were used to fit the model from March 20, 2020 to May 10, 2020. A tool MATLAB is used for numerical simulation and validation of analytical results. It is found that the coronavirus infection rate depends on the compactness of the population i.e., the infection rate in a particular region either increases or decreases depending on the population density of the region. The infection dynamics of coronavirus are also reduced when masks are worn and social distance is maintained.

A SARS-CoV-2 Fractional-Order Mathematical Model via the Modified Euler Method

This article develops a within-host viral kinetics model of SARS-CoV-2 under the Caputo fractional-order operator. We prove the results of the solution’s existence and uniqueness by using the Banach mapping contraction principle. Using the next-generation matrix method, we obtain the basic reproduction number. We analyze the model’s endemic and disease-free equilibrium points for local and global stability. Furthermore, we find approximate solutions for the non-linear fractional model using the Modified Euler Method (MEM). To support analytical findings, numerical simulations are carried out.

COVID-19 outbreak: a predictive mathematical study incorporating shedding effect.

In this paper, a modified SEIR epidemic model incorporating shedding effect is proposed to analyze transmission dynamics of the COVID-19 virus among different individuals’ classes. The direct impact of pathogen concentration over susceptible populations through the shedding of COVID-19 virus into the environment is investigated. Moreover, the threshold value of shedding parameters is computed which gives information about their significance in decreasing the impact of the disease. The basic reproduction number (R_{0}) is calculated using the next-generation matrix method, taking shedding as a new infection. In the absence of disease, the condition for the equilibrium point to be locally and globally asymptotically stable with R_{0}<1 are established. It has been shown that the unique endemic equilibrium point is globally asymptotically stable under the condition R_{0}>1. Bifurcation theory and center manifold theorem imply that the system exhibit backward bifurcation at R_{0}=1. The sensitivity indices of R_{0} are computed to investigate the robustness of model parameters. The numerical simulation is demonstrated to illustrate the results.

SEIR-FMi: A coronavirus disease epidemiological model based on intra-city movement, inter-city movement and medical resource investment

In this paper, we propose a coronavirus disease (COVID-19) epidemiological model called SEIR-FMi (Susceptible-Exposed-Infectious-Recovery with Flow and Medical investments) to study the effects of intra-city population movement, inter-city population movement, and medical resource investment on the spread of the COVID-19 epidemic. We theoretically derived the reproduction number of the SEIR-FMi model by using the next-generation matrix method and empirically simulate the individual impacts of population movement and medical resource investment on epidemic control. We found that intra- and inter-city population movements will increase the risk of epidemic spread, and the effect of inter-city population movement on low-risk areas is higher than that on high-risk areas. Increasing medical resource investment can not only speed up the recover rate of patients but also reduce the growth rate of infected cases and shorten the spread duration of the epidemic. We collected data on intra-city population movement, inter-city population movement, medical resource investment, and confirmed cases in the cities of Wuhan, Jingzhou, and Xiangyang, Hubei Province, China, from January 15 to March 15, 2020. Using the collected data, we validated that the proposed SEIR-FMi model performs well in simulating the spread of COVID-19 in the three cities. Meanwhile, this study confirms that three non-pharmaceutical interventions, namely community isolation, population mobility control, and medical resource aid, applied during the epidemic period are indispensable in controlling the spread of COVID-19 in the three cities.

The impact of official rumor-refutation information on the dynamics of rumor spread

The rapid development of the internet has resulted in a significant rise in online rumor spreading. Therefore, effectively reducing or eliminating the harm resulting from these rumors has become a new research focus. This paper proposed a variant of classic Susceptible–Infected–Recovered (SIR) disease transmission model to explore the influence of official rumor-refutation information quantity and content on rumor spread. A nonlinear differential equation model was constructed to describe the rumor propagation process and a next-generation matrix method employed to calculate rumor propagation threshold R0. Two types of equilibrium points are solved, and it is proven that when R0<1, the rumor-free equilibrium is globally asymptotically stable, and when R0>1, the rumor propagation equilibrium is locally asymptotically stable. A sensitivity analysis was performed on R0, and the impact of official rumor-refutation information content on the sensitivity of the R0 related parameters is considered. Then certain numerical values were assigned to the relevant parameters and the evolutionary mechanism simulated in different situations, with the theoretical results being verified using numerical simulation. It was found that the quantity of official rumor-refutation information played a positive but non-critical role in the rumor dissemination process, it even has a backfire effect in rumor controlling process under certain conditions, while the official rumor-refutation information content was a key factor affecting the dissemination threshold; therefore, official rumor-refutation information content needs to be improved to effectively control rumor spread. Finally, some policy suggestions that can contribute to rumor management in an emergency event were given.

Mathematical modeling to study the impact of immigration on the dynamics of the COVID-19 pandemic: A case study for Venezuela

We propose two different mathematical models to study the effect of immigration on the COVID-19 pandemic. The first model does not consider immigration, whereas the second one does. Both mathematical models consider five different subpopulations: susceptible, exposed, infected, asymptomatic carriers, and recovered. We find the basic reproduction number R0 using the next-generation matrix method for the mathematical model without immigration. This threshold parameter is paramount because it allows us to characterize the evolution of the disease and identify what parameters substantially affect the COVID-19 pandemic outcome. We focus on the Venezuelan scenario, where immigration and emigration have been important over recent years, particularly during the pandemic. We show that the estimation of the transmission rates of the SARS-CoV-2 are affected when the immigration of infected people is considered. This has an important consequence from a public health perspective because if the basic reproduction number is less than unity, we can expect that the SARS-CoV-2 would disappear. Thus, if the basic reproduction number is slightly above one, we can predict that some mild non-pharmaceutical interventions would be enough to decrease the number of infected people. The results show that the dynamics of the spread of SARS-CoV-2 through the population must consider immigration to obtain better insight into the outcomes and create awareness in the population regarding the population flow.

Mathematical analysis of the transmission dynamics of COVID-19 infection in the presence of intervention strategies

ABSTRACT The novel Coronavirus (COVID-19) infection has become a global public health issue, and it has been a cause for morbidity and mortality of more people throughout the world. In this paper, we investigated the impacts of vaccination, other protection measures, home quarantine with treatment, and hospital quarantine with treatment strategies simultaneously using a deterministic mathematical modelling approach. No one has considered these intervention strategies simultaneously in his/her modelling approach. We examined all the qualitative properties of the model such as the positivity and boundedness of the model solutions, the disease-free and endemic equilibrium points, the effective reproduction number using next-generation matrix method, local stabilities of equilibrium points using the Routh–Hurwitz method. Using the Centre Manifold criteria, we have shown the existence of backward bifurcation whenever the COVID-19 effective reproduction number is less than unity. Moreover, we have analysed both sensitivity and numerical simulation using parameter values taken from published literature. The numerical results show that the transmission rate is the most sensitive parameter we have to control. Also vaccination, other protection measures, home quarantine with treatment, and hospital quarantine with treatment have great effects to minimize the COVID-19 transmission in the community.

A Fractional-Order Epidemic Model with Quarantine Class and Nonmonotonic Incidence: Modeling and Simulations.

In any outbreak of infectious disease, the timely quarantine of infected individuals along with preventive measures strategy are the crucial methods to control new infections in the population. Therefore, this study aims to provide a novel fractional Caputo derivative-based susceptible-infected-quarantined-recovered-susceptible epidemic mathematical model along with a nonmonotonic incidence rate of infection. A new quarantined individual compartment is incorporated into the susceptible-infected-recovered-susceptible compartmental model by dividing the total population into four subpopulations. The nonmonotonic incidence rate of infection is considered as Monod–Haldane functional type to understand the psychological effects in the population. Qualitative analysis of the study shows that the model solutions are well-posed i.e., they are nonnegative and bounded in an attractive region. It is revealed that the model has two equilibria, namely, disease-free (DFE) and endemic (EE). The stability analysis of equilibria is investigated for local as well as global behaviors. Mathematical analysis of the model reveals that DFE is locally asymptotically stable when the basic reproduction number ({R}_{0}) is lower than one. The basic reproduction number {R}_{0} is computed using the next-generation matrix method. The existence of EE is shown and it is investigated that EE is locally asymptotically stable when {R}_{0}>1 under some appropriate conditions. Moreover, the global stability behaviors of DFE and EE are analyzed under some conditions using {R}_{0}. Finally, some numerical simulations are performed to interpret the theoretical findings.

Male Circumcision: A Means to Reduce HIV Transmission between Truckers and Female Sex Workers in Kenya

Kenya records over 1.5 million cases of HIV-infected people with a prevalence of 4.8% among adultsin 2019, ranking Kenya as the seventh-largest HIV population in the world. A recent study showed that55.9% of Kenyan truckers pay for sex in while 46.6% had a regular partner along their trucking route inaddition to a wife or girlfriend at home. The complexity in the sexual network of Truckers, which can be aconduit for the widespread of HIV, necessitated the need to better understand the dynamics of transmissionof HIV/AIDS between truckers and female sex workers. In this study, a model is formulated for HIV/AIDSdynamics along the Northern corridor highway in Kenya. The reproduction number, disease-free equilibriumand endemic equilibrium points were determined and their stabilities were also determined using the nextgenerationmatrix method. The disease-free equilibrium is stable when R0u &lt; 1, R0c &lt; 1 and R0f &lt; 1 whilethe endemic equilibrium point is stable when R0u &gt; 1, R0c &gt; 1 and R0f &gt; 1. It is found that circumcision canbe used as an intervention to minimize the infection of HIV among truckers and female sex workers.

Mathematical Modeling and Numerical Simulation for the Outbreak of COVID-19 Involving Loss of Immunity and Quarantined Class.

For the analysis of the recent deadly pandemic Sars-Cov-2, we constructed the mathematical model containing the whole population, partitioned into five different compartments, represented by SEIQR model. This current model especially contains the quarantined class and the factor of loss of immunity. Further, we discussed the stability of the SEIQR model (constructed on the basis of system of coupled differential equations). The basic reproduction that indicates the behavior of the disease is also estimated by the use of next-generation matrix method. Numerical simulation of this model is provided, the results are analyzed by theoretically strong numerical methods, and computationally known tool MATLAB Simulink is also used for visualization of the results. Validation of results by Simulink software and numerical methods shows that our model and adopted methodology are appropriate and accurate and could be used for further predictions on COVID-19. Our results suggest that the isolation of the active cases and strong immunization of patients or individuals play a major role to fight against the deadly Sars-Cov-2.

Correlation between mumps and meteorological factors in Xiamen City, China: A modelling study.

ObjectiveMumps is a seasonal infectious disease, always occurring in winter and spring. In this study, we aim to analyze its epidemiological characteristics, transmissibility, and its correlation with meteorological variables.MethodA seasonal Susceptible–Exposed–Infectious/Asymptomatic–Recovered model and a next-generation matrix method were applied to estimate the time-dependent reproduction number (Rt).ResultsThe seasonal double peak of annual incidence was mainly in May to July and November to December. There was high transmission at the median of Rt = 1.091 (ranged: 0 to 4.393). Rt was seasonally distributed mainly from February to April and from September to November. Correlations were found between temperature (Pearson correlation coefficient [r] ranged: from 0.101 to 0.115), average relative humidity (r = 0.070), average local pressure (r = -0.066), and the number of new cases. In addition, average local pressure (r = 0.188), average wind speed (r = 0.111), air temperature (r ranged: -0.128 to -0.150), average relative humidity (r = -0.203) and sunshine duration (r = -0.075) were all correlated with Rt.ConclusionA relatively high level of transmissibility has been found in Xiamen City, leading to a continuous epidemic of mumps. Meteorological factors, especially air temperature and relative humidity, may be more closely associated with mumps than other factors.

Optimal control design incorporating vaccination and treatment on six compartment pandemic dynamical system

In this paper, a mathematical model of the COVID-19 pandemic with lockdown that provides a more accurate representation of the infection rate has been analyzed. In this model, the total population is divided into six compartments: the susceptible class, lockdown class, exposed class, asymptomatic infected class, symptomatic infected class, and recovered class. The basic reproduction number (R0) is calculated using the next-generation matrix method and presented graphically based on different progression rates and effective contact rates of infective individuals. The COVID-19 epidemic model exhibits the disease-free equilibrium and endemic equilibrium. The local and global stability analysis has been done at the disease-free and endemic equilibrium based on R0. The stability analysis of the model shows that the disease-free equilibrium is both locally and globally stable when R0<1, and the endemic equilibrium is locally and globally stable when R0>1 under some conditions. A control strategy including vaccination and treatment has been studied on this pandemic model with an objective functional to minimize. Finally, numerical simulation of the COVID-19 outbreak in India is carried out using MATLAB, highlighting the usefulness of the COVID-19 pandemic model and its mathematical analysis.

A noninteger order SEITR dynamical model for TB

This research paper designs the noninteger order SEITR dynamical model in the Caputo sense for tuberculosis. The authors of the article have classified the infection compartment into four different compartments such as newly infected unrecognized individuals, diagnosed patients, highly infected patients, and patients with delays in treatment which provide better detail of the TB infection dynamic. We estimate the model parameters using the least square curve fitting and demonstrate that the proposed model provides a good fit to tuberculosis confirmed cases of India from the year 2000 to 2020. Further, we compute the basic reproduction number as Re _{0} approx 1.73 of the model using the next-generation matrix method and the model equilibria. The existence and uniqueness of the approximate solution for the SEITR model is validated using the generalized Adams–Bashforth–Moulton method. The graphical representation of the fractional order model is given to validate the result using the numerical simulation. We conclude that the fractional order model is more realistic than the classical integer order model and provide more detailed information about the real data of the TB disease dynamics.

Spatial dynamics of dengue fever spreading for the coexistence of two serotypes with an application to the city of São Paulo, Brazil

Background and objective:Dengue fever is a disease in which individuals’ spatial distribution and Aedes aegypti mosquitoes breeding places are important factors for the disease dynamics. Typically urban, dengue is a problem for least developed countries due to the ineffectiveness in controlling the vector and disorderly urbanization processes. The result is a composition of urban sanitation problems and areas with high demographic densities and intense flows of people. This paper explores the spatial distribution of vector breeding places to evaluate introducing a new dengue serotype to a population at equilibrium for a pre-existing serotype. The paper’s objective is to analyze the spatial dynamics of dengue using variations of the basic reproduction number.Methods:A model based on probabilistic cellular automata is proposed to permitting the necessary flexibility to consider some spatial distributions of vector breeding places. Then, ordinary differential equations are used as a mean-field approach of the model, and the basic reproduction number (R0) is derived considering the next-generation matrix method. A spatial approach for R0 is also proposed, and the model is tested in a neighbourhood from the city of São Paulo, Brazil, to examine the potential risks of vector breeding cells distribution.Results:The results indicated that the more spread out these places, the higher are the values of R0. When the model is applied to a neighbourhood in São Paulo, residential areas may boost the infections and must be under public vigilance to combat vector breeding sites.Conclusions:Considering the mean-field approximation of the cellular automata model by ordinary differential equations, the basic reproduction number derived returned an estimative of the disease dynamics in the population. However, the spatial basic reproduction number was more assertive in showing areas with a higher disease incidence. Moreover, the model could be easily adapted to be used in real maps enabling simulations closer to real problems.