Next Generation Matrix Method Research Articles

Modelling and analysis of fractional-order vaccination model for control of COVID-19 outbreak using real data.

In this paper, we construct the SV1V2EIR model to reveal the impact of two-dose vaccination on COVID-19 by using Caputo fractional derivative. The feasibility region of the proposed model and equilibrium points is derived. The basic reproduction number of the model is derived by using the next-generation matrix method. The local and global stability analysis is performed for both the disease-free and endemic equilibrium states. The present model is validated using real data reported for COVID-19 cumulative cases for the Republic of India from 1 January 2022 to 30 April 2022. Next, we conduct the sensitivity analysis to examine the effects of model parameters that affect the basic reproduction number. The Laplace Adomian decomposition method (LADM) is implemented to obtain an approximate solution. Finally, the graphical results are presented to examine the impact of the first dose of vaccine, the second dose of vaccine, disease transmission rate, and Caputo fractional derivatives to support our theoretical results.

MATHEMATICAL MODEL OF TIPHOID FEVER SPREAD USING SATURATED INCIDENCE RATE

Typhoid fever is a bacterial infectious disease caused by Salmonella typhi, transmitted through food or water contaminated by feces or urine of people whowas infected by Salmonella typhi. In this research , mathematical model of typhoid fever with satura t ed incidence rate that when infected population increase, people more aware and inhibit the spread of the disease. According to model analysis, there was two equilibrium point, free disease and endemic. The basic reproduction number has been computed using next-generation matrix method . We have shown that the disease free equilibrium point of the model is globally asymptotic stable when basic reproduction number is less than unity and endemic equilibrium point is locally asymptotic stable when basic reproduction number is greater than unity. Numerical simulation was shown around e q uilibrium point. The addition of the saturated incidence rate inhibits the spread of typh oid . When the basic reproduction number is less than 1, typh oid will disappear and if the basic reproduction number is greater than 1, typh oid will remain.

Impact of media coverage on a fractional-order SIR epidemic model

In this work, we formulated and analyzed a fractional-order epidemic model of infectious disease (such as SARS, 2019-nCoV and COVID-19) concerning media effect. The model is based on classical susceptible-infected-recovered (SIR) model. Basic properties regarding positivity, boundedness and non-negative solutions are discussed. Basic reproduction number [Formula: see text] of the system has been calculated using next-generation matrix method and it is seen that the disease-free equilibrium is locally as well as globally asymptotically stable if [Formula: see text], otherwise unstable. The existence of endemic equilibrium point is established using the Lambert W function. The condition for global stability has been derived. Numerical simulation suggests that fractional order and media have a large effect on our system dynamics. When media impact is stronger enough, our fractional-order system stabilizes the oscillation.

Global dynamics of ILSR rumor spreading model with general nonlinear spreading rate in multi-lingual environment

In this paper, under the multi-lingual environment, a new ILSR rumor spreading model is firstly proposed, combining with the incubation mechanism, cross-transmitted mechanism and general nonlinear spreading rate. Firstly, the basic reproduction number are obtained by using the next-generation matrix method. Secondly, the global stability of ILSR rumor spreading model are analysed in the bi-lingual environment via using Lyapunov functions and graph theories. Further, the results are extended to the multi-lingual environment. Moreover, an optimal control method is proposed by using Pontryagin’s maximum principle, which can achieve high control effect with low cost. Finally, the correctness of theoretical results is verified by numerical simulation of several models with classical nonlinear spreading rate, such as the saturated spreading rate, the non-monotone spreading rate and so on.

Role of pine wilt disease based on optimal control strategy at multiple scales: A case study of Korea

Pine wilt disease is one of the most serious conifer diseases: this is because pine trees contribute greatly to the economy and domestic wealth in Korea. Our model of this disease is based on the parametrisation of infectious pine trees in Korea for the period of 2010-2019. The model captures the growth in case onsets and the estimated results are almost compatible with the reported data. To control the spread of this disease to the whole pine tree community, we found a threshold parameter called 'basic reproduction number' using the nextgeneration matrix method. During the analysis of the model, equilibrium points were first computed: there are two points -one has no disease class and other has all the disease classes. For the global behaviour of the mathematical model of these two points, Lypunove functional theory was used for disease-free and endemic equilibrium. Sensitivity analysis was performed to observe the relative importance of these parameters to the transmission and prevalence of pine wilt disease. To control the dissemination of the disease, we formulated an optimal control problem. Strategies used to control this disease were based on the consequences of the significant effects of the estimated parameters on the basic reproduction number. We re-examined the mathematical system to determine the agreement between numerically and analytically calculated outcomes. After analysing the problem numerically, we can discern that the numerical findings support the results calculated analytically.

Mathematical Modeling and Analysis of the Dynamics of Chikungunya in Bangladesh

Chikungunya is one of the major public health problems in Bangladesh and the effect of the disease is more virulent over the country. In this paper, a compartmental mathematical model has been proposed that describes the transmission of chikungunya disease in Bangladesh. Nonlinear incidence rate is considered for disease transmission and the system of nonlinear differential equations is developed to represent the model. Furthermore, the treatment term has been introduced in the model. The basic reproduction number, which is a biological threshold parameter for this disease, has been computed for the model by the method of next generation matrix. Disease free and endemic equilibrium points are calculated and the stability of the model has been analyzed using the basic reproduction number. The global stability has been established using the Lyapunov function theory. In this study, we have calculated the numerical value of the basic reproduction number and performed numerical simulations based on the real data collected from several health institutes of Bangladesh. Both analytical and numerical results provide a pattern about the dynamics of the disease in Bangladesh. Based on the effects, some important policies and insights are provided that might help in reducing the number of infected cases of chikungunya.

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

The SARS-CoV-2 continues to spread across the world. During this COVID-19 pandemic, several variants of the SARS-CoV-2 have been found. Some of these new variants like the VOC-202012/01 of lineage B.1.1.7 or the most recently B.1.617 emerging in India have a higher infectiousness than those previously prevalent. We propose a mathematical model based on ordinary differential equations to investigate potential consequences of the appearance of a new more transmissible SARS-CoV-2 strain in a given region. The proposed mathematical model incorporates the presymptomatic and asymptomatic subpopulations in addition to the more usual susceptible, exposed, infected, and recovered subpopulations. This is important from a realistic point of view since it has been found recently that presymptomatic and asymptomatic individuals are relevant spreaders of the SARS-CoV-2. Using the next-generation matrix method, we find the basic reproduction number, {mathcal {R}}_{0}, an important threshold parameter that provides insight regarding the evolution and outcome of a certain instance of the COVID-19 pandemic. The local and global stability of system equilibria are also presented. In particular, for the global stability we construct a Lyapunov functional and use the LaSalle invariant principle to prove that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. On the other hand, if {mathcal {R}}_{0}>1 the endemic equilibrium is globally asymptotically stable. Finally, we present numerical simulations to numerically support the analytic results and to show the impact of the introduction of a new more contagious SARS-CoV-2 variant in a population.

Dynamics of Tuberculosis (TB) with Drug Resistance to First-Line Treatment and Leaky Vaccination: A Deterministic Modelling Perspective.

A deterministic model was formulated and employed in the analysis of the dynamics of tuberculosis with a keen emphasis on vaccination and drug resistance as the first line of treatment. It was assumed that some of the susceptible population were vaccinated but with temporal immunity. This is due to the fact that vaccines do not confer permanent immunity. Moreover, part of the infected individual after treatment grows resistance to the drug. Infective immigrants were also considered to be part of the population. The basic reproductive number for the model is estimated using the next-generation matrix method. The equilibrium points of the TB model and their local and global stability were determined. It was established that if the basic reproductive number was less than unity (R0 < 1), then the disease free equilibrium is stable and unstable if R0 > 1. Furthermore, we investigated the optimal prevention, treatment, and vaccination as control measures for the disease. As the objective functional was optimised, there have been a significant reduction in the number of infections and an increase in the number of recovery. The best control measure in combating tuberculosis infections is prevention and vaccination of the susceptible population.

Optimal Control and Cost Effectiveness Analysis of SIRS Malaria Disease Model with Temperature Variability Factor

In this study, we proposed and analyzed the optimal control and cost-effectiveness strategies for malaria epidemics model with impact of temperature variability. Temperature variability strongly determines the transmission of malaria. Firstly, we proved that all solutions of the model are positive and bounded within a certain set with initial conditions. Using the next-generation matrix method, the basic reproductive number at the present malaria-free equilibrium point was computed. The local stability and global stability of the malaria-free equilibrium were depicted applying the Jacobian matrix and Lyapunov function respectively when the basic reproductive number is smaller than one. However, the positive endemic equilibrium occurs when the basic reproductive number is greater than unity. A sensitivity analysis of the parameters was conducted; the model showed forward and backward bifurcation. Secondly, using Pontryagin’s maximum principle, optimal control interventions for malaria disease reduction are described involving three control measures, namely use of insecticide-treated bed nets, treatment of infected humans using anti-malarial drugs, and indoor residual insecticide spraying. An analysis of cost-effectiveness was also conducted. Finally, based on the simulation of different control strategies, the combination of treatment of infected humans and insecticide spraying was proved to be the most efficient and least costly strategy to eradicate the disease.

Estimating the HIV Effective Reproduction Number in the United States and Evaluating HIV Elimination Strategies.

The reproduction number is a fundamental epidemiologic concept used to assess the potential spread of infectious diseases and whether they can be eliminated. We estimated the 2017 United States HIV effective reproduction number, Re, the average number of secondary infections from an infected person in a partially infected population. We analyzed the potential effects on Re of interventions aimed at improving patient flow rates along different stages of the HIV care continuum. We also examined these effects by individual transmission groups. We used the HIV Optimization and Prevention Economics (HOPE) model, a compartmental model of disease progression and transmission, and the next-generation matrix method to estimate Re. We then projected the impact of changes in HIV continuum-of-care interventions on the continuum-of-care flow rates and the estimated Re in 2020. United States. The HOPE model simulated the sexually active US population and persons who inject drugs, aged 13 to 64 years, which was stratified into 195 subpopulations by transmission group, sex, race/ethnicity, age, male circumcision status, and HIV risk level. The estimated value of Re in 2017 and changes in Re in 2020 from interventions affecting the continuum-of-care flow rates. Our estimated HIV Re in 2017 was 0.92 [0.82, 0.94] (base case [min, max across calibration sets]). Among the interventions considered, the most effective way to reduce Re substantially below 1.0 in 2020 was to maintain viral suppression among those receiving HIV treatment. The greatest impact on Re resulted from changing the flow rates for men who have sex with men (MSM). Our results suggest that current prevention and treatment efforts may not be sufficient to move the country toward HIV elimination. Reducing Re to substantially below 1.0 may be achieved by an ongoing focus on early diagnosis, linkage to care, and sustained viral suppression especially for MSM.

Stability Analysis of a Mathematical Model SI$_{u}$I$_{a}$QR for COVID-19 with the Effect of Contamination Control (Filiation) Strategy

In this study, using a system of delay nonlinear ordinary differential equations, we introduce a new compartmental epidemic model considered the effect of filiation (contamination) control strategy to the spread of Covid-19. Firstly, the formulation of this new $SI_{u}I_{a}QR$ epidemic model with delay process and the parameters arised from isolation and filiation is formed. Then the disease-free and endemic equilibrium points of the model is obtained. Also, the basic reproduction number $\mathcal{R}_{0}$ is found by using the next-generation matrix method, and the results on stabilities of the disease-free and endemic equilibrium points are investigated. Finally some examples are presented to show the effect of filiation control strategy.

Stability analysis of SCPUR mathematical model for the spread of COVID-19 (corona virus disease-19)

This article discusses SCPUR mathematical model for the spread of COVID-19 using data of people with COVID-19 in Makassar City. In this model, the population class is divided into five classes: susceptible, asymptomatic infectious, reported symptomatic infectious, unreported symptomatic infectious, and recovered classes. The proportion of body immunity to the increase of infected individuals, the proportion of large-scale social restrictions, and the proportion of quarantine as a healing process are also added. The research begins by determining the equilibrium point of the model, namely the disease-free equilibrium point and the endemic equilibrium point. Then, the stability test is carried out using linearization method and the eigenvalues are determined. The value of the basic reproduction number is obtained using next-generation matrix method, where the initial state of the basic reproduction number value R0>1 mean COVID-19 will still exist in Makassar City. Treatment is carried out so that R0<1 which means Makassar City will be free of COVID-19.

Global dynamics of a tuberculosis model with sensitivity of the smear microscopy

Sputum smear microscopy and chest X-ray are the key TB diagnosis methods available in resource-constrained health settings of many developing countries worldwide. The test has moderate sensitivity towards the detection of pulmonary tuberculosis (PTB) cases. However, the undetected cases are also capable of transmitting the disease with a reduced transmission possibility. In this work, a five-dimensional compartmental model is formulated considering the infectivity of both smear-positive and negative individuals. The next-generation matrix method yields the expression of basic reproduction number R0. The global asymptotic stability of the disease-free equilibrium point ξ0 for R0<1 and that of endemic equilibrium point for R0>1 are established with suitably constructed Lyapunov functions. The sensitivity indices of the associated parameters of R0 are obtained with a suitable choice of parameter values. It has been found that neglecting the transmission capacity of smear-negative individuals underestimates the value of R0, whereas ignoring the smear-negative compartment overestimates the same quantity.

Modeling and optimal control analysis of malaria epidemic in the presence of temperature variability

In this paper, we propose and analyze a nonlinear deterministic malaria disease model for the impact of temperature variability on malaria epidemics. Firstly, we analyzed the invariant region and the positivity solution of the model. The basic reproduction number with respect to disease free-equilibrium is calculated by the next-generation matrix method. The local stability and global stability of the equilibrium points are shown using the Routh–Hurwitz criterion and the Lyapunov function, respectively. A disease-free equilibrium point is globally asymptotically stable if the basic reproduction number is less than one and endemic equilibrium exists otherwise. Moreover, we have shown the sensitivity analysis of the basic reproduction number and the model exhibits forward and backward bifurcation. Secondly, we apply the optimal control theory to describe the model with incorporates three controls, namely using treated bed nets, treatment of infected with anti-malaria drugs and for vector killing using insecticide spray strategy. Pontraygin’s maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the simulation result of optimal control problem and analysis of cost-effectiveness show that a combination of using treated bed nets and treatment is the most effective and least-cost strategy to prevent the malaria disease.

Stability Analysis of Fractional SIR Model Related to Delay in State and Control Variables

The study of a nonlinear mathematical fractional SIR (Susceptible - Infected - Recovered) epidemiological model related to the delay in state and control variables in terms of time is the focus of this paper. The existence of a bounded solution for the fractional SIR epidemic model has been demonstrated, and it is unique. A new set of infection-free equilibrium points has been discovered, and their local stability has been investigated. In addition, using the next-generation matrix method, the basic reproductive number Ro was calculated

Analysis of epidemic vaccination strategies on heterogeneous networks: Based on SEIRV model and evolutionary game

Nowadays, vaccination is the most effective way to control the epidemic spreading. In this paper, an epidemic SEIRV (susceptible-exposed-infected-removed -vaccinated) model and an evolutionary game model are established to analyze the difference between mandatory vaccination method and voluntary vaccination method on heterogeneous networks. Firstly, we divide the population into four categories, including susceptible individuals, exposed individuals, infected individuals and removed individuals. Based on the mean field approximation theory, differential equations are developed to characterize the changes of the proportions of the four groups over time under mandatory vaccination. Then through the analysis of the differential equations, the disease-free equilibrium point (DFE) and the endemic disease equilibrium point (EDE) are obtained. Also, the basic reproduction number is obtained by the next-generation matrix method and the stability analysis of the equilibrium points is performed. Next, by considering factors such as vaccination cost, treatment cost and government subsidy rate, differential equations are established to represent the change of vaccination rate over time. By analyzing the final vaccination coverage rate, we can get the minimum vaccination cost to make infectious disease disappear. Finally, the Monte Carlo method is used for numerical simulation to verify the results obtained from the theoretical analysis. Using the SARS-Cov-2 pandemic data from Wuhan, China, the experimental results show that when the effectiveness rate of vaccination is 0.75, the vaccination cost is not higher than 0.886 so that the vaccination strategy can be spread among the population. If mandatory vaccination is adopted, the minimum vaccination rate is 0.146.

Mathematical modelling of the COVID-19 pandemic with demographic effects

In this paper, a latent infection susceptible–exposed–infectious–recovered model with demographic effects is used to understand the dynamics of the COVID-19 pandemics. We calculate the basic reproduction number ({R}_{0}) by solving the differential equations of the model and also using next-generation matrix method. We also prove the global stability of the model using the Lyapunov method. We showed that when the {R}_{0}<1 or {R}_{0}le 1 and {R}_{0}>1 or {R}_{0}ge 1 the disease-free and endemic equilibria asymptotic stability exist theoretically. We provide numerical simulations to demonstrate the detrimental impact of the direct and latent infections for the COVID-19 pandemic.

Mathematical Modeling and Analysis of Khat-Chewing Dynamics

Khat is a green leaf and greenish plant where its branches and leaves are chewed to discharge liquid having active chemicals that change the user’s mood. The purpose of this article is to develop and analyze a mathematical model that can be used to understand the dynamics of chewing Khat. The proposed model monitors the dynamics of five compartments, namely, a group of people who do not chew Khat, designated as N t ; a group of people who are surrounded by Khat chewers but do not chew at present and may chew Khat in the future, denoted this as Σ t ; a group of people who chew Khat, which is represented in C t ; a group of people contains individuals who consumed Khat quite temporarily for social, spiritual, and recreational purposes, and we describe this group in T t ; and a group of people those who constantly chew Khat, and they are denoted by H t . We determined the Khat chewing generation number R c 0 using the next-generation matrix method, and we have examined the biological meaningfulness, mathematical wellposedness, and stability of both Khat chewing-free and Khat chewing-present equilibrium points of the model analytically. Numerical simulations were presented by solving our dynamical system using Matlabode45 to check the analytical results by considering parameter estimations. The results of this study show that, for R c 0 = .00039 , the Khat chewing-free equilibrium point is stable, and it is unstable for R c 0 = 1.194 , and the Khat chewing-present equilibrium point is stable if R c 0 = 1.194 , and it is unstable if R c 0 = .00039 . The stability of both equilibrium points implies that, for a high rate of conversion from non-Khat chewer to exposed groups ρ , the inflow of an insignificant number of Khat chewers to the community produces a significant number of Khat chewers , and if the return back from Khat chewing to the exposed group because of socio-economic, environmental, and religious influences α 2 grows exponentially, the inflow of an insignificant number of Khat chewers to the community produces an insignificant number of Khat chewers. It is found that increasing the rate of conversion from non-Khat chewer to exposed groups ρ makes the disease eradication more challenging. We, therefore, strongly urge religious leaders, social committee leaders, elders, and health experts to teach their followers to reduce their Khat-chewing habits.

The effect of quarantine and treatment in COVID-19 transmission: from mathematical modeling perspective

Corona-virus disease (COVID-19) is caused by the novel-virus (SARS-COV2). This disease mainly targets human respiratory system. COVID-19 (Coronavirus) has affected day to day life and is slowing down the global economy. This pandemic has affected thousands of peoples, who are either sick or are being killed due to the spread of this disease. In this paper we developed an eight compartmental model with quarantine and treatment of COVID-19. After proposing the model, we analysed the qualitative behaviors of the model, like the disease free and endemic equilibrium points and their stability analysis. Moreover, we obtained the basic reproduction number using next-generation matrix method and we performed the sensitivity analysis to identify the most affecting parameters in terms of disease control and spread. To investigate the detail effect of each major parameters, we performed numerical simulation. We obtained that using both quarantine and treatment is best way in combating COVID-19 in the community. Therefore, stakeholders and policy makers of a government should use both quarantine and treatment simultaneously in fighting the pandemic from the population.

Exploring the linkages between HIV/AIDS and poverty: insights through a mathematical model

Poverty continues to impact people living with HIV/AIDS negatively and it drives HIV/AIDS epidemics mostly in poor resource countries such as Malawi. In this paper, a mathematical model for HIV/AIDS that provides linkages between HIV/AIDS and poverty is formulated and analysed using nonlinear ordinary differential equations. The theoretical evaluations identified four equilibria namely; an equilibrium in which there is no disease and per capita income is zero, disease-free, an equilibrium in which per capita income is zero and endemic. An epidemic threshold value, RM, called the reproduction number is determined using the next-generation matrix method. The analysis of the model shows that the disease-free and endemic equilibria are globally stable for RM 1 respectively. Sensitivity analysis of RM is performed in order to determine how the parameters affect the dynamics of HIV/AIDS epidemics. Numerical simulations are done to show the role of some key model parameters on HIV/AIDS as well as to verify some analytical results. The results of the model suggest that income has the propensity to control the spread of HIV/AIDS because it has the capacity of decreasing unemployed people's vulnerability to sexually transmitted diseases such as HIV/AIDS.