Next Generation Matrix Method Research Articles

Dynamic analysis and optimal control of a novel fractional-order 2I2SR rumor spreading model

In this paper, a novel fractional-order 2I2SR rumor spreading model is investigated. Firstly, the boundedness and uniqueness of solutions are proved. Then the next-generation matrix method is used to calculate the threshold. Furthermore, the stability of rumor-free/spreading equilibrium is discussed based on fractional-order Routh–Hurwitz stability criterion, Lyapunov function method, and invariance principle. Next, the necessary conditions for fractional optimal control are obtained. Finally, some numerical simulations are given to verify the results.

Epidemiological characteristics and transmission dynamics of the COVID-19 outbreak in Hohhot, China: a time-varying SQEIAHR model analysis.

On September 28, 2022, the first case of Omicron subvariant BF.7 was discovered among coronavirus disease 2019 (COVID-19) infections in Hohhot, China, and then the epidemic broke out on a large scale during the National Day holiday. It is imminently necessary to construct a mathematical model to investigate the transmission dynamics of COVID-19 in Hohhot. In this study, we first investigated the epidemiological characteristics of COVID-19 cases in Hohhot, including the spatiotemporal distribution and sociodemographic distribution. Then, we proposed a time-varying Susceptible-Quarantined Susceptible-Exposed-Quarantined Exposed-Infected-Asymptomatic-Hospitalized-Removed (SQEIAHR) model to derive the epidemic curves. The next-generation matrix method was used to calculate the effective reproduction number (Re). Finally, we explored the effects of higher stringency measures on the development of the epidemic through scenario analysis. Of the 4,889 positive infected cases, the vast majority were asymptomatic and mild, mainly concentrated in central areas such as Xincheng District. People in the 30-59 age group primarily were affected by the current outbreak, accounting for 53.74%, but females and males were almost equally affected (1.03:1). Community screening (35.70%) and centralized isolation screening (26.28%) were the main ways to identify positive infected cases. Our model predicted the peak of the epidemic on October 6, 2022, the dynamic zero-COVID date on October 15, 2022, a number of peak cases of 629, and a cumulative number of infections of 4,963 (95% confidential interval (95%CI): 4,692 ~ 5,267), all four of which were highly consistent with the actual situation in Hohhot. Early in the outbreak, the basic reproduction number (R0) was approximately 7.01 (95%CI: 6.93 ~ 7.09), and then Re declined sharply to below 1.0 on October 6, 2022. Scenario analysis of higher stringency measures showed the importance of decreasing the transmission rate and increasing the quarantine rate to shorten the time to peak, dynamic zero-COVID and an Re below 1.0, as well as to reduce the number of peak cases and final affected population. Our model was effective in predicting the epidemic trends of COVID-19, and the implementation of a more stringent combination of measures was indispensable in containing the spread of the virus.

Within-host models of dengue virus transmission with immune response

AbstractDengue fever is an infectious viral fever. The complex behavior of the virus within the body can be explained through mathematical models to understand the virus’s dynamics. We propose two different with-in host models of dengue virus transmission with humoral immune response. The proposed models differ from one another because one of the models assumes that newly formed viruses infect healthy cells again. To understand the dynamics of the proposed models, we perform a comparative study of stability analysis, numerical simulation, and sensitivity analysis. The basic reproduction number (BRN) of the two models is computed using next-generation matrix method. The local stability (l.s) analysis is discussed using the linearization method. The Lyapunov’s direct method is used to check the global stability (g.s) of the models. It has been found that both the equilibrium states for both the models, namely, virus-free equilibrium state and endemic equilibrium state, are globally stable, based on the value of BRN. Results show the influence of immune response on the cell dynamics and virus particles. The virus neutralization rate by antibodies and rate that affects the antibody growth are highly sensitive for the two models. Optimal control is applied to explore the possible control strategies to prevent virus spread in the host system. It is evident from the results that the strategy to administrate antibiotic drugs and home remedies slow down the virus spread in the host.

Effects of Community Connectivity on the Spreading Process of Epidemics.

Community structure exists widely in real social networks. To investigate the effect of community structure on the spreading of infectious diseases, this paper proposes a community network model that considers both the connection rate and the number of connected edges. Based on the presented community network, a new SIRS transmission model is constructed via the mean-field theory. Furthermore, the basic reproduction number of the model is calculated via the next-generation matrix method. The results reveal that the connection rate and the number of connected edges of the community nodes play crucial roles in the spreading process of infectious diseases. Specifically, it is demonstrated that the basic reproduction number of the model decreases as the community strength increases. However, the density of infected individuals within the community increases as the community strength increases. For community networks with weak strength, infectious diseases are likely not to be eradicated and eventually will become endemic. Therefore, controlling the frequency and range of intercommunity contact will be an effective initiative to curb outbreaks of infectious diseases throughout the network. Our results can provide a theoretical basis for preventing and controlling the spreading of infectious diseases.

Stability and optimal control analysis for studying the transmission dynamics of a fractional-order MSV epidemic model

In this research paper, we study the transmission dynamics of the maize streak virus disease in agriculture using fractional calculus. Our fractional-order model (FOM) consists of fractional differential equations within Caputo fractional derivative (CFD). Further, we investigate the mathematical analysis of this FOM such as the positivity and boundedness of the projected solutions, compute the control reproduction number (Rc) by the next-generation matrix method, examine the local stability by using Routh–Hurwitz conditions and global stability by using fractional LaSalle’s invariance principle for all possible equilibrium points. A sensitivity analysis is conducted on the parameters integrated into Rc to evaluate how accurately the parameter estimation fits. Additionally, we present further numerical results on the FOM to demonstrate the obtained analytical results. Then, we formulate the fractional optimal control problem (FOCP) depending on the suggested FOM with three control efforts u1, u2 and u3 called respectively, prevention, quarantine and insecticide chemical. Also, we study this FOCP analytically to derive the fractional necessary optimality conditions (NOCs) based on a kind of Pontryagin’s maximum principle. After that, we solve these fractional NOCs numerically where the state and co-state variables are subject to the left CFD. We present various strategies with different combinations of suggested controls to examine the best strategy for decreasing the transmission of MSV in maize plants, where each one of these strategies is able to reduce the maize streak disease (MSD) at the period of control time. We perform simulations of outcomes using MATLAB by developing the forward–backward sweep method utilizing the fractional Euler method. Moreover, some figures are presented to show the impact of both control interventions, change in weight factors and fractional order within the suggested strategies. We can observe that the fractional order plays a significant role in limiting the spread of MSD where the obtained results illustrated that the MSD can be reduced through the suggested FOM, which has a notable influence on plant epidemiology.

A Fractional-Order Density-Dependent Mathematical Model to Find the Better Strain of Wolbachia

The primary objective of the current study was to create a mathematical model utilizing fractional-order calculus for the purpose of analyzing the symmetrical characteristics of Wolbachia dissemination among Aedesaegypti mosquitoes. We investigated various strains of Wolbachia to determine the most sustainable one through predicting their dynamics. Wolbachia is an effective tool for controlling mosquito-borne diseases, and several strains have been tested in laboratories and released into outbreak locations. This study aimed to determine the symmetrical features of the most efficient strain from a mathematical perspective. This was accomplished by integrating a density-dependent death rate and the rate of cytoplasmic incompatibility (CI) into the model to examine the spread of Wolbachia and non-Wolbachia mosquitoes. The fractional-order mathematical model developed here is physically meaningful and was assessed for equilibrium points in the presence and absence of disease. Eight equilibrium points were determined, and their local and global stability were determined using the Routh–Hurwitz criterion and linear matrix inequality theory. The basic reproduction number was calculated using the next-generation matrix method. The research also involved conducting numerical simulations to evaluate the behavior of the basic reproduction number for different equilibrium points and identify the optimal CI value for reducing disease spread.

Threshold quantities and Lyapunov functions for ordinary differential equations epidemic models with mass action and standard incidence functions

This paper presents a novel algebraic method for the construction of Lyapunov functions to study global stability of the disease-free equilibrium points of deterministic epidemic ordinary differential equation models with mass action and standard incidence functions. The method is named as Jacobian-Determinant method. In our method, a direct algebraic procedure that also relies only on determinant of the Jacobian matrix of the infected subsystem is developed to determine a threshold quantity, R0′ akin to the basic reproduction number, R0 of such class of models. The developed technique is applied on a wide variety of models to construct Lyapunov functions to study the global stability of the infection-free critical points. Further, implementation of our method reveals that the threshold quantity is the same as (or the square) of the basic reproduction numbers as obtained using the next-generation matrix method. It is further observed that even for models that do not use the standard or mass action incidence, the threshold quantity is still related to the basic reproduction numbers as obtained with the next-generation matrix method.

Vector-Reservoir Transmission in A Japanese Encephalitis Model

In this paper, a model for characterizing the dynamics of vector-borne diseases is put out, emphasizing Japanese encephalitis. The susceptible-infectious-recovered (SIR) model for the host population and the susceptible-infectious (SI) model for the vector and reservoir populations are used to examine the role of host-vector-reservoir dynamics and their interplay. The standard incidence rate represents the probability of an actual disease contact. The model has two equilibrium points: an endemic equilibrium point that only exists under specific circumstances and a disease-free equilibrium point that always exists. The stability analysis of the model’s equilibrium point has been established. The basic reproduction number is calculated using the next-generation matrix method. A sensitivity analysis on models supported by numerical simulations is provided to demonstrate the critical parameter that affects the spread of disease. Our findings indicate that vector-reservoir transmission is the primary cause of endemic. Controlling vector-reservoir transmission lowers the likelihood of human infection and creates disease-free settings.

Stability analysis of a fractional‐order SEIR epidemic model with general incidence rate and time delay

In this paper, we investigate the qualitative behavior of a class of fractional SEIR epidemic models with a more general incidence rate function and time delay to incorporate latent infected individuals. We first prove positivity and boundedness of solutions of the system. The basic reproduction number of the model is computed using the method of next generation matrix, and we prove that if , the healthy equilibrium is locally asymptotically stable, and when , the system admits a unique endemic equilibrium which is locally asymptotically stable. Moreover, using a suitable Lyapunov function and some results about the theory of stability of differential equations of delayed fractional‐order type, we give a complete study of global stability for both healthy and endemic steady states. The model is used to describe the COVID‐19 outbreak in Algeria at its beginning in February 2020. A numerical scheme, based on Adams–Bashforth–Moulton method, is used to run the numerical simulations and shows that the number of new infected individuals will peak around late July 2020. Further, numerical simulations show that around 90% of the population in Algeria will be infected. Compared with the WHO data, our results are much more close to real data. Our model with fractional derivative and delay can then better fit the data of Algeria at the beginning of infection and before the lock and isolation measures. The model we propose is a generalization of several SEIR other models with fractional derivative and delay in literature.

Modeling the effect of Wolbachia to control malaria transmission

Female Anopheles mosquitoes are the primary vectors responsible for malaria transmission. The control of malaria is essential for humankind. In this study, we introduce a nonlinear deterministic model to control malaria disease by deploying Wolbachia bacteria into the Anopheles mosquitoes. Further, we study the effect of Wolbachia on temperature variation in tropical and subtropical regions of the world. We analyze the model for its positivity and boundedness with an initial condition in a specific set so that the model is well-defined mathematically and meaningful epidemiologically. We find the malaria-free equilibrium points of the model and analyze the local stability of all the equilibrium points. We calculate the basic reproduction number by the next-generation matrix method, which shows the stability of malaria-free equilibrium. Then we explore malaria present endemic equilibrium in both cases: Wolbachia-infected and Wolbachia-free possibilities. We observe that the number of infections is less in the Wolbachia-infected equilibrium case, and we find that Wolbachia can reduce malaria transmission in all temperature conditions. The sensitivity analysis of all different parameters is also calculated. We demonstrate the optimum temperature (from 22 °C to 28 °C) by numerical simulation where the malaria exposure and transmission rate is high. We can apply the work in all subtropical and tropical regions to control malaria transmission worldwide.

Mathematical Model of Foot and Mouth Disease Considering Vaccination Disinfection And Early Quarantine

Foot and Mouth Disease (FMD) is a highly contagious disease that can infect all cloven hoofed animals. FMD is one of the most important infectious viral diseases in the world. This disease is responsible for international losses in livestock production and trade. In this article, we formulate a mathematical model for the spread of FMD in goat farms that considers early quarantine, vaccination, and disinfection interventions. The model constructed is expressed as a system of ordinary differential equations consisting of eight state variables. We determine the basic reproduction number using the next-generation matrix method. Our numerical simulation results show that vaccination has a very crucial role in controlling the spread of FMD and reducing the basic reproduction number. In addition, quarantine interventions for latent and infectious goats are also effective in controlling the spread of FMD. The basic reproduction number can also be suppressed by disinfection and quarantine interventions. Therefore, we recommend implementing routine vaccination, screening and disinfection programs. In addition, farmer should be given education so that they have the ability to identify goats exposed to FMDV so that quarantine can be carried out early.

Study on an SIR rumor propagation model with an interaction mechanism on WeChat networks

The spread of rumors does great harm to society. It will not only disrupt the social order but also damage public interests and even cause public safety and social shock. With the development of social platforms, it has become a fertile ground for rumors to breed and spread. The WeChat group as a very popular social platform is an important medium of rumor spreading with accurate and fast characteristics. In this paper, in order to describe the dynamic characteristics of rumors spread in WeChat groups, a susceptible–infected–removed rumor propagation model with an interaction mechanism and population dynamics on WeChat groups is proposed. In order to analyze the dynamic characteristics of the system, the next-generation matrix method is used to calculate the basic reproduction number of the proposed model. Then, the stability of a disease-free equilibrium point and a positive equilibrium point of the system is analyzed in detail. Finally, the accuracy of the analysis results is verified by numerical simulation. The research results are of great significance for controlling the rumors spread in WeChat groups.

Dynamic analysis and optimal control of worm propagation model with saturated incidence rate

In order to prevent the propagation of computer worms effectively, based on the latent character of worms, the exposed compartments of computer and USB device are introduced respectively, and a series of computer worm propagation models with saturation incidence rate are proposed. The qualitative behavior of the proposed model is studied. Firstly, the threshold R0 of the model is derived by using the next-generation matrix method, which completely characterized the stability of disease free equilibrium and endemic equilibrium. If R0 < 1, the disease free equilibrium is asymptotically stable, implying that the worm dies out eventually and its attack remains under control; if R0 > 1, the asymptotic stability of endemic equilibrium under certain conditions is proved, which means that the worm is always persistent and uncontrollable under such conditions. Secondly, the theoretical results are verified by numerical study, in which the relative importance of each parameter in worm prevalence is evaluated by sensitivity analysis. Finally, so as to minimize the number of computer and USB device carrying computer worms in short span of time, the worm propagation model is extended to incorporate three control strategies. The Pontryagin’s maximum principle is used to characterize the controls’ optimal levels. According to the control effect diagram, the combined strategy is effective in minimizing the transmission dynamics of worm virus in computer and USB devices populations respectively.

MODELING AND ANALYZING QUARANTINE STRATEGIES OF EPIDEMIC ON TWO-LAYER NETWORKS: GAME THEORY APPROACH

The quarantine strategy plays a crucial role in the prevention and control of infectious disease. In this paper, a two-layer network model coupling the transmission of infectious diseases and the dynamics of human behavior based on game theory is proposed. The basic reproduction number of the infectious disease in our proposed model is obtained by the next-generation matrix method and the stability of the disease-free equilibrium is analyzed. Theoretical results show that the spread of infectious diseases can be controlled when the voluntary quarantined individuals reach a certain proportion. The sensitivities of the parameters are analyzed by simulations, and the results show that increasing propaganda can directly accelerate quarantine, and reducing the relative cost of quarantine has a significant effect on preventing the infectious diseases. Increasing the detection rate will lead to overestimating the proportion of undiagnosed infected individuals, and can also promote individuals to quarantine.

Mathematical Model of COVID-19 with Quarantine and Vaccination

Abstract We present a mathematical model of COVID-19 disease by modifying the SEIR model. The model considers two additional compartments, quarantine (Q) and vaccination (V) which aim to control the spread of COVID-19. Based on the model, we obtained a disease-free equilibrium point and an endemic equilibrium point. The basic reproduction numbers were calculated using the next-generation matrix method. In this model, we analyzed the stability conditions that must be satisfied by the defining parameters. We perform data on the spread of COVID-19 in Indonesia for estimation to provide the parameter value in the model. Based on the result, there is an influence of changes in several parameter values on the number of individuals infected with COVID-19.

Threshold dynamics of a nonlocal diffusion West Nile virus model with spatial heterogeneity

<abstract><p>In this study, we investigated the threshold dynamics of a spatially heterogeneous nonlocal diffusion West Nile virus model. By employing semigroup theory and continuous Fréchet-differentiable, we established the well-posedness of the solution. The expression for the basic reproduction number derived using the next-generation matrix method. The authors demonstrated the threshold dynamics of the system by constructing a Lyapunov function and applying the comparison principle. Finally, numerical simulations were used to validate the theorem results. It can be suggested that to control disease development rapidly, measures should be taken to reduce the spread of mosquitoes and birds.</p></abstract>

Coronavirus dynamics, infections and preventive interventions using fractional-calculus analysis

<abstract><p>In this research work, we construct an epidemic model to understand COVID-19 transmission vaccination and therapy considerations. The model's equilibria were examined, and the reproduction parameter was calculated via a next-generation matrix method, symbolized by $ \mathcal{R}_0 $. We have shown that the infection-free steady state of our system is locally asymptotically stable for $ \mathcal{R}_0 < 1 $. Also, the local asymptotic stability of the endemic steady state has been established for $ \mathcal{R}_0 > 1 $. We have used a partial rank correlation coefficient method for sensitivity analysis of the threshold parameter $ \mathcal{R}_0 $. The contribution of vaccination to the threshold parameter is explored through graphical results. In addition to this, the uniqueness and existence of the solution to the postulated model of COVID-19 infection is shown. We ran various simulations of the proposed COVID-19 dynamics with varied input parameters to scrutinize the complex dynamics of COVID-19 infection. We illustrated the variation in the dynamical behavior of the system with different values of the input parameters. The key factors of the system are visualized for the public health officials for the control of the infection.</p></abstract>

OPTIMAL CONTROL ANALYSIS ON MATHEMATICAL MODEL OF DYNAMICAL TRANSMISSION OF HIV-MALARIA CO-INFECTION

Malaria and HIV are two of the most serious global health challenges affecting developing countries. To have a deeper understanding of the two diseases, we developed a deterministic mathematical model with fourteen compartments on the transmission dynamics of HIV-Malaria Co-infections. The boundary of the model solution is performed, and we proceed to determine both the disease-free equilibrium (DFE) and the endemic equilibrium (EE) for the system of the equations. The next-generation matrix method is used to determine the basic reproductive number R 0 necessary for disease control. The basic model is extended into an optimal control problem by incorporating four control strategies, The Pontryagin’s maximum principle is employed to derive the necessary conditions for the existence of optimal control. Numerical results are displayed graphically. The results of the analysis show that the best approach to control the HIV-malaria co-infection is to combine both HIV control measures and malaria disease control measures on HIV-malaria co-infection patients. Keywords: HIV-Malaria Co-Infection, Pontryagin’s Maximum Principle, Hamiltonian DOI： https://doi.org/10.35741/issn.0258-2724.58.1.44

Modelling and analysis of the HIV/AIDS epidemic with fast and slow asymptomatic infections in China from 2008 to 2021.

The aim of this paper is to investigate the spread of the HIV/AIDS epidemic in China during 2008-2021. A new mathematical model is proposed to study the dynamics of HIV transmission with acute infection, fast asymptomatic infections, and slow asymptomatic infections. The basic reproduction number is obtained by the next-generation matrix method. A quantitative analysis of the model, including the local behavior, global behavior, and permanence, is performed. Numerical simulations are presented to enhance the results of these analyses. The behavior or the model's parameters are estimated from real data. A sensitivity analysis shows that the proportion of asymptomatic infections co-infected with other diseases significantly affects the basic reproduction number. We further analyze the impact of implementing single and multiple measure(s) in parallel with the epidemic. The study results conclude that multiple measures are more effective in controlling the spread of AIDS compared to just one. The HIV epidemic can be effectively curbed by reducing the contact rate between fast asymptomatic infected individuals and susceptible populations, increasing the early diagnosis and screening of HIV-infected individuals co-infected with other diseases, and treating co-infected patients promptly.

Optimal Control and Bifurcation Analysis of HIV Model.

In this study, a very crucial stage of HIV extinction and invisibility stages are considered and a modified mathematical model is developed to describe the dynamics of infection. Moreover, the basic reproduction number R 0 is computed using the next-generation matrix method whereas the stability of disease-free equilibrium is investigated using the eigenvalue matrix stability theory. Furthermore, if R 0 ≤ 1, the disease-free equilibrium is stable both locally and globally whereas if R 0 > 1, based on the forward bifurcation behavior, the endemic equilibrium is locally and globally asymptotically stable. Particularly, at the critical point R 0 = 1, the model exhibits forward bifurcation behavior. On the other hand, the optimal control problem is constructed and Pontryagin's maximum principle is applied to form an optimality system. Further, forward fourth-order Runge-Kutta's method is applied to obtain the solution of state variables whereas Runge-Kutta's fourth-order backward sweep method is applied to obtain solution of adjoint variables. Finally, three control strategies are considered and a cost-effective analysis is performed to identify the better strategies for HIV transmission and progression. In advance, prevention control measure is identified to be the better strategy over treatment control if applied earlier and effectively. Additionally, MATLAB simulations were performed to describe the population's dynamic behavior.