A mathematical investigation of an "SVEIR" epidemic model for the measles transmission

A mathematical model of COVID-19 with multiple variants of the virus under optimal control in Ghana

In this paper, we suggest a mathematical model of COVID-19 with multiple variants of the virus under optimal control. Mathematical modeling has been used to gain deeper insights into the transmission of COVID-19, and various prevention and control strategies have been implemented to mitigate its spread. Our model is a SEIR-based model for multi-strains of COVID-19 with 7 compartments. We also consider the circulatory structure to account for the termination of immunity for COVID-19. The model is established in terms of the positivity and boundedness of the solution and the existence of equilibrium points, and the local stability of the solution. As a result of fitting data of COVID-19 in Ghana to the model, the basic reproduction number of the original virus and Delta variant was estimated to be 1.9396, and the basic reproduction number of the Omicron variant was estimated to be 3.4905, which is 1.8 times larger than that. We observe that even small differences in the incubation and recovery periods of two strains with the same initial transmission rate resulted in large differences in the number of infected individuals. In the case of COVID-19, infections caused by the Omicron variant occur 1.5 to 10 times more than those caused by the original virus. In terms of the optimal control strategy, we formulate three control strategies focusing on social distancing, vaccination, and testing-treatment. We have developed an optimal control model for the three strategies outlined above for the multi-strain model using the Pontryagin's Maximum Principle. Through numerical simulations, we analyze three optimal control strategies for each strain and also consider combinations of the two control strategies. As a result of the simulation, all control strategies are effective in reducing disease spread, in particular, vaccination strategies are more effective than the other two control strategies. In addition the combination of the two strategies also reduces the number of infected individuals by 1/10 compared to implementing one strategy, even when mild levels are implemented. Finally, we show that if the testing-treatment strategy is not properly implemented, the number of asymptomatic and unidentified infections may surge. These results could help guide the level of government intervention and prevention strategy formulation.

Mathematical modeling and dynamics of immunological exhaustion caused by measles transmissibility interaction with HIV host

This paper mainly addressed the study of the transmission dynamics of infectious diseases and analysed the effect of two different types of viruses simultaneously that cause immunodeficiency in the host. The two infectious diseases that often spread in the populace are HIV and measles. The interaction between measles and HIV can cause severe illness and even fatal patient cases. The effects of the measles virus on the host with HIV infection are studied using a mathematical model and their dynamics. Analysing the dynamics of infectious diseases in communities requires the use of mathematical models. Decisions about public health policy are influenced by mathematical modeling, which sheds light on the efficacy of various control measures, immunization plans, and interventions. We build a mathematical model for disease spread through vertical and horizontal human population transmission, including six coupled nonlinear differential equations with logistic growth. The fundamental reproduction number is examined, which serves as a cutoff point for determining the degree to which a disease will persist or die. We look at the various disease equilibrium points and investigate the regional stability of the disease-free and endemic equilibrium points in the feasible region of the epidemic model. Concurrently, the global stability of the equilibrium points is investigated using the Lyapunov functional approach. Finally, the Runge-Kutta method is utilised for numerical simulation, and graphic illustrations are used to evaluate the impact of different factors on the spread of the illness. Critical factors that effect the dynamics of disease transmission and greatly affect the rate and range of the disease's spread in the population have been determined through a thorough analysis. These factors are crucial in determining the expansion of the disease.

Dynamics analysis of an SVEIR epidemic model in a patchy environment

In this paper, we propose a multi-patch SVEIR epidemic model that incorporates vaccination of both newborns and susceptible populations. We determine the basic reproduction number $ R_{0} $ and prove that the disease-free equilibrium $ P_{0} $ is locally and globally asymptotically stable if $ R_{0} < 1, $ and it is unstable if $ R_{0} > 1. $ Moreover, we show that the disease is uniformly persistent in the population when $ R_{0} > 1. $ Numerical simulations indicate that vaccination strategies can effectively control disease spread in all patches while population migration can either intensify or prevent disease transmission within a patch.

Modelling and Analysis of a Measles Epidemic Model with the Constant Proportional Caputo Operator

Despite the existence of a secure and reliable immunization, measles, also known as rubeola, continues to be a leading cause of fatalities globally, especially in underdeveloped nations. For investigation and observation of the dynamical transmission of the disease with the influence of vaccination, we proposed a novel fractional order measles model with a constant proportional (CP) Caputo operator. We analysed the proposed model’s positivity, boundedness, well-posedness, and biological viability. Reproductive and strength numbers were also verified to examine how the illness dynamically behaves in society. For local and global stability analysis, we introduced the Lyapunov function with first and second derivatives. In order to evaluate the fractional integral operator, we used different techniques to invert the PC and CPC operators. We also used our suggested model’s fractional differential equations to derive the eigenfunctions of the CPC operator. There is a detailed discussion of additional analysis on the CPC and Hilfer generalised proportional operators. Employing the Laplace with the Adomian decomposition technique, we simulated a system of fractional differential equations numerically. Finally, numerical results and simulations were derived with the proposed measles model. The intricate and vital study of systems with symmetry is one of the many applications of contemporary fractional mathematical control. A strong tool that makes it possible to create numerical answers to a given fractional differential equation methodically is symmetry analysis. It is discovered that the proposed fractional order model provides a more realistic way of understanding the dynamics of a measles epidemic.

Mathematical analysis of a two-tiered microbial food-web model for the anaerobic digestion process

In this research paper, we presented a four-dimensional mathematical system modeling the anaerobic mineralization of phenol in a two-step microbial food-web. The inflowing concentrations of the hydrogen and the phenol are considered in our model. We considered the case of general class of nonlinear growth kinetics, instead of Monod kinetics. Due to some conservative relations, the proposed model was reduced to a two-dimensional system. The stability of the steady states was carried out. Based on the species growth rates and the three main operating parameters of the model, represented by the dilution rate and input concentrations of the phenol and the hydrogen, we showed that the system can have up to four steady states. We gave the necessary and sufficient conditions ensuring the existence and the stability for each feasible equilibrium state. We showed that in specific cases, the positive steady state exists and is stable. We gave numerical simulations validating the obtained results.

Global investigation for an "SIS" model for COVID-19 epidemic with asymptomatic infection

In this paper, we analyse a dynamical system taking into account the asymptomatic infection and we consider optimal control strategies based on a regular network. We obtain basic mathematical results for the model without control. We compute the basic reproduction number (R) by using the method of the next generation matrix then we analyse the local stability and global stability of the equilibria (disease-free equilibrium (DFE) and endemic equilibrium (EE)). We prove that DFE is LAS (locally asymptotically stable) when R<1 and it is unstable when R>1. Further, the existence, the uniqueness and the stability of EE is carried out. We deduce that when R>1, EE exists and is unique and it is LAS. By using generalized Bendixson-Dulac theorem, we prove that DFE is GAS (globally asymptotically stable) if R<1 and that the unique endemic equilibrium is globally asymptotically stable when R>1. Later, by using Pontryagin's maximum principle, we propose several reasonable optimal control strategies to the control and the prevention of the disease. We mathematically formulate these strategies. The unique optimal solution was expressed using adjoint variables. A particular numerical scheme was applied to solve the control problem. Finally, several numerical simulations that validate the obtained results were presented.

Modelling the impact of perfect and imperfect vaccination strategy against SARS CoV-2 by assuming varied vaccine efficacy over India

Background

The outbreak of Coronavirus disease (COVID-19) has swiftly spread globally and caused public health and socio-economic disruption in many countries. An epidemiological modelling studies in the susceptible-infectious-removed (SIR) has played an important role for making effective public health policy to mitigate the spread of COVID-19. The aim of the present study is to investigate the optimal vaccination strategy to control the COVID-19 pandemic in India.

Methods

We have applied compartment mathematical model susceptible-vaccination-infectious-removed (SVIR) with different range of vaccine efficacy scenarios and predicted the population to be covered for vaccination per day in India as well as state level was performed.

Results

The model assumed that a vaccine has 100% efficacy, predicted that >5 million populace to be vaccinated per day to flatten the epidemic curve in India. Similarly, different vaccination mechanisms such as ‘all-or-nothing’ (AoN) and leaky vaccines does not have potential discordance in their effectiveness at higher efficacies (>70%). However, AoN vaccine was found to be marginally effective than leaky at lower efficacies (<70%) when administered at the higher coverage strategies. Further state level analyses were performed and it was found that 0.3, 0.3, 0.2 and 1 million vaccinations required per day in Andhra Pradesh, Gujarat, Kerala and Maharashtra as it assumes that the vaccine efficacy is 70%.

Conclusion

The proposed modelling approach shows a range of assumptions on the efficacy of vaccine which helps the health authorities to prioritize the vaccination strategies to prevent the transmission as well as disease.

Exploring HIV Dynamics and an Optimal Control Strategy

In this paper, we propose a six-dimensional nonlinear system of differential equations for the human immunodeficiency virus (HIV) including the B-cell functions with a general nonlinear incidence rate. The compartment of infected cells was subdivided into three classes representing the latently infected cells, the short-lived productively infected cells, and the long-lived productively infected cells. The basic reproduction number was established, and the local and global stability of the equilibria of the model were studied. A sensitivity analysis with respect to the model parameters was undertaken. Based on this study, an optimal strategy is proposed to decrease the number of infected cells. Finally, some numerical simulations are presented to illustrate the theoretical findings.