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

<abstract><p>A generalized "SVEIR" epidemic model with general nonlinear incidence rate has been proposed as a candidate model for measles virus dynamics. The basic reproduction number $ \mathcal{R} $, an important epidemiologic index, was calculated using the next generation matrix method. The existence and uniqueness of the steady states, namely, disease-free equilibrium ($ \mathcal{E}_0 $) and endemic equilibrium ($ \mathcal{E}_1 $) was studied. Therefore, the local and global stability analysis are carried out. It is proved that $ \mathcal{E}_0 $ is locally asymptotically stable once $ \mathcal{R} $ is less than. However, if $ \mathcal{R} &gt; 1 $ then $ \mathcal{E}_0 $ is unstable. We proved also that $ \mathcal{E}_1 $ is locally asymptotically stable once $ \mathcal{R} &gt; 1 $. The global stability of both equilibrium $ \mathcal{E}_0 $ and $ \mathcal{E}_1 $ is discussed where we proved that $ \mathcal{E}_0 $ is globally asymptotically stable once $ \mathcal{R}\leq 1 $, and $ \mathcal{E}_1 $ is globally asymptotically stable once $ \mathcal{R} &gt; 1 $. The sensitivity analysis of the basic reproduction number $ \mathcal{R} $ with respect to the model parameters is carried out. In a second step, a vaccination strategy related to this model will be considered to optimise the infected and exposed individuals. We formulated a nonlinear optimal control problem and the existence, uniqueness and the characterisation of the optimal solution was discussed. An algorithm inspired from the Gauss-Seidel method was used to resolve the optimal control problem. Some numerical tests was given confirming the obtained theoretical results.</p></abstract>

Sensitivity, Optimal Control, and Cost-Effectiveness Analysis of Intervention Strategies of Filariasis

In this work, sensitivity, optimal control, and cost-effectiveness of several intervention strategies of filariasis are discussed. We study the intervention strategies that are related to bednet use, insecticide, and the combination of bed-net use and insecticide. We use Pontryagin’s maximum principle to characterize the optimal controls. The Average Cost-Effectiveness Ratio (ACER) and Infection Averted Ratio (IAR) are used to identify the most cost-effective strategy. We also determine the basic reproduction number and investigate the sensitivity of the basic reproduction number on the parameters that are related to bed-net use and insecticide. Based on the ACER values, the most cost-effective strategy to control filariasis is insecticide intervention. On the other hand, the IAR values indicates that bed-net use intervention is the most cost-effective strategy. Furthermore, it is also the most effective strategy to eliminate filariasis. The sensitivity analysis results show that the control parameter related to bed net use and treatment have a central role in reducing the basic reproduction number and filariasis spread.

Dynamics of COVID-19 Epidemic Model with Asymptomatic Infection, Quarantine, Protection and Vaccination

We discuss the dynamics of new COVID-19 epidemic model by considering asymptomatic infections and the policies such as quarantine, protection (adherence to health protocols), and vaccination. The proposed model contains nine subpopulations: susceptible (S), exposed (E), symptomatic infected (I), asymptomatic infected (A), recovered (R), death (D), protected (P), quarantined (Q), and vaccinated (V ). We first show the non-negativity and boundedness of solutions. The equilibrium points, basic reproduction number, and stability of equilibrium points, both locally and globally, are also investigated analytically. The proposed model has disease-free equilibrium point and endemic equilibrium point. The disease-free equilibrium point always exists and is globally asymptotically stable if basic reproduction number is less than one. The endemic equilibrium point exists uniquely and is globally asymptotically stable if the basic reproduction number is greater than one. These properties have been confirmed by numerical simulations using the fourth order Runge-Kutta method. Numerical simulations show that the disease transmission rate of asymptomatic infection, quarantine rates, protection rate, and vaccination rates affect the basic reproduction number and hence also influence the stability of equilibrium points.

Forward Bifurcation with Hysteresis Phenomena from Atherosclerosis Mathematical Model

Atherosclerosis is a non-communicable disease (NCDs) which appears when the blood vessels in the human body become thick and stiff. The symptoms range from chest pain, sudden numbness in the arms or legs, temporary loss of vision in one eye, or even kidney failure, which may lead to death. Treatment in cases with severe symptoms requires surgery, in which the number of doctors or hospitals is limited in some countries, especially countries with low health levels. This article aims to propose a mathematical model to understand the impact of limited hospital resources on the success of the control program of atherosclerosis spreads. The model was constructed based on a deterministic model, where the hospitalization rate is defined as a time-dependent saturated function concerning the number of infected individuals. The existence and stability of all possible equilibrium points were shown analytically and numerically, along with the basic reproduction number. Our analysis indicates that our model may exhibit various types of bifurcation phenomena, such as forward bifurcation, backward bifurcation, or a forward bifurcation with hysteresis depending on the value of hospitalization saturation parameter and the infection rate for treated infected individuals. These phenomenon triggers a complex and tricky control program of atherosclerosis. A forward bifurcation with hysteresis auses a possible condition of having more than one stable endemic equilibrium when the basic reproduction number is larger than one, but close to one. The more significant value of hospitalization saturation rate or the infection rate for treated infected individuals increases the possibility of the stable endemic equilibrium point even though the disease-free equilibrium is stable. Furthermore, the Pontryagin Maximum Principle was used to characterize the optimal control problem for our model. Based on the results of our analysis, we conclude that atherosclerosis control interventions should prioritize prevention efforts over endemic reduction scenarios to avoid high intervention costs. In addition, the government also needs to pay great attention to the availability of hospital services for this disease to avoid the dynamic complexity of the spread of atherosclerosis in the field.

Serial interval and basic reproduction number of SARS-CoV-2 Omicron variant in South Korea

South Korea is experiencing the community transmission of the SARS-CoV-2 Omicron variant (B.1.1.529). We estimated that the mean of the serial interval was 2.22 days, and the basic reproduction number was 1.90 (95% Credible Interval, 1.50-2.43) for the Omicron variant outbreak in South Korea.

Stability Analysis of SEIL Tuberculosis Epidemic Model with Logistic Growth in Susceptible Compartment

This article discusses modifications to the SEIL model that involve logistical growth. This model is used to describe the dynamics of the spread of tuberculosis disease in the population. The existence of the model's equilibrium points and its local stability depends on the basic reproduction number. If the basic reproduction number is less than unity, then there is one equilibrium point that is locally asymptotically stable. The equilibrium point is a disease-free equilibrium point. If the basic reproduction number ranges from one to three, then there are two equilibrium points. The two equilibrium points are disease-free equilibrium and endemic equilibrium points. Furthermore, for this case, the endemic equilibrium point is locally asymptotically stable.

Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model

The aim of this work is to study qualitative dynamical properties of a generalized hepatitis B epidemic model and its dynamically consistent discrete model. Positivity, boundedness, the basic reproduction number and asymptotic stability properties of the model are analyzed rigorously. By the Lyapunov stability theory and the Poincare-Bendixson theorem in combination with the Bendixson-Dulac criterion, we show that a disease-free equilibrium point is globally asymptotically stable if the basic reproduction number $\mathcal{R}_0 \leq 1$ and a disease-endemic equilibrium point is globally asymptotically stable whenever $\mathcal{R}_0 > 1$. Next, we apply the Mickens’ methodology to propose a dynamically consistent nonstandard finite difference (NSFD) scheme for the continuous model. By rigorously mathematical analyses, it is proved that the constructed NSFD scheme preserves essential mathematical features of the continuous model for all finite step sizes. Finally, numerical experiments are conducted to illustrate the theoretical findings and to demonstrate advantages of the NSFD scheme over standard ones. The obtained results in this work not only improve but also generalize some existing recognized works.

A two-sex model of human papillomavirus infection: Vaccination strategies and a case study

Vaccination is effective in preventing human papillomavirus (HPV) infection. It still remains debatable whether males should be included in a vaccination program and unclear how to allocate the vaccine in genders to achieve the maximum benefits. In this paper, we use a two-sex model to assess HPV vaccination strategies and use the data from Guangxi Province in China as a case study. Both mathematical analysis and numerical simulations show that the basic reproduction number, an important indicator of the transmission potential of the infection, achieves its minimum when the priority of vaccination is given to the gender with a smaller recruit rate. Given a fixed amount of vaccine, splitting the vaccine evenly usually leads to a larger basic reproduction number and a higher prevalence of infection. Vaccination becomes less effective in reducing the infection once the vaccine amount exceeds the smaller recruit rate of the two genders. In the case study, we estimate the basic reproduction number is 1.0333 for HPV 16/18 in people aged 15-55. The minimal bivalent HPV vaccine needed for the disease prevalence to be below 0.05% is 24050 per year, which should be given to females. However, with this vaccination strategy it would require a very long time and a large amount of vaccine to achieve the goal. In contrast with allocating the same vaccine amount every year, we find that a variable vaccination strategy with more vaccine given in the beginning followed by less vaccine in later years can save time and total vaccine amount. The variable vaccination strategy illustrated in this study can help to better distribute the vaccine to reduce the HPV prevalence. Although this work is for HPV infection and the case study is for a province in China, the model, analysis and conclusions may be applicable to other sexually transmitted diseases in other regions or countries.

Reassessment of contact restrictions and testing campaigns against COVID-19 via spatio-temporal modeling

Since the earliest outbreak of COVID-19, the disease continues to obstruct life normalcy in many parts of the world. The present work proposes a mathematical framework to improve non-pharmaceutical interventions during the new normal before vaccination settles herd immunity. The considered approach is built from the viewpoint of decision makers in developing countries where resources to tackle the disease from both a medical and an economic perspective are scarce. Spatial auto-correlation analysis via global Moran’s index and Moran’s scatter is presented to help modulate decisions on hierarchical-based priority for healthcare capacity and interventions (including possible vaccination), finding a route for the corresponding deployment as well as landmarks for appropriate border controls. These clustering tools are applied to sample data from Sri Lanka to classify the 26 Regional Director of Health Services (RDHS) divisions into four clusters by introducing convenient classification criteria. A metapopulation model is then used to evaluate the intra- and inter-cluster contact restrictions as well as testing campaigns under the absence of confounding factors. Furthermore, we investigate the role of the basic reproduction number to determine the long-term trend of the regressing solution around disease-free and endemic equilibria. This includes an analytical bifurcation study around the basic reproduction number using Brouwer Degree Theory and asymptotic expansions as well as related numerical investigations based on path-following techniques. We also introduce the notion of average policy effect to assess the effectivity of contact restrictions and testing campaigns based on the proposed model’s transient behavior within a fixed time window of interest.

Generalized SEIR Epidemic Model for COVID-19 in a Multipatch Environment

In this paper, an n -patch SEIR epidemic model for the coronavirus disease 2019 (COVID-19) is presented. It is shown that there is unique disease-free equilibrium for this model. Then, the dynamic behavior is studied by the basic reproduction number. The transmission of COVID-19 is fitted based on actual data. The influence of quarantined rate and population migration rate on the spread of COVID-19 is also discussed by simulation.