This paper presents a comparative analysis of two computational approaches to optimizing vaccination strategy: a new nonsmooth Newton method (NNM) and the classical Pontryagin’s maximum principle (PMP). The algorithms are contrasted on three regimes of constraints: an unconstrained reference, a control-only constraints (vaccination rate limits) and mixed state-control constraints (such as vaccine supplies and susceptible population limits). The SEIR compartmental model is used in assessing the efficacy of the vaccination strategy. The NNM is a direct method that employs the discretize-then-optimize strategy to transform the optimal control problem as a nonlinear system using Fischer–Burmeister complementarity and solve it economically with a Newton-type approach. Although the method exhibits local quadratic convergence, this property is contingent upon the initial guess being sufficiently close to the solution, hence, a globalization mechanism is essential to guarantee convergence from distant starting points. The PMP is an indirect method, solving a two-point boundary value problem by iterative shooting techniques, which are usually initialization-sensitive and time-consuming. Numerical experiments show that NNM converges faster and has lower optimization costs compared to PMP. They show the effectiveness of the proposed NNM compared to the traditional PMP-based techniques.
In last two decades, people battled with three coronaviruses, namely Severe Acute Respiratory Syndrome (SARS) in November 2002; Middle East Respiratory Syndrome (MERS) in 2012 and finally the devastating SARS-CoV-2 infection or COVID-19 pandemic in December, 2019. Though the pandemic is now under control, we need to know whether it is completely eradicated or not. For this purpose, we have formulated a mathematical model incorporating three doses of corona vaccine and studied the impact of these vaccines on the disease dynamics. In our study, we have sought the conditions for eradication of the virus from the environment. To understand the consistency of the model, we have studied first its positivity and boundedness. Then we have carried out its steady-state analysis. The stability of the COVID-free equilibrium (CFE) and COVID-endemic equilibrium (CEE) points has been analyzed with the help of the basic reproduction number [Formula: see text], which is derived by the next-generation matrix approach. Further, this model has been validated by real data reported from India and estimated model parameters. Most effective model parameters are found by sensitivity analysis. Our findings show that first and second doses of COVID vaccines are not sufficient for eradication of the disease from the population, but the third (or booster) dose is able to control the disease if no comorbidity is considered.
In this paper, we assume that two key parameters follow Ornstein–Uhlenbeck processes and incorporate treatment interventions into the model to gain further insight into the transmission dynamics. We first prove the existence and uniqueness of a positive solution. By constructing a suitable Lyapunov function and exploiting the ergodicity of the Ornstein–Uhlenbeck process, we derive sufficient conditions for the existence of a stationary distribution. Under the same conditions that guarantee local asymptotic stability of the endemic equilibrium, we then obtain an explicit expression for the covariance matrix of the probability density around the quasi-endemic equilibrium of the stochastic system. We also establish sufficient conditions for extinction. Numerical simulations are performed to validate the theoretical results.
Our study proposes a delay differential equation (DDE) model for the dynamics of tumor cells, immune cells, oncolytic viruses, and chemotherapy. Two discrete delays associated with immune activation and chemotherapy response are incorporated into the model. Positive and bounded solutions are established, several equilibria are determined, and delay-dependent stability conditions are derived. A Hopf bifurcation may occur when the immune-activation delay crosses an equilibrium-dependent critical threshold. To identify time-dependent chemotherapy and virotherapy schedules that reduce tumor burden while preserving immune activity, we formulate an optimal control problem. Based on numerical simulations, combined chemo-virotherapy provides better tumor control than either treatment alone, particularly when treatment timing is appropriately selected.
The integration of human vaccination into malaria models offers a novel tool for leveraging high-efficacy vaccines like R21/Matrix-M in blocking transmission. Concurrently, incorporating mosquito chemotaxis is critical for simulating the directed movement of vectors that drives spatial hotspot formation. To this end, we develop a novel reaction-diffusion framework integrating both mechanisms. We establish the well-posedness of the model, derive the basic reproduction number R-0 and analyze the associated threshold dynamics. Furthermore, an optimal control framework is formulated under the assumption of no chemotaxis to minimize the density of infected humans and mosquitoes, and the cost of vaccination, bed-net usage, treatment and insecticide application. Numerical simulations validate the analytical results and reveal that vaccination substantially reduces disease burden, and that increasing chemotaxis induces heterogeneous spatial distribution of mosquitoes, driving their aggregation in areas of high human density. Additionally, in malaria control, strategies must account for geographical heterogeneity to enable spatially targeted interventions that effectively suppress transmission while minimizing operational costs.
