Malaria remains a significant global health challenge, with sub-Saharan Africa bearing the majority of the burden. While vector control measures such as pyrethroid-based insecticidal nets and indoor residual spraying have significantly reduced malaria incidence, the emergence of insecticide resistance in Anopheles mosquito populations threatens these gains. Resistance develops through genetic mutations under prolonged selection pressure, complicating control efforts and necessitating a deeper understanding of its evolutionary dynamics. This study introduces a novel mathematical framework to investigate the emergence and spread of insecticide resistance in mosquito populations. By modelling insecticide resistance as a continuous (quantitative) trait influenced by multiple genes, we capture its variability and evolutionary transient dynamics. We propose an age-structured mosquito population model using integro-differential equations, where the resistance trait influences life-history parameters such as mortality and reproduction. Our approach provides new insights into how resistance emerges and spreads within mosquito populations over time. We analyse the model’s properties, including the existence of a unique maximal bounded semiflow, and derive conditions for the existence and stability of steady states. Through parameterization and simulations, we explore the transient and long-term dynamics of resistance evolution under different scenarios. The results offer valuable insights into the evolutionary mechanisms driving insecticide resistance and inform the design of sustainable vector control strategies.

In this paper we investigate the population dynamics of a species with age structure in the case where the diffusion and death rates of the matured population are both age-dependent. We develop a new application of the age-structure technique in terms of an integral equation. For unbounded spatial domains, we study the existence of travelling waves, whilst in bounded domains, we investigate the existence of positive steady-state solutions and their stability.

We introduce a model of the optimal education policy at the macro level, allowing for heterogeneity of the workforce with respect to its age and qualification skills. Within this framework we study the optimal education rate in the context of changes in labor demand (as represented by the elasticity of substitution across ages and qualification) and labor supply (as represented by a change in the population growth rates). Applying an age-structured optimal-control model, we derive features of the optimal age-specific education rate. Our results show that the relation between the elasticities of substitution of labor across ages plays a crucial role in the way the demographic changes affect (both in the short and in the long run) the optimal educational policy. We also show that under imperfect substitutability across age and qualification groups, the optimal educational policy is adjusted in advance to any change in the labor supply.