Reproduction Number Research Articles

Nonlinear dynamics in a fear‐driven predator–prey system: Bistability, bifurcations, hydra effect, and optimal harvesting

The impact of predator‐driven fear on ecosystems is significant and can encompass both trophic (direct) and nontrophic (indirect) effects. Previous studies have shown that nontrophic fear effects have an important role in predator–prey dynamics. This study investigates the nontrophic fear effect on prey caused by generalist predators and explores optimal harvesting. We assume that the reproduction rate of prey is reduced due to the fear effect, and generalist predator follows Holling type II foraging strategy for predation. Additionally, we assume that predators are commercially valuable and harvested proportionately to their density. We demonstrate the existence of equilibrium points, their local and global stability, and bifurcation analysis. We observe that the model system undergoes a sequence of codimension one and codimension two bifurcations. Our results show that in the absence of predator harvesting, increasing levels of fear destabilize the predator–prey system, and controlled harvesting is beneficial for the coexistence of both populations. Also, the harvesting of predators may produce hydra and multiple hydra effects. We identify different types of bistability phenomena, which emphasize the importance of initial population size. Further, an optimal harvesting policy is also explored by formulating an optimal control problem (OCP). The harvesting cost‐functional is designed by incorporating the bionomic equilibrium state. We use Pontryagin's maximum principle and solve the OCP numerically. It is observed that implementing optimal harvesting not only contributes to the ecological benefits by maintaining a sustainable balance of predator–prey evolution but also plays a significant role in maximizing the economic benefits.

Ebola disease outbreak caused by the Sudan virus in Uganda, 2022: a descriptive epidemiological study

Uganda has had seven Ebola disease outbreaks, between 2000 and 2022. On Sept 20, 2022, the Ministry of Health declared a Sudan virus disease outbreak in Mubende District, Central Uganda. We describe the epidemiological characteristics and transmission dynamics. For this descriptive study, cases were classified as suspected, probable, or confirmed using Ministry of Health case definitions. We investigated all reported cases to obtain data on case-patient demographics, exposures, and signs and symptoms, and identified transmission chains. We conducted a descriptive epidemiological study and also calculated basic reproduction number (Ro) estimates. Between Aug 8 and Nov 27, 2022, 164 cases (142 confirmed, 22 probable) were identified from nine (6%) of 146 districts. The median age was 29 years (IQR 20-38), 95 (58%) of 164 patients were male, and 77 (47%) patients died. Symptom onsets ranged from Aug 8 to Nov 27, 2022. The case fatality rate was highest in children younger than 10 years (17 [74%] of 23 patients). Fever (135 [84%] of 160 patients), vomiting (93 [58%] patients), weakness (89 [56%] patients), and diarrhoea (81 [51%] patients) were the most common symptoms; bleeding was uncommon (21 [13%] patients). Before outbreak identification, most case-patients (26 [60%] of 43 patients) sought care at private health facilities. The median incubation was 6 days (IQR 5-8), and median time from onset to death was 10 days (7-23). Most early cases represented health-care-associated transmission (43 [26%] of 164 patients); most later cases represented household transmission (109 [66%]). Overall Ro was 1·25. Despite delayed detection, the 2022 Sudan virus disease outbreak was rapidly controlled, possibly thanks to a low Ro. Children (aged <10 years) were at the highest risk of death, highlighting the need for targeted interventions to improve their outcomes during Ebola disease outbreaks. Initial care-seeking occurred at facilities outside the government system, showing a need to ensure that private and public facilities receive training to identify possible Ebola disease cases during an outbreak. Health-care-associated transmission in private health facilities drove the early outbreak, suggesting gaps in infection prevention and control. None.

Global stability for age-structured heroin model with general nonlinear incidence rate and time delay

In this article, we consider a mathematical model on the consumption of Heroin, it is an epidemiological model describing the interactions between heroin consumers and the rest of the population. The model is a system of differential equations with delay, integro-differential equations and partial differential equations. The first equation describes the evolution of healthy individuals, who do not consume Heroin. The second equation describes the evolution of consumer individuals, and in the last equation we find the evolution of individuals under treatment, so that they become healthy. In this work, we are interested in the qualitative study of the system considered. We examine the effects of a nonlinear incidence function with lag on the dynamics of an age-of-infection structured Heroin consumption model in the framework of mathematical epidemiology. We prove that incorporating the lag into the model does not change the asymptotic behavior of its equilibrium states. By defining the basic reproduction number R0 as a critical threshold parameter, we show that if R0 &lt; 1, the disease-free equilibrium (DFE) is globally stable regardless of the lag τ. Conversely, if R0 &lt; 1, the model exhibits an endemic equilibrium (EE) that remains locally and globally asymptotically stable for any lag τ. Our results highlight the robustness of the equilibrium stability in response to variations in the lag and provide important insights into the long-term dynamics of Heroin consumption, providing a clearer understanding of the conditions under which epidemic outcomes are stable or unstable.

Global stability analysis, stabilization and synchronization of an SEIR epidemic model

In this paper, we investigate an SEIR epidemic model in which recruitment is assumed to be regulated by a constant function. The SEIR model is a widely-used epidemiological framework that categorises the population into four distinct groups: Susceptible, Exposed, Infectious, and Removed. It employs a system of differential equations to describe the transitions of individuals among these groups, making it a valuable tool for understanding and modelling the dynamics of infectious diseases. Mathematical analysis is used to study the dynamic behaviour of this model. We first determine the basic reproduction number, denoted as R 0 , which governs the dynamics of the SEIR model. Our findings reveal that when R 0 is less than 1, the unique disease-free equilibrium is asymptotically stable, and there is no endemic equilibrium. However, when R 0 exceeds 1, the disease-free equilibrium becomes unstable, and a unique, asymptotically stable endemic equilibrium emerges. Furthermore, by constructing appropriate Lyapunov functions, we have studied the global asymptotic stability of the two equilibrium points. On the other hand, we have used the backstepping method to design a control law that stabilises the SEIR model towards the disease-free equilibrium, which means the eradication of the disease from the target population. Next, we have used the same method to synchronise the SEIR drive system and the SEIR response system. This method is used to stabilise the synchronisation error dynamics arising from the mismatch between the drive and response systems. Finally, we presented numerical simulations to illustrate the theoretical results.

On nonlinear dynamical analysis of a fractional-order two-strains Nipah virus model

The Nipah virus (NiV) is one of the most lethal viruses which can infect humans and lead to fatal encephalitis. The recent significant awareness of NiV is due to its elevated death rate and effective transmission capabilities among humans. With its recurrent outbreaks and exceptionally high mortality rate, the NiV infections have emerged as one of the most concerning hazards to public health. The exploration of NiV and its characteristics revealed that NiV has two distinct strains, namely, the Malaysia (NiVM) strain and the Bangladesh (NiVB) strain. In this paper, we propose a novel Caputo fractional order mathematical model to simulate the dynamics of the two strains NiV. The positivity and boundedness of the model’s solutions are investigated. The existence and asymptotic stability of the equilibrium points of the model are examined. The analysis of basic reproduction number is presented to determine whether the infection will die out or persist. The effects of key parameters of the model on its dynamical behaviors are also explored. Finally, efficient numerical technique is used to confirm the analytical results through detailed numerical simulations.

Sustained Spread of HIV-1 CRF55_01B in Its Place of Initial Origin: Dynamics and Hotspots.

Shenzhen, a city with a substantial mobile population, was identified as the first discovered region of HIV-1 CRF55_01B and epicenter of its severe epidemic. During the implementation of venue-based behavioral interventions and the "treat-all" policy, discerning the spread patterns and transmission hotspots of CRF55_01B is imperative. In this study, 1,450 partial pol sequences, with demographic information, were collected from all newly diagnosed CRF55_01B infections in Shenzhen from 2008 to 2020. Molecular networks were constructed using the maximum likelihood and time-resolve phylogenies. Transmission rates, effective reproduction numbers (Re) of clusters and viral dispersal were evaluated using Bayesian inference. In total, 526 sequences formed 114 clusters, including seven large clusters. The status and size of clusters were strongly correlated with age, ethnicity, occupation and CD4+ T cell counts. The transmission rates of clusters were significantly higher than the national epidemic estimate. Four large clusters had Re exceeding 1 at the end of sampling period. Immigrants from Guangdong and Hunan, along with local residents, were identified as the transmission hubs, with heterosexual men being the main source and MSM being the main destination. The virus exhibited a high movement frequency from individuals aged 30-49 years toward diverse age groups. This study demonstrated the hidden CRF55_01B transmissions continued despite current combined interventions in Shenzhen, and special at-risk individuals susceptible to infection or transmission were identified, potentially serving as targets for more effective prevention and control of the local epidemic, thereby mitigating cross-regional spread nationwide due to population migration.

Methodology for assessing the impact of emergencies on the spread of infectious diseases

The spread of infectious diseases is significantly influenced by emergencies, particularly military conflicts, which disrupt healthcare systems and increase the risks of epidemics. The full-scale Russian invasion of Ukraine has exacerbated these challenges, causing environmental damage, mass displacement, and the breakdown of healthcare services, all of which contribute to the spread of infectious diseases. This study aims to develop a comprehensive methodology for assessing the impact of emergencies on the spread of infectious diseases, focusing on the full-scale invasion of Ukraine. The object of this study is to address epidemic threats posed by emergencies, particularly the increased spread of infectious diseases due to war-related disruptions. The subject of this study is methods and models of infectious disease transmission under conditions of emergencies, emphasizing the Russian full-scale invasion of Ukraine. The tasks of this study are to provide an analysis of the current state of research and develop a methodology for assessing the impact of emergencies on the spread of infectious diseases. The proposed methodology includes several key components. Comprehensive data from public health organizations includes infectious disease statistics, demographic shifts, healthcare disruptions, and environmental factors exacerbated by emergencies. Data preprocessing removes inconsistencies, standardization of formats, and normalization for population size differences. Machine learning models, including convolutional neural networks and recurrent neural networks, have been developed to simulate the spread of diseases based on demographic, environmental, and healthcare-related variables. Deep learning models analyze spatial and temporal patterns, whereas compartmental models such as SIR estimate changes in reproductive numbers (R₀ and Re). Additionally, models of excess mortality incorporate mixed effects to account for regional and time-based variations. The methodology incorporates real-time monitoring of epidemic threats using real-time data from multiple sources, enabling dynamic assessments of disease spread and facilitating predictive modeling. The models were trained on historical data and validated using cross-validation techniques to ensure robustness and reliability, with a specific focus on the pre- and post-invasion phases in Ukraine.Results: The study provides a comprehensive framework for collecting and processing data on infectious diseases and epidemic threats in emergencies. The proposed model introduces advanced machine learning and epidemiological models trained on pre- and post-invasion data to analyze disease transmission patterns and forecast future epidemic dynamics. Conclusion: The proposed methodology addresses current gaps in infectious disease during emergencies by integrating real-time data and machine learning techniques. This research improves decision-making in public health management and biosafety during crises, particularly in war-affected regions like Ukraine.

Dynamics of Activation and Regulation of the Immune Response to Attack by Viral Pathogens Using Mathematical Modeling

In this paper, a mathematical model is developed to simulate the activation of regulatory T lymphocytes dynamics. The model considers the adaptive immune response and consists of epithelial cells, infected cells, free virus particles, helper and cytotoxic T lymphocytes, B lymphocytes, and regulatory T lymphocytes. A mathematical analysis was carried out to discuss the conditions of existence and stability of equilibrium solutions in terms of the basic reproductive number. In addition, the definitions and properties necessary to preserve the positivity and stability of the model are shown. The precision of these mathematical models can be affected by numerous sources of uncertainty, partly due to the balance between the complexity of the model and its predictive capacity to depict the biological process accurately. Nevertheless, these models can provide remarkably perspectives on the dynamics of infection and assist in identification specific immunological traits that improve our comprehension of immune mechanisms. The theoretical results are validated by numerical simulations using data reported in the literature. The construction, analysis, and simulation of the developed models demonstrate that the increased induced regulatory T lymphocytes effectively suppress the inflammatory response in contrast to similar cells at lower contents, playing a key role in maintaining self-tolerance and immune homeostasis.

Mathematical modeling of Ebola using delay differential equations

Nonlinear delay differential equations (NDDEs) are essential in mathematical epidemiology, computational mathematics, sciences, etc. In this research paper, we have presented a delayed mathematical model of the Ebola virus to analyze its transmission dynamics in the human population. The delayed Ebola model is based on the four human compartments susceptible, exposed, infected, and recovered (SEIR). A time-delayed technique is used to slow down the dynamics of the host population. Two significant stages are analyzed in the said model: Ebola-free equilibrium (EFE) and Ebola-existing equilibrium (EEE). Also, the reproduction number of a model with the sensitivity of parameters is studied. Furthermore, the local asymptotical stability (LAS) and global asymptotical stability (GAS) around the two stages are studied rigorously using the Jacobian matrix Routh–Hurwitz criterion strategies for stability and Lyapunov function stability. The delay effect has been observed in the model in inverse relation of susceptible and infected humans (it means the increase of delay tactics that the susceptibility of humans increases and the infectivity of humans decreases eventually approaches zero which means that Ebola has been controlled into the population). For the numerical results, the Euler method is designed for the system of delay differential equations (DDEs) to verify the results with an analytical model analysis.

Optimal control strategies for infectious diseases with consideration of behavioral dynamics

We present a simple SVIR (susceptible, vaccinated, infected, recovered) model to analyze the spread of COVID‐19, incorporating the influence of the population's caution on the transmission rate, which is considered nonlinear in current number of infected. Demonstrating a positive bound solution confirms the model's biological relevance. Through a formula for the basic reproduction number, we explore the local asymptotic stability of the disease‐free equilibrium (DFE) and endemic equilibrium (EE), showing that the existence of the EE relies on the basic reproduction number. Furthermore, we establish the global stability of the DFE by constructing a Lyapunov function. We present an optimal control problem for vaccination, demonstrating the existence and uniqueness of the optimal strategy. Our simulations indicate that optimal vaccination is effective in reducing infections and costs. We also investigate the effect of integrating education into the model to underscore its importance in decreasing disease transmission rates and reducing the necessity for vaccine uptake.

Modeling and analysis of a two-strain immuno-epidemiological model with reinfection

In this paper, we formulate a two-strain model with reinfection that combines immunological and epidemiological dynamics across scales, using the COVID-19 pandemic as a case study. Firstly, we conduct a qualitative analysis of both within-host and between-host models. For the within-host model, we prove the existence and stability of equilibria, and Hopf bifurcation occurs from the infection equilibrium with immune response. This implies that, under specific immune states, the virus within an infected individual may persist, and its concentration may also oscillate periodically. For the between-host model, the disease-free equilibrium always exists and is locally asymptotically stable when the epidemiological basic reproduction number ℜ0<1. In addition, the model can have boundary equilibria of strain 1 or strain 2, which are locally asymptotically stable under specific conditions. However, the co-existence equilibrium does not exist. Secondly, to explore the infection and transmission mechanisms of two strain models and obtain reliable parameter values, we utilize statistical data to fit the immuno-epidemiological model. Simultaneously, we conduct an identifiability analysis of the immuno-epidemiological model to ensure the robustness of the fitted parameters. The results demonstrate the reliable estimation of parameter ranges for structurally unidentifiable parameters with minor measurement errors using the affine invariant ensemble Markov Chain Monte Carlo algorithm (GWMCMC). Moreover, simulations illustrate that enhancing treatment of patients infected with BA.2 strains to inhibit the number of viruses released by infected cells can significantly reduce disease spread.

Global dynamic analyzes of the discrete SIS models with application to daily reported cases

Emerging infectious diseases, such as COVID-19, manifest in outbreaks of varying magnitudes. For large-scale epidemics, continuous models are often employed for forecasting, while discrete models are preferred for smaller outbreaks. We propose a discrete susceptible-infected-susceptible model that integrates interaction between parasitism and hosts, as well as saturation recovery mechanisms, and undertake a thorough theoretical and numerical exploration of this model. Theoretically, the model incorporating nonlinear recovery demonstrates complex behavior, including backward bifurcations and the coexistence of dual equilibria. And the sufficient conditions that guarantee the global asymptotic stability of a disease-free equilibrium have been obtained. Considering the challenges posed by saturation recovery in theoretical analysis, we then consider the case of linear recovery. Bifurcation analysis for of the linear recovery model displays a variety of bifurcations at the endemic equilibrium, such as transcritical, flip, and Neimark–Sacker bifurcations. Numerical simulations reveal complex dynamic behavior, including backward and fold bifurcations, periodic windows, period-doubling cascades, and multistability. Moreover, the proposed model could be used to fit the daily COVID-19 reported cases for various regions, not only revealing the significant advantages of discrete models in fitting, evaluating, and predicting small-scale epidemics, but also playing an important role in evaluating the effectiveness of prevention and control strategies. Furthermore, sensitivity analyses for the key parameters underscore their significant impact on the effective reproduction number during the initial months of an outbreak, advocating for better medical resource allocation and the enforcement of social distancing measures to curb disease transmission.

Demographic analysis and biotic potential of Spodoptera frugiperda (J.E. Smith) (Lepidoptera: Noctuidae) on pea.

The fall armyworm (FAW), Spodoptera frugiperda (J.E. Smith) (Lepidoptera: Noctuidae) is a highly destructive polyphagous pest that primarily damages maize. Maize is considered a most versatile crop for growing intercrops due to the wide row it needs. Maize-pea intercropping is preferred by small and marginal farmers worldwide due to various advantages including higher yield and improved economic benefits. However, the success of this intercropping system may be hampered if pea could sustain the FAW population. Thus, to clarify the fitness and potential effect of S. frugiperda on pea, we analysed the survival and development of S. frugiperda fed on pea leaves in the laboratory and constructed age-stage and two-sex life tables. Results showed that FAW successfully completed its life cycle when fed on pea and produced fertile offspring. The pre-adult duration was significantly higher on pea than maize. The net reproductive rate, intrinsic and finite rate of population increase on pea (135.06 offspring per individual, 0.12 offspring per individual per day and 1.13 times per day) were all significantly different from those on maize (417.64 offspring per individual, 0.19 offspring per individual per day and 1.21 times per day). The probability of survival of S. frugiperda at each stage was lower when fed on pea leaves than that of maize-fed larvae. Due to the overlapping growth periods of the maize and pea, S. frugiperda can easily proliferate throughout the year by shifting between adjacent crops. Thus, this study revealed the adaptability of S. frugiperda for pea and provides the foundation for further assessment of FAW risk to other inter-crops.

Stationary distribution and mean extinction time in a generalist prey–predator model driven by Lévy noises

In this study, a delayed generalist prey–predator model incorporating the logistic-like growth and Holling type-III functional response for predator subjected to two Lévy noises is established. Such environmental fluctuations can describe abrupt change in population density. First, we conduct stability analysis and determine the critical value of time delay when Hopf bifurcation takes place. Then, the existence of the global positive solution and stability in distribution of stochastic model are studied. Numerical results are given to illustrate the analytical findings, and mainly dedicated to the consequences of time delay and Lévy noises on the stationary distribution, mean extinction time and reproduction rate of prey–predator populations, via stationary probability distribution function (PDF), mean first passage time (MFPT) and relaxation time, respectively. Our findings indicate that the decreases in time delay, noise intensities (Di,i=1,2), and stability indexes within the range of αi∈[1,2] are more likely to maintain the population in a high density state. Phase transition occurs when the system is in the suitable parameter regime. Additionally, there exist optimal noise intensities that slow down switching of prey species from a stable state to an extinction state. Smaller time delay τ and stability indexes αi inhibit population extinction, whereas smaller skewness parameter β speed up prey extinction. More importantly, two noises can tune the enhanced stability characteristic of the system.

On a new species of ostracod from the Brazilian Amazon and its potential for experimental studies in laboratory culture

Ostracods are taxonomically and ecologically diverse small crustaceans that have recently gained prominence in laboratory studies and environmental impact assessment. In this context, the present study aims to assess the applicability of a new freshwater ostracod for experimental studies in the laboratory, and we provide the formal description of Strandesia rondoniensisn. sp. The original specimens for setting the laboratory cultures originated from the Natural Park of Porto Velho, in the Amazon region (Rondônia State, Brazil). The growth and reproductive rates of eleven adult individuals of S. rondoniensisn. sp. were analyzed. The results showed a high morphological resemblance with Neostrandesia striata and Bradleytriebella lineata, even though the new species belongs to Strandesia, indicating evolutionary convergence. The life cycle analysis showed that individuals of S. rondoniensisn. sp. have fast growth and high reproductive rates, which favour their use in laboratory studies. Besides contributing to the knowledge about ostracods in the Amazon region, which has been poorly studied, the life cycle experiment characterizations provided here should promote the use of this new species as a model organism for laboratory studies.

Modeling the Transmission Dynamics and Optimal Control Strategy for Huanglongbing

Huanglongbing (HLB), also known as citrus greening disease, represents a severe and imminent threat to the global citrus industry. With no complete cure currently available, effective control strategies are crucial. This article presents a transmission model of HLB, both with and without nutrient injection, to explore methods for controlling disease spread. By calculating the basic reproduction number (R0) and analyzing threshold dynamics, we demonstrate that the system remains globally stable when R0&lt;1, but persists when R0&gt;1. Sensitivity analyses reveal factors that significantly impact HLB spread on both global and local scales. We also propose a comprehensive optimal control model using the pontryagin minimum principle and validate its feasibility through numerical simulations. Results show that while removing infected trees and spraying insecticides can significantly reduce disease spread, a combination of measures, including the production of disease-free budwood and nursery trees, nutrient solution injection, removal of infected trees, and insecticide application, provides superior control and meets the desired control targets. These findings offer valuable insights for policymakers in understanding and managing HLB outbreaks.

Global Analysis of a Fractional‐Order Hepatitis B Virus Model Under Immune Response in the Presence of Cytokines

AbstractThis research proposes and investigates an epidemiological model to study the dynamic behaviors of the Hepatitis B virus (HBV) under immune response and cytokine influence. The model's stability, positivity, boundedness, and equilibria are analyzed using Lyapunov functional methods and the Routh–Hurwitz criterion under Caputo fractional derivative. The study evaluates nucleoside analogues and interferon treatments, determining critical drug efficiencies. Equilibria, including infection‐free and endemic states, are analyzed using the fundamental reproduction number, , to predict disease elimination. Numerical simulations utilize the fractional Adams method and the L1 scheme, capturing memory traces as the fractional order changes. Results show the L1 scheme effectively captures memory traces, providing empirical support for the theoretical findings. Furthermore, Ulam–Hyers stability is treated according to the equilibrium point, which describes relationships between functions. Notably, the findings of the study yielded profound insights. They revealed that the HBV system remains locally asymptotic stable at disease‐free and the endemic point when . At the same time, the simulations illustrated a correlation between the rate of infection and the rise in infected individuals, indicating the feasibility of eradicating and effectively managing HBV infections through a multifaceted approach and various measures such as vaccination and effective drug administration protocols. The proposed framework can guide medical professionals and decision‐makers in developing effective strategies to limit and eliminate the spread of HBV in the population.

Implications of cypermethrin exposure on population dynamics and fitness traits of laboratory-selected resistant and susceptible populations of Spodoptera litura (Fabricius)

The tobacco cutworm, Spodoptera litura (Fabricius) (Lepidoptera: Noctuidae) is an economically important agricultural polyphagous pest worldwide. It has shown high resistance to several insecticides, including cypermethrin, a synthetic pyrethroid that is used in large-scale commercial agricultural applications. The present study investigated the development of selection-induced resistance to cypermethrin and associated fitness costs in S. litura. After continuous exposure to cypermethrin for consecutive fifteen generations, the cypermethrin-selected population (CYP-Sel) of S. litura developed a 21.2-fold resistance. The CYP-Sel strain had a relative fitness of 0.16 when treated with LC50, prolonged larval duration, and development time. Meanwhile, the strain also showed shorter adult duration, lower fecundity, and hatchability compared with the Unsel-Lab population. CYP-Sel population showed a significant disadvantage in intrinsic rate of natural increase (rm), net reproductive rate (Ro), and finite rate of increase (λ) when compared to the Unsel-Lab population. This knowledge could help to design resistance management strategies against this particular pest, along with potential management strategies to overcome the development of resistance.

Stability and Hopf Bifurcation of a Cytokine-Enhanced HIV Infection Model with Saturation Incidence and Three Delays

To discuss the influence of cytokine-enhanced rate and antibody delay on viral infection, in this paper, we propose a cytokine-enhanced HIV model with saturation incidence, antibody immune response and time delays. Multiple time delays, namely intracellular delay, virus replication delay and antibody delay, are included. First, we prove that the model is well-posed. Then, we obtain two basic reproductive numbers of the model. Next, the stability of the equilibria is investigated by using Lyapunov functions and LaSalle’s invariance principle. The results show that intracellular delay and virus replication delay do not impact on the stability of the three equilibria. However, if antibody immune response delay is positive, we determine some conditions for stability switches of the endemic equilibrium by using antibody immune response delay as a bifurcation parameter. It is concluded that with the increase of antibody immune response delay, the endemic equilibrium loses its stability and the system generates Hopf bifurcation. Finally, the numerical simulation results also show the correctness of the theoretical results.

Insufficient and excessive Ca2+ intake negatively impact the life history performance and disrupt the hemolymph metabolism of Spodoptera litura

Calcium ions (Ca2+), essential as second messengers in all cells, play a pivotal role as micronutrients in insects. However, few studies have explored the effects of both insufficient and excessive Ca2+ intake on life history performance and population parameters. This study examines the impact of varying Ca2+ intake levels—insufficient (0 mg/kg), appropriate (100 mg/kg), and excessive (250 mg/kg)—on the life history performance and population parameters of Spodoptera litura using two-sex life tables. Insufficient and excessive Ca2+ intakes significantly extended the preadult development period and decreased the preadult survival rates of S. litura, compared to those on an appropriate Ca2+ intake. The population parameters (Intrinsic rate of increase (r), Finite rate of increase (λ), and Net reproductive rate (R0)) of S. litura on a 100 mg/kg diet (r = 0.1364, λ = 1.1462, R0 = 390) were significantly higher than those on a 0 mg/kg diet (r = 0.1091, λ = 1.1153, R0 = 130.52). Additionally, untargeted metabolomics analysis revealed that inappropriate Ca2+ levels (either insufficient or excessive) triggered significant up-regulation of 71.1 % and 92.8 % of the metabolites in the hemolymph, respectively, compared to the appropriate Ca2+ intake. Notably, disruptions in metabolite balance affected critical components such as melatonin and melanin within the tryptophan and tyrosine metabolism pathways. These findings underscore that both insufficient and excessive Ca2+ intakes adversely affect the life history performance and disrupt hemolymph metabolic balance in S. litura.