Cervical intraepithelial neoplasia (CIN) is the development of abnormal cells on the surface of the cervix, caused by an infection with human papillomavirus. Although in most cases it is resolved by the immune system, a small percentage of people may develop to more severe CIN, which, if left untreated, can progress to cervical cancer. Cervical cancer is the fourth most common cancer among women worldwide. This work intends to develop a nonlinear mathematical model that describes the dynamics at the microscopic level of the epithelial cells (healthy, dysplastic, cancer and immune), and the viral particles, while accounting for a dual immunotherapy. Using the two-scale convergence method, we derive the effective macroscopic model, thus highlighting the influence of the microscopic properties of the tumor on the overall dynamics of CIN. The resulting model is new in the literature and is simpler to handle as far as the computational aspects are concerned.

Cervical cancer in women remains a major public health challenge in sub-Saharan Africa, where both the late diagnoses and limited financial resources hinder effective prevention and treatment. We propose an economic-epidemiological model that integrates the care pathway for cervical lesions and the financial ability to access different treatment stages. Analytical results establish the existence of a basic economic threshold $ \mathcal{R}_0 $, whose value determines whether the system sustains preventive care or collapses into diagnostic failure. We show that a backward bifurcation can occur when $ \mathcal{R}_0 < 1 $, leading to the coexistence of two stable equilibrium points. Furthermore, we establish the global stability of the system collapse equilibrium when $ \mathcal{R}_0 \leq 1 $ under specific conditions. Sensitivity analysis identifies key economic parameters such as screening subsidy, treatment cost, and patient contribution to the billing account that critically influence model dynamics. Numerical simulations estimate that a minimum subsidy level of approximately 13.1% of the human papillomavirus (HPV) test cost is required to ensure system viability ($ \mathcal{R}_0 > 1 $) and prevent long-term collapse of screening coverage. We further decompose the total treatment cost across the care stages and formulate an optimization problem to determine the optimal allocation of government subsidies, minimizing treatment costs while ensuring 90% coverage of women who test positive for HPV. The results show that the optimal strategy requires the government to cover 21.65% of the total treatment cost, mainly focusing on the eligibility test and Large loop excision of the transformation zone (LLETZ) treatment. These findings highlight the importance of targeted subsidies to guarantee both accessibility and financial sustainability in the treatment of precancerous cervical lesions.

Human papillomavirus (HPV) is the leading sexually transmitted infection and the primary cause of cervical cancer, a major contributor to cancer-related morbidity and mortality among women in sub-Saharan Africa, including Tanzania. This persistent burden highlights the need for effective, evidence-based, and cost-efficient control strategies. Deterministic compartmental ODE model describing HPV transmission in both sexes and progression to cervical cancer, incorporating screening, vaccination, and treatment is developed. Using graph theoretic approach, Re, is derived and used to characterize disease dynamics. Stability analysis shows that the HPV-free equilibrium is globally stable when Re≤1, while an endemic equilibrium exists when Re>1. Model parameters are calibrated using Bayesian MCMC methods with Tanzanian data (2018–2023), ensuring local relevance. Optimal control and ICER analysis are then used to assess intervention efficiency.Results show that screening alone reduces infections (Re=0.96), vaccination alone substantially reduces transmission (Re=0.19) but has limited impact on existing infections, while the combined strategy achieves the greatest reduction and drives Re<1, indicating herd immunity. Cost-effectiveness analysis identifies screening alone as the least costly per infection averted, whereas screening combined with vaccination provides the best overall balance of epidemiological impact and economic efficiency.We recommend prioritizing HPV vaccination for girls aged 9–14 years and integrating it with routine cervical cancer screening within primary healthcare systems, while ensuring timely treatment only for screen-positive cases. This combined strategy should be guided by, Re, as a monitoring threshold to support efficient, cost-effective, and population-level HPV control policies.

Like other viruses, human papillomavirus genotypes can remain dormant for years or decades and later reactivate due to some well-known factors. The activation of such a dormant infection years later can cause many health and behavioural problems at the individual and societal levels, respectively. From a personalised health perspective, reactivation of a dormant HPV, especially high-risk genotypes such as HPV 16 or 18 can worsen the health condition of the infected person, should they be (or happen to be newly) infected with a different HPV genotype. However, detailed mechanisms that impact the health outcome of an infected person due to rebounding dormant HPV have not been mathematically investigated. The core of this paper is to study the dynamics of an in-host high-risk HPV infection, taking into account cell-mediated immunity and the consequences of reactivation of HPV-infected dormant cells using compartmental modelling. The local and global stabilities of the equilibria are established using Lyapunov and LaSalle techniques, and the bifurcation analysis is performed. Compared to the model without the provision of virus particles due to the reactivation of infected dormant cells, our results suggest that such reactivation can exacerbate the health condition of an infected person by promoting the persistence of infection, which further weakens the active immune response and favours the progression of infection to HPV-induced cancers. In addition, we provide the sensitivity analysis of threshold parameters, state variables with respect to model parameters, and numerical simulations are used to illustrate the theoretical results.

Human Papillomavirus (HPV) is a group of contagious viruses primarily transmitted through sexual contact and is a major cause of severe health issues, including cervical cancer. In SubSaharan Africa, including Tanzania, cervical cancer is the leading cause of cancer-related deaths among women of all ages. In 2022, there were 125,699 new cases and 80,614 deaths, making cervical cancer the second most common cancer. Of these, Tanzania recorded 10,868 cases and 6,832 deaths. To reduce the number of girls and female affected by HPV infections, particularly those vulnerable to cervical cancer, we have developed and analyzed a mathematical model for HPV transmission dynamics that incorporates vaccination. The analysis demonstrates the presence of both HPV-free and endemic equilibrium states. By applying the Graph Theoretic method, the reproduction number R-e was computed. The results indicate that the HPV-free equilibrium is globally asymptotically stable when R-e <= 1, while the endemic equilibrium is globally asymptotically stable when R-e > 1. We employed a Markov Chain Monte Carlo (MCMC) method for model calibration, which highlighted several key factors. The interaction between vaccination rates for young girls and older females suggests long-term benefits from vaccinating both groups, contributing to increased herd immunity. Additionally, the strong identifiability of the recovery rate emphasizes its critical role in reducing HPV prevalence and cervical cancer progression. The correlations observed indicate the dual role of vaccination in both preventing infection and promoting recovery. On the other hand, the poor identifiability of the mortality rate points to gaps in understanding the long-term burden of cervical cancer. However, since the data used are synthetic, the uncertainties highlight how important it is to use real data and break it into groups to better understand how different factors affect the results. The herd immunity threshold was calculated to be 0.4417, recommending that at least 55.83% of the population be vaccinated to halt HPV transmission and reduce cervical cancer incidence.

We construct and analyse nonstandard finite difference (NSFD) schemes for two epidemic optimal control problems. Firstly, we consider the well-known MSEIR system that can be used to model childhood diseases such as the measles, with the vaccination as a control intervention. The second optimal control problem is related to the 2014–2016 West Africa Ebola Virus Disease (EVD) outbreak, that came with the unprecedented challenge of the disease spreading simultaneously in three different countries, namely Guinea, Liberia and Sierra Leone, where it was difficult to control the considerable migrations and travels of people inbound and outbound. We develop an extended SEIRD metapopulation model modified by the addition of compartments of quarantined and isolated individuals. The control parameters are the exit screening of travelers and the vaccination of the susceptible individuals. For the two optimal control problems, we provide the results on: (i) the (global) stability of the disease-free and/or endemic equilibria of the state variable systems; (ii) the positivity and boundedness of solutions of the state variables systems; (iii) the existence, uniqueness and characterization of the optimal control solutions that minimizes the cost functional. On the other hand: (iv) we design Euler-based nonstandard finite difference versions of the Forward-Backward Sweep Method (NSFD-FBSM) that are dynamically consistent with the state variable systems; (v) we provide numerical simulations that support the theory and show the superiority of the nonstandard approach over the classical FBSM. The numerical simulations suggest that significantly increasing the coverage of the vaccine with its implementation for adults as well is essential if the recurrence of measles outbreaks is to be stopped in South Africa. They also show that the optimal control vaccination for the 2014-2016 EVD is more efficient than the exit screening intervention.

Bacteriophages, or phages (viruses of bacteria), play significant roles in shaping the diversity of bacterial communities within the human gut. A phage-infected bacterial cell can either immediately undergo lysis (virulent/lytic infection) or enter a stable state within the host as a prophage (lysogeny) until a trigger event, called prophage induction, initiates the lysis process. We develop an approach based on a model structured in terms of time since bacterial infection. We derive important threshold parameters for the asymptotic dynamics of the system and demonstrate that the model's qualitative behavior can range from the extinction of all bacterial types to the persistence of a single type (either lysogenic or nonlysogenic bacteria) or the coexistence of all populations at a positive steady state. We highlight the existence of critical time delay values that lead to the coexistence of all states through periodic oscillations. We also conduct a global sensitivity analysis for an effective bacterial clearance. In scenarios where antibiotics are not sufficiently effective, we identify four key phage parameter traits: (i) the phage induction probability, describing the capacity of prophages to be induced, (ii) the probability of absorption, describing the phages' ability to invade susceptible bacteria, (iii) the reproduction number of susceptible bacteria in the absence of antibiotics, and (iv) the latent period, describing the time since absorption. The obtained results emphasize the effective therapeutic potential of selected phages.

Poliomyelitis is a deadly viral disease caused by various strains and highly contagious. The wild poliovirus (WPV) is the most harmful, attacking the host’s nervous system, leading to paralysis and death within hours. Vaccination remains the main effective prevention measure which protect the population at high risk composed mainly of children under 5 years of age. In Cameroon, a fourth-dose vaccination campaign was introduced to eradicate the disease, and the country was declared polio-free in 2020 by WHO. However, in 2021, two cases were detected again. This paper evaluates the effectiveness of the multi-dose vaccination strategy to eliminate Polio in Cameroon, using mathematical modelling. The study formulates and analyses a compartmental mathematical model with four vaccination classes, focusing on the administration of the oral polio vaccine (OPV) and the inactivated polio vaccine (IPV). Using combined data on reported cases and vaccine coverage, we calibrate and validate the model, estimate the basic reproduction number ℛ_0 , and make predictions about the dynamics of the disease until 2035. The model exhibits a backward bifurcation phenomenon, which is a potential explanation why poliomyelitis is persistent in Cameroon with sporadic cases despite being declared polio-free. Sensitivity analysis reveals that asymptomatic carriers spread the disease more than symptomatic cases. our model suggests that, though the current vaccination scheme may eliminate Polio, a vaccine efficacy 85

Malaria remains a significant public health burden in Sub-Saharan African countries, hindering their development. With no effective vaccine currently available, controlling the malaria vector population remains the most effective preventive measure. A promising strategy to combat this disease involves the use of the bacterium Microsporidia MB to reduce/replace disease-transmitting wild mosquito population. In this paper, we develop a dynamic model to analyze the transmission dynamics of Microsporidia MB within the wild mosquito population, considering both imperfect maternal and horizontal transmissions. In this paper, we develop a dynamic mathematical model to analyze the transmission dynamics of Microsporidia MB within wild mosquito populations, accounting for both imperfect maternal and horizontal transmission. We derive and analyze a threshold condition to assess the potential for Microsporidia MB-infected mosquitoes to invade and persist in the wild population. By establishing conditions for the local stability of equilibrium points, we explore the influence of maternal transmission on the spread of Microsporidia MB. Additionally, we employ a cascade reduction approach to simplify the model into a two-dimensional system and formulate an optimal control problem. Using Pontryagin’s Maximum Principle, we determine optimal release strategies for infected mosquitoes to either replace the wild population or promote coexistence with a significantly reduced wild mosquito population. Through theoretical analysis and numerical simulations, our findings contribute to the understanding and development of effective strategies for malaria control using Microsporidia MB.

Microsporidia MB is an endosymbiont which naturally infects Anopheles mosquitoes. Due to its ability to block Plasmodium transmission, it shows potential as a bio-based agent for the control of malaria. Its self-sustainability is promising, as it can spread through both vertical and horizontal transmissions. However, its low prevalence in mosquito populations remains a challenge. We develop an eco-epidemiological mathematical model describing the co-dynamics of Microsporidia MB (within mosquito population) and malaria (within human population). The model is used to assess the potential of Microsporidia MB-infected mosquitoes on the control of malaria infection. The results on the basic reproduction numbers, the stability of the equilibria, and the existence of bifurcations are obtained, providing conditions for the extinction and persistence of MB-infected mosquitoes. We highlight relevant threshold parameters for the elimination and persistence of MB-infected mosquitoes and malaria-infected individuals. Using real data from Kenya, we found that, given a horizontal transmission rate between 0 and 0.5, a minimum vertical rate of 0.55 is required to avoid extinction of MB-infected mosquitoes. The predicted prevalence of MB-infected mosquitoes using transmission rates reported from lab experiments align with the observed low prevalence of MB-infected mosquitoes in the field, thereby validating our model and results. Finally, predictions indicate that increasing MB mosquito infection could effectively control malaria, with target prevalence varying by region: 15% in Highland, 40% on the coast, and 70% in the Lake region. This study offers insights into the use of bio-based vector population replacement solutions to reduce malaria incidence in regions where Microsporidia MB is prevalent.

Human Papillomavirus (HPV) is a group of contagious viruses primarily transmitted through sexual contact and is a major cause of severe health issues, including cervical cancer. In Sub-Saharan Africa, including Tanzania, cervical cancer is the leading cause of cancer-related deaths among women of all ages. In 2022, there were 125,699 new cases and 80,614 deaths, making cervical cancer the second most common cancer. Of these, Tanzania recorded 10,868 cases and 6,832 deaths. To reduce the number of girls and female affected by HPV infections, particularly those vulnerable to cervical cancer, we have developed and analyzed a mathematical model for HPV transmission dynamics that incorporates vaccination. The analysis demonstrates the presence of both HPV-free and endemic equilibrium states. By applying the Graph Theoretic method, the reproduction number Re was computed. The results indicate that the HPV-free equilibrium is globally asymptotically stable when Re≤1, while the endemic equilibrium is globally asymptotically stable when Re>1. We employed a Markov Chain Monte Carlo (MCMC) method for model calibration, which highlighted several key factors. The interaction between vaccination rates for young girls and older females suggests long-term benefits from vaccinating both groups, contributing to increased herd immunity. Additionally, the strong identifiability of the recovery rate emphasizes its critical role in reducing HPV prevalence and cervical cancer progression. The correlations observed indicate the dual role of vaccination in both preventing infection and promoting recovery. On the other hand, the poor identifiability of the mortality rate points to gaps in understanding the long-term burden of cervical cancer. However, since the data used are synthetic, the uncertainties highlight how important it is to use real data and break it into groups to better understand how different factors affect the results. The herd immunity threshold was calculated to be 0.4417, recommending that at least 55.83% of the population be vaccinated to halt HPV transmission and reduce cervical cancer incidence.

In this paper, we propose a novel mathematical model for indirectly transmitted typhoid fever disease that incorporates the use of modern and traditional medicines as modes of treatment. Theoretically, we provide two Lyapunov functions to prove the global asymptotic stability of the disease-free equilibrium (DFE) and the endemic equilibrium (EE) when the basic reproduction number $ (\mathcal{R}_0) $ is less than one and greater than one, respectively. The model is calibrated using the number of cumulative cases reported in the Penka-Michel health district in Cameroon. The parameter estimates thus obtained give a value of $ \mathcal{R}_0 $ = 1.2058 > 1, which indicates that the disease is endemic in the region. The forecast of the outbreak up to November 2026 suggests that the number of cases will be 21,270, which calls for urgent attention on this endemic disease. A sensitivity analysis with respect to the basic reproduction number is conducted, and the main parameters that impact the widespread of the disease are determined. The analysis highlights that the environmental transmission rate $ \beta $ and the decay rate $ \mu_b $ of the bacteria in the environment are the most influential parameters for $ \mathcal{R}_0 $. This underscores the urgent need for potable water and adequate sanitation within this area to reduce the spread of the disease. Numerically, we illustrate the usefulness of recourse to any mode of treatment to lessen the number of infected cases and the necessity of switching from modern treatment to the traditional treatment, a useful adjuvant therapy. Conversely, we show that the relapse phenomenon increases the burden of the disease. Hence adopting a synergistic therapy approach will significantly mitigate typhoid disease cases and overcome the cycle of poverty within the afflicted communities.

Human papillomavirus (HPV) is a highly prevalent sexually transmitted infection and the primary cause of cervical cancer, which remains a leading cause of cancer-related mortality among women globally. Despite ongoing vaccination efforts, challenges such as latency, persistent infections, and imperfect vaccine coverage complicate disease control. In this study, we develop a novel fractional-order compartmental model using Caputo derivatives to capture the memory and non-local transmission effects inherent in HPV dynamics. We analyze the model’s epidemiological properties by proving positivity, boundedness, and deriving the effective reproduction number (Re) via a Graph Theoretic approach. Stability of disease-free and endemic equilibria is established through Lyapunov theory, complemented by Hyers–Ulam stability to ensure robustness. Parameter estimation is performed using Markov Chain Monte Carlo (MCMC), and sensitivity analysis utilizes Partial Rank Correlation Coefficients (PRCC) to identify key drivers of transmission. Our results indicate that achieving 56% vaccination coverage with 45.5% efficacy can reduce Re below one, supporting herd immunity. Numerical simulations demonstrate that vaccination coverage, timely treatment, and vaccine efficacy critically reduce infection prevalence and disease burden. Furthermore, higher fractional orders accelerate convergence to equilibrium without changing equilibrium values. This work lies in integrating fractional calculus with time-dependent vaccination and treatment controls to realistically model HPV progression and intervention impact. This approach provides a more accurate representation of HPV transmission dynamics, especially the long-term memory effects, thereby offering valuable insights for optimizing public health strategies.

Smallholder farmers rely on their farm earnings to cover operating costs and generate income. That is not an easy task because of the pests, which reduce yields and generate plant protection costs. The farm yield and plant protection depend on the budget capacity of the farmer. In this work, we want to explore conditions for a sustainable and self-financing cabbage farm. We propose then a non-linear mathematical model for cabbage crops by considering the current account of the plantation as a dynamic variable. We assume that this variable increases due to the sale of cabbages, and provides for the seedling purchase, the plant protection costs, and the grower's income. In the first part, we analyze the model without pest management. We determine how the budget must be spent and we show the existence of a double transcritical bifurcation. We quantify the seasonal yield and income, and estimate the damage due to pest herbivory. In the second part, we analyze a slightly simplified version of our model and obtain the existence of a backward bifurcation. Furthermore, we show that botanical pesticides can be used to prevent pest spread with relatively low plant protection costs.

We construct anew metapopulation model for the transmission dynamics and control of the Ebola Virus Disease (EVD) in an environment characterized by considerable migrations and travels of people. It is an extended SEIR model modified by the addition of Quarantine and Isolated compartments to account for travelers who undergo the exit screening. The model is well-fitted by using the reported cases from the neighboring countries Guinea, Liberia and Sierra Leone where the 2014-2016 Ebola outbreak simultaneously arose. We show that the unique disease-free equilibrium (DFE) of the model is unstable or locally asymptotically stable (LAS) depending on whether the control reproduction number is larger or less than unity. In the latter case, we prove that the DFE is globally asymptotically stable (GAS) provided that the exit screening is 100% negative. We also prove the GAS of the DFE by introducing more explicit thresholds, thanks to which the existence of at least one boundary equilibrium is established. We design two new nonstandard finite difference (NSFD) schemes, which preserve the dynamics of the continuous model. Numerical simulations that support the theory highlight that exit screening is useful to mitigate the infection. They also suggest that the disease is controlled or the explicit threshold is less than unity provided that the migration and the exit screening parameters are above a critical value.

The coffee berry borer (CBB) Hypothenemus hampei (Coleoptera: Scolytidae) is the most important insect pest affecting coffee production worldwide and generating huge economic losses. As most of its life cycle occurs inside the coffee berry, its control is extremely difficult. To tackle this issue, we solve an optimal control problem based on a berry age-structured dynamical model that describes the infestation dynamics of coffee berries by CBB during a cropping season. This problem consists in applying a bio-insecticide at discrete times in order to maximise the economic profit of healthy coffee berries, while minimising the CBB population for the next cropping season. We derive analytically the first-order necessary optimality conditions of the control problem. Numerical simulations are provided to illustrate the effectiveness of the optimal control strategy.

Ivermectin (IVM), used alongside mass treatment strategies, has been suggested as a potential tool for reducing malaria transmission. The effectiveness of IVM in shortening vector lifespan depends on the time elapsed between the administration of IVM to the host and the blood meal taken by the vector. This effectiveness is measured by the median effective dose ( $ {\rm ED}_{50} $ ED50), the IVM concentration required to kill 50% of mosquitoes after a specific host exposure period. We use a mathematical model structured by human and vector exposure times to IVM and the model's well-posedness is established through semigroups theory. We calculate the basic reproduction number, linking it to epidemiological dynamics, and show steady states bifurcate at $ \mathcal {R}_0=1 $ R0=1, governed by a constant $ C_{\rm bif} $ Cbif. We identify the optimal human exposure to IVM and intervention interval to reduce prevalence by 10% to 20%. This depends on the IVM formulation ( $ {\rm ED}_{50} $ ED50) and the target number of campaigns in the host population.

In this paper, we propose a two-group deterministic COVID-19 model which takes into account educational campaigns and the fact that people infected with COVID-19 may choose either modern (allopathic) medicine, traditional medicine or may combine the two modes of treatment. The model is analysed in the case where modern medicine is the only mode of treatment and when traditional medicine is taken as an adjuvant (or another mode of treatment). We prove in the first case that the model has a disease-free equilibrium (DFE), globally asymptotically stable when the control reproduction number is less than one and whenever it is greater than one, we prove the local asymptotic stability of the endemic equilibrium. In the second case, we prove that, misconceptions in the population lead to a backward bifurcation phenomenon, which makes the control of the disease more difficult. We derive using the Lyapunov method that a threshold T $\mathcal{T}$ ensures the global asymptotic stability of DFE in some cases when its value is less than one. Both models are fitted using daily COVID-19 cumulative cases reported from January to February 2022 in South Africa. We found a control reproduction number less than one, meaning that COVID-19 will be eliminated. Comparison of the two models fits highlights that misconceptions should be taken into account to accurately describe the dynamics of COVID-19 in South Africa. Numerically, we prove that educational campaigns should focus on preventive measures and both traditional and allopathic medicine health care systems should complement each other in the fight against COVID-19.

Schistosomiasis is classified by WHO as a neglected tropical disease. Recent research works have shown that large-scale development projects involving massive population displacement and water irrigation, such as the construction of dams, lakes, and the development of agricultural areas, favour the proliferation of bilharzia. These observations motivate us to propose a reaction–diffusion model to assess the role of the displacements of humans, snails, cercaria, miracidia in the transmission dynamics of Schistosomiasis. The model incorporates a general non-linear contact functions and density-dependent parameters. The aim is to better understanding the role of spatial interactions on the spread of Schistosomiasis, in order to propose appropriate recommendations for the control of that silent threat. We characterize the basic reproduction number R0 of the model. The uniform persistence theory, the maximum principle are used to conduct an in-depth analysis of both the homogeneous and heterogeneous models. Theoretical results are illustrated through numerical simulations.