Optimal control of a two-group malaria transmission model with vaccination.

Malaria is a vector-borne disease that poses major health challenges globally, with the highest burden in children less than 5 years old. Prevention and treatment have been the main interventions measures until the recent groundbreaking highly recommended malaria vaccine by WHO for children below five. A two-group malaria model structured by age with vaccination of individuals aged below 5 years old is formulated and theoretically analyzed. The disease-free equilibrium is globally asymptotically stable when the disease-induced death rate in both human groups is zero. Descarte's rule of signs is used to discuss the possible existence of multiple endemic equilibria. By construction, mathematical models inherit the loss of information that could make prediction of model outcomes imprecise. Thus, a global sensitivity analysis of the basic reproduction number and the vaccination class as response functions using Latin-Hypercube Sampling in combination with partial rank correlation coefficient are graphically depicted. As expected, the most sensitive parameters are related to children under 5 years old. Through the application of optimal control theory, the best combination of interventions measures to mitigate the spread of malaria is investigated. Simulations results show that concurrently applying the three intervention measures, namely: personal protection, treatment, and vaccination of childreen under-five is the best strategy for fighting against malaria epidemic in a community, relative to using either single or any dual combination of intervention(s) at a time.

Mathematical modeling and optimal control of SARS-CoV-2 and tuberculosis co-infection: a case study of Indonesia.

A new mathematical model incorporating epidemiological features of the co-dynamics of tuberculosis (TB) and SARS-CoV-2 is analyzed. Local asymptotic stability of the disease-free and endemic equilibria are shown for the sub-models when the respective reproduction numbers are below unity. Bifurcation analysis is carried out for the TB only sub-model, where it was shown that the sub-model undergoes forward bifurcation. The model is fitted to the cumulative confirmed daily SARS-CoV-2 cases for Indonesia from February 11, 2021 to August 26, 2021. The fitting was carried out using the fmincon optimization toolbox in MATLAB. Relevant parameters in the model are estimated from the fitting. The necessary conditions for the existence of optimal control and the optimality system for the co-infection model is established through the application of Pontryagin's Principle. Different control strategies: face-mask usage and SARS-CoV-2 vaccination, TB prevention as well as treatment controls for both diseases are considered. Simulations results show that: (1) the strategy against incident SARS-CoV-2 infection averts about 27,878,840 new TB cases; (2) also, TB prevention and treatment controls could avert 5,397,795 new SARS-CoV-2 cases. (3) In addition, either SARS-CoV-2 or TB only control strategy greatly mitigates a significant number of new co-infection cases.

Dynamic of a two-strain COVID-19 model with vaccination.

COVID-19 is a respiratory illness caused by an ribonucleic acid (RNA) virus prone to mutations. In December 2020, variants with different characteristics that could affect transmissibility emerged around the world. To address this new dynamic of the disease, we formulate and analyze a mathematical model of a two-strain COVID-19 transmission dynamics with strain 1 vaccination. The model is theoretically analyzed and sufficient conditions for the stability of its equilibria are derived. In addition to the disease-free and endemic equilibria, the model also has single-strain 1 and strain 2 endemic equilibria. Using the center manifold theory, it is shown that the model does not exhibit the phenomenon of backward bifurcation, and global stability of the model equilibria are proved using various approaches. Simulations to support the model theoretical results are provided. We calculate the basic reproductive number R 1 and R 2 for both strains independently. Results indicate that - both strains will persist when R 1 > 1 and R 2 > 1 - Stain 2 could establish itself as the dominant strain if R 1 < 1 and R 2 > 1 , or when R 2 > R 1 > 1 . However, because of de novo herd immunity due to strain 1 vaccine efficacy and provided the initial stain 2 transmission threshold parameter R 2 is controlled to remain below unity, strain 2 will not establish itself/persist in the community.

HIV and COVID-19 co-infection: A mathematical model and optimal control.

A new mathematical model for COVID-19 and HIV/AIDS is considered to assess the impact of COVID-19 on HIV dynamics and vice-versa. Investigating the epidemiologic synergy between COVID-19 and HIV is important. The dynamics of the full model is driven by that of its sub-models; therefore, basic analysis of the two sub-models; HIV-only and COVID-19 only is carried out. The basic reproduction number is computed and used to prove local and global asymptotic stability of the sub-models' disease-free and endemic equilibria. Using the fmincon function in the Optimization Toolbox of MATLAB, the model is fitted to real COVID-19 data set from South Africa. The impact of intervention measures, namely, COVID-19 and HIV prevention interventions and COVID-19 treatment are incorporated into the model using time-dependent controls. It is observed that HIV prevention measures can significantly reduce the burden of co-infections with COVID-19, while effective treatment of COVID-19 could reduce co-infections with opportunistic infections such as HIV/AIDS. In particular, the COVID-19 only prevention strategy averted about 10,500 new co-infection cases, with similar number also averted by the HIV-only prevention control.

Modeling COVID-19 daily cases in Senegal using a generalized Waring regression model.

The rapid spread of the COVID-19 pandemic has triggered substantial economic and social disruptions worldwide. The number of infection-induced deaths in Senegal in particular and West Africa in general are minimal when compared with the rest of the world. We use count regression (statistical) models such as the generalized Waring regression model to forecast the daily confirmed COVID-19 cases in Senegal. The generalized Waring regression model has an advantage over other models such as the negative binomial regression model because it considers factors that cannot be observed or measured, but that are known to affect the number of daily COVID-19 cases. Results from this study reveal that the generalized Waring regression model fits the data better than most of the usual count regression models, and could better explain some of the intrinsic characteristics of the disease dynamics.

Optimal control analysis of a COVID-19 and tuberculosis co-dynamics model.

Tuberculosis and COVID-19 are among the diseases with major global public health concern and great socio-economic impact. Co-infection of these two diseases is inevitable due to their geographical overlap, a potential double blow as their clinical similarities could hamper strategies to mitigate their spread and transmission dynamics. To theoretically investigate the impact of control measures on their long-term dynamics, we formulate and analyze a mathematical model for the co-infection of COVID-19 and tuberculosis. Basic properties of the tuberculosis only and COVID-19 only sub-models are investigated as well as bifurcation analysis (possibility of the co-existence of the disease-free and endemic equilibria). The disease-free and endemic equilibria are globally asymptotically stable. The model is extended into an optimal control system by incorporating five control measures. These are: tuberculosis awareness campaign, prevention against COVID-19 (e.g., face mask, physical distancing), control against co-infection, tuberculosis and COVID-19 treatment. Five strategies which are combinations of the control measures are investigated. Strategy B which focuses on COVID-19 prevention, treatment and control of co-infection yields a better outcome in terms of the number of COVID-19 cases prevented at a lower percentage of the total cost of this strategy.

Impact of environmental transmission and contact rates on Covid-19 dynamics: A simulation study.

The emergence of the COVID-19 pandemic has been a major social and economic challenge globally. Infections from infected surfaces have been identified as drivers of Covid-19 transmission, but many epidemiological models do not include an environmental component to account for indirect transmission. We formulate a deterministic Covid-19 model with both direct and indirect transmissions. The computed basic reproduction number R 0 represents the average number of secondary direct human-to-human infections, and the average number of secondary indirect infections from the environment. Using Partial Rank Correlation Coefficient, we compute sensitivity indices of the basic reproductive number R 0 . As expected, the most significant parameter to reduce initial disease transmission is the natural death rate of pathogens in the environment. Variation of the basic reproduction number for different values of direct and indirect transmissions are numerically investigated. Decreasing the effective direct human-to-human contact rate and indirect transmission from human-to-environment will decrease the spread of the disease as R 0 decreases and vice versa. Since the effective contact rate often accounted for as a factor of the force of infection and other interventions measures such as treatment rate are prominent features of infectious diseases, we consider several functional forms of the incidence function, and numerically investigate their potential impact on the long-term dynamics of the disease. Simulations results revealed some differences for the time and infection to reach its peak. Thus, the choice of the functional form of the force of infection should mainly be influenced by the specifics of the prevention measures being implemented.

COVID-19 and dengue co-infection in Brazil: optimal control and cost-effectiveness analysis.

A mathematical model for the co-interaction of COVID-19 and dengue transmission dynamics is formulated and analyzed. The sub-models are shown to be locally asymptotically stable when the respective reproduction numbers are below unity. Using available data sets, the model is fitted to the cumulative confirmed daily COVID-19 cases and deaths for Brazil (a country with high co-endemicity of both diseases) from February 1, 2021 to September 20, 2021. The fitting was done using the fmincon function in the Optimization Toolbox of MATLAB. Parameters denoting the COVID-19 contact rate, death rate and loss of infection acquired immunity to COVID-19 were estimated using the two data sets. The model is then extended to include optimal control strategies. The appropriate conditions for the existence of optimal control and the optimality system for the co-infection model are established using the Pontryagin's Principle. Different control strategies and their cost-effectiveness analyses were considered and simulated for the model, which include: controls against incident dengue and COVID-19 infections, control against co-infection with a second disease and treatment controls for both dengue and COVID-19. Highlights of the simulation results show that: (1) dengue prevention strategy could avert as much as 870,000 new COVID-19 infections; (2) dengue only control strategy or COVID-19 only control strategy significantly reduces new co-infection cases; (3) the strategy implementing control against incident dengue infection is the most cost-effective in controlling dengue and COVID-19 co-infections.

A Mathematical Model of COVID-19 with Vaccination and Treatment.

We formulate and theoretically analyze a mathematical model of COVID-19 transmission mechanism incorporating vital dynamics of the disease and two key therapeutic measures-vaccination of susceptible individuals and recovery/treatment of infected individuals. Both the disease-free and endemic equilibrium are globally asymptotically stable when the effective reproduction number R 0(v) is, respectively, less or greater than unity. The derived critical vaccination threshold is dependent on the vaccine efficacy for disease eradication whenever R 0(v) > 1, even if vaccine coverage is high. Pontryagin's maximum principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions to optimally mitigate the spread of the disease. The model is fitted with cumulative daily Senegal data, with a basic reproduction number R 0 = 1.31 at the onset of the epidemic. Simulation results suggest that despite the effectiveness of COVID-19 vaccination and treatment to mitigate the spread of COVID-19, when R 0(v) > 1, additional efforts such as nonpharmaceutical public health interventions should continue to be implemented. Using partial rank correlation coefficients and Latin hypercube sampling, sensitivity analysis is carried out to determine the relative importance of model parameters to disease transmission. Results shown graphically could help to inform the process of prioritizing public health intervention measures to be implemented and which model parameter to focus on in order to mitigate the spread of the disease. The effective contact rate b, the vaccine efficacy ε, the vaccination rate v, the fraction of exposed individuals who develop symptoms, and, respectively, the exit rates from the exposed and the asymptomatic classes σ and ϕ are the most impactful parameters.

Malaria and COVID-19 co-dynamics: A mathematical model and optimal control.

Malaria, one of the longest-known vector-borne diseases, poses a major health problem in tropical and subtropical regions of the world. Its complexity is currently being exacerbated by the emerging COVID-19 pandemic and the threats of its second wave and looming third wave. We formulate and analyze a mathematical model incorporating some epidemiological features of the co-dynamics of both malaria and COVID-19. Sufficient conditions for the stability of the malaria only and COVID-19 only sub-models' equilibria are derived. The COVID-19 only sub-model has globally asymptotically stable equilibria while under certain condition, the malaria-only could undergo the phenomenon of backward bifurcation whenever the sub-model reproduction number is less than unity. The equilibria of the dual malaria-COVID19 model are locally asymptotically stable as global stability is precluded owing to the possible occurrence of backward bifurcation. Optimal control of the full model to mitigate the spread of both diseases and their co-infection are derived. Pontryagin's Maximum Principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions for optimal control of the diseases. Though this is not a case study, simulation results to support theoretical analysis of the optimal control suggests that concurrently applying malaria and COVID-19 protective measures could help mitigate their spread compared to applying each preventive control measure singly as the world continues to deal with this unprecedented and unparalleled COVID-19 pandemic.

A Mathematical Model for the Transmission Dynamics of Lymphatic Filariasis with Intervention Strategies.

This manuscript considers the transmission dynamics of lymphatic filariasis with some intervention strategies in place. Unlike previously developed models, our model takes into account both the exposed and infected classes in both the human and mosquito populations, respectively. We also consider vaccinated, treated and recovered humans in the presented model. The global dynamics of the proposed model are completely determined by the basic ([Formula: see text]) and effective reproduction numbers ([Formula: see text]). We then use Lyapunov function theory to find the sufficient conditions for global stability of both the disease-free equilibrium and endemic equilibrium. The Lyapunov functions show that when the basic reproduction number is less than or equal to unity, the disease-free equilibrium is globally asymptotically stable, and when it is greater than unity then the endemic equilibrium is also globally asymptotically stable. Finally, numerical simulations are carried out to investigate the effects of the intervention strategies and key parameters to the spread of lymphatic filariasis. The numerical simulations support the analytical results and illustrate possible model behavioral scenarios.

Mathematical studies on the sterile insect technique for the Chikungunya disease and Aedes albopictus.

Chikungunya is an arthropod-borne disease caused by the Asian tiger mosquito, Aedes albopictus. It can be an important burden to public health and a great cause of morbidity and, sometimes, mortality. Understanding if and when disease control measures should be taken is key to curtail its spread. Dumont and Chiroleu (Math Biosc Eng 7(2):315-348, 2010) showed that the use of chemical control tools such as adulticide and larvicide, and mechanical control, which consists of reducing the breeding sites, would have been useful to control the explosive 2006 epidemic in Réunion Island. Despite this, chemical control tools cannot be of long-time use, because they can induce mosquito resistance, and are detrimental to the biodiversity. It is therefore necessary to develop and test new control tools that are more sustainable, with the same efficacy (if possible). Mathematical models of sterile insect technique (SIT) to prevent, reduce, eliminate or stop an epidemic of Chikungunya are formulated and analysed. In particular, we propose a new model that considers pulsed periodic releases, which leads to a hybrid dynamical system. This pulsed SIT model is coupled with the human population at different epidemiological states in order to assess its efficacy. Numerical simulations for the pulsed SIT, using an appropriate numerical scheme are provided. Analytical and numerical results indicate that pulsed SIT with small and frequent releases can be an alternative to chemical control tools, but only if it is used or applied early after the beginning of the epidemic or as a preventive tool.

Mathematical analysis of a cholera model with public health interventions.

Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. We formulate a mathematical model that captures some essential dynamics of cholera transmission to study the impact of public health educational campaigns, vaccination and treatment as control strategies in curtailing the disease. The education-induced, vaccination-induced and treatment-induced reproductive numbers R(E), R(V), R(T) respectively and the combined reproductive number R(C) are compared with the basic reproduction number R(0) to assess the possible community benefits of these control measures. A Lyapunov functional approach is also used to analyse the stability of the equilibrium points. We perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Graphical representations are provided to qualitatively support the analytical results.

Modeling gonorrhea and HIV co-interaction.

A mathematical model was designed to explore the co-interaction of gonorrhea and HIV in the presence of antiretroviral therapy and gonorrhea treatment. Qualitative and comprehensive mathematical techniques have been used to analyse the model. The gonorrhea-only and HIV-only sub-models are first considered. Analytic expressions for the threshold parameter in each sub-model and the co-interaction model are derived. Global dynamics of this co-interaction shows that whenever the threshold parameter for the respective sub-models and co-interaction model is less than unity, the epidemics dies out, while the reverse results in persistence of the epidemics in the community. The impact of gonorrhea and its treatment on HIV dynamics is also investigated. Numerical simulations using a set of reasonable parameter values show that the two epidemics co-exists whenever their reproduction numbers exceed unity (with no competitive exclusion). Further, simulations of the full HIV-gonorrhea model also suggests that an increase in the number of individuals infected with gonorrhea (either singly or dually with HIV) in the presence of treatment results in a decrease in gonorrhea-only cases, dual-infection cases but increases the number of HIV-only cases.

A theoretical assessment of the effects of smoking on the transmission dynamics of tuberculosis.

Smoking has long being associated with tuberculosis. We present a tuberculosis dynamics model taking into account the fact that some people in the population are smoking in order to assess the effects of smoking on tuberculosis transmission. The epidemic thresholds known as the reproduction numbers and equilibria for the model are determined and stabilities analyzed. Qualitative analysis of the model including positivity and persistence of solutions are presented. The model is numerically analyzed to assess the effects of smoking on the transmission dynamics of tuberculosis. Numerical simulations of the model show that smoking enhances tuberculosis transmission, progression to active disease and in a population of smokers, tuberculosis cannot be controlled even when treatment success is assumed to be as high as 88%. Further, analysis of the reproduction numbers indicates that the number of active tuberculosis cases increases as the number of smokers increase.

Optimal control and sensitivity analysis of an influenza model with treatment and vaccination.

We formulate and analyze the dynamics of an influenza pandemic model with vaccination and treatment using two preventive scenarios: increase and decrease in vaccine uptake. Due to the seasonality of the influenza pandemic, the dynamics is studied in a finite time interval. We focus primarily on controlling the disease with a possible minimal cost and side effects using control theory which is therefore applied via the Pontryagin's maximum principle, and it is observed that full treatment effort should be given while increasing vaccination at the onset of the outbreak. Next, sensitivity analysis and simulations (using the fourth order Runge-Kutta scheme) are carried out in order to determine the relative importance of different factors responsible for disease transmission and prevalence. The most sensitive parameter of the various reproductive numbers apart from the death rate is the inflow rate, while the proportion of new recruits and the vaccine efficacy are the most sensitive parameters for the endemic equilibrium point.

HIV/AIDS dynamics: impact of economic classes with transmission from poor clinical settings.

We formulate and analyze a nonlinear deterministic HIV/AIDS model with two social classes, namely the poor and the rich including transmission from poor clinical settings with a randomly variable population. Four sub-models are derived from the full model, the disease threshold parameters are computed, and it is shown that the disease will die down if these initial threshold parameters are less than unity and will persist if they exceed unity. The impact of economic classes (along with transmission from poor/inadequate clinical settings) on the disease dynamics is assessed, and we observe that even with a single sexual partner, the reproduction number is slightly greater than unity, implying that the additional transmission can only be from clinical settings. Stability (local and global) of both the disease-free and endemic equilibria are then investigated using various techniques of dynamical systems such as Centre Manifold theory and Lyapunov's second method. Analysis on the bifurcation parameter is carried out to assess the impact of related HIV transmission from poor clinical settings. We estimate some of the model parameter values and numerical simulations of the model are represented graphically. Our results show that the prevalence of HIV in rich communities seems to be higher than that in the poor, but the disease develops faster in impoverished individuals.

An age-structured model with delay mortality.

Many species experience aperiodic mortality. Yet, there is little or no understanding of how this event affects population dynamics. We have considered one of the most simple class of age-structured models, namely, the MacKendrick Von Foerster type equations with suitable modifications to suit the purpose of this study. The main result shows the effect of delay in the estimate of the population. If the delay parameter is taken as a period, then the model equations describe the dynamics of seasonal insects such as locusts whose large population decreases very fast.