Browsing by Author "Kuznetsov, Dmitry"

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Backward Bifurcation and Optimal Control Analysis of a Trypanosoma brucei rhodesiense Model
(MDPI, 2019-10-14) Helikumi, Mlyashimbi; Kgosimore, Moatlhodi; Kuznetsov, Dmitry; Mushayabasa, Steady
In this paper, a mathematical model for the transmission dynamics of Trypanosoma brucei rhodesiense that incorporates three species—namely, human, animal and vector—is formulated and analyzed. Two controls representing awareness campaigns and insecticide use are investigated in order to minimize the number of infected hosts in the population and the cost of implementation. Qualitative analysis of the model showed that it exhibited backward bifurcation generated by awareness campaigns. From the optimal control analysis we observed that optimal awareness and insecticide use could lead to effective control of the disease even when they were implemented at low intensities. In addition, it was noted that insecticide control had a greater impact on minimizing the spread of the disease compared to awareness campaigns
Desirable Dog-Rabies Control Methods in an Urban setting in Africa - a Mathematical Model
(Modern Education and Computer Science Press, 2020-02-08) Renalda, Edwiga Kishinda; Kuznetsov, Dmitry; Kreppel, Katharina
Rabies is a fatal, zoonotic, viral disease that causes an acute inflammation of the brain in humans and other mammals. It is transmitted through contact with bodily fluids of infected mammals, usually via bites or scratches. In this paper, we formulate a deterministic model which measures the effects of different rabies control methods (mass-culling and vaccination of dogs) for urban areas near wildlife, using the Arusha region in Tanzania as an example. Values for various parameters were deduced from five years’ worth of survey data on Arusha’s dog population. Data included vaccination coverage, dog bites and rabies deaths recorded by a local non-governmental organization and the Ministry of Agriculture, Livestock Development and Fisheries of the United Republic of Tanzania. The basic reproduction number R0 and effective reproduction number Re were computed and found to be 1.9 and 1.2 respectively. These imply that the disease is endemic in Arusha. The numerical simulation of the reproduction number shows that vaccination is the most appropriate control method for rabies transmission in urban areas near wildlife reservoirs. The disease free equilibrium ε0 is also computed. If the effective reproduction number Re is computed and found to be less than 1, it implies that it is globally asymptotically stable in the feasible region Φ. If Re > 1 it is implied that there is one equilibrium point which is endemic and it is locally asymptotically stable.
Dividend Maximization Under a Set Ruin Probability Target in the Presence of Proportional and Excess-of-loss Reinsurance
(Applications and Applied Mathematics, 2020-06) Kasumo, Christian; Kasozi, Juma; Kuznetsov, Dmitry
We study dividend maximization with set ruin probability targets for an insurance company whose surplus is modelled by a diffusion perturbed classical risk process. The company is permitted to enter into proportional or excess-of-loss reinsurance arrangements. By applying stochastic control theory, we derive Volterra integral equations and solve numerically using block-by-block methods. In each of the models, we have established the optimal barrier to use for paying dividends provided the ruin probability does not exceed a predetermined target. Numerical examples involving the use of both light- and heavy-tailed distributions are given. The results show that ruin probability targets result in an improvement in the optimal barrier to be used for dividend payouts. This is the case for light- and heavy-tailed distributions and applies regardless of the risk model used.
Dynamic modelling and optimal control analysis of a fractional order chikungunya disease model with temperature effects
(Elsevier, 2023-01-19) Lusekelo, Eva; Helikumi, Mlyashimbi; Kuznetsov, Dmitry; Mushayabasa, Steady
Approximately 1.3 billion inhabitants in 94 countries are estimated to be at risk of chikungunya virus infection. A mechanistic compartmental model based on fractional calculus, the Caputo derivative has been proposed to evaluate the effects of temperature and multiple disease control measures (larvicides use, insecticides and physical barriers) during an outbreak. The proposed model was calibrated based on data from literature and validated with daily chikungunya fever cases reported at Kadmat primary health centre, India. The transmission potential of the disease was examined. Sensitive analyses were conducted through computing partial rank correlation coefficients. Memory effects which are often neglected when mechanistic models are used to model the transmission dynamics of infectious diseases, were found to have a significant effect on the dynamics of chikungunya.
Dynamical and optimal control analysis of a seasonal Trypanosoma brucei rhodesiense model
(AIMS Press, 2020-02-27) Helikumi, Mlyashimbi; Kgosimore, Moatlhodi; Kuznetsov, Dmitry; Mushayabasa, Steady
The e ects of seasonal variations on the epidemiology of Trypanosoma brucei rhodesiense disease is well documented. In particular, seasonal variations alter vector development rates and behaviour, thereby influencing the transmission dynamics of the disease. In this paper, a mathematical model for Trypanosoma brucei rhodesiense disease that incorporates seasonal e ects is presented. Owing to the importance of understanding the e ective ways of managing the spread of the disease, the impact of time dependent intervention strategies has been investigated. Two controls representing human awareness campaigns and insecticides use have been incorporated into the model. The main goal of introducing these controls is to minimize the number of infected host population at low implementation costs. Although insecticides usage is associated with adverse e ects to the environment, in this study we have observed that by totally neglecting insecticide use, e ective disease management may present a formidable challenge. However, if human awareness is combined with low insecticide usage then the disease can be e ectively managed.
A fractional-order Trypanosoma brucei rhodesiense model with vector saturation and temperature dependent parameters
(Springer Nature, 2020-06-11) Helikumi, Mlyashimbi; Kgosimore, Moatlhodi; Kuznetsov, Dmitry; Mushayabasa, Steady
Temperature is one of the integral environmental drivers that strongly affect the distribution and density of tsetse fly population. Precisely, ectotherm performance measures, such as development rate, survival probability and reproductive rate, increase from low values (even zero) at critical minimum temperature, peak at an optimum temperature and then decline to low levels (even zero) at a critical maximum temperature. In this study, a fractional-order Trypanosoma brucei rhodesiense model incorporating vector saturation and temperature dependent parameters is considered. The proposed model incorporates the interplay between vectors and two hosts, humans and animals. We computed the basic reproduction number and established results on the threshold dynamics. Meanwhile, we explored the effects of vector control and screening of infected host on long-term disease dynamics. We determine threshold levels essential to reducing the basic reproduction number to level below unity at various temperature levels. Our findings indicate that vector control and host screening could significantly control spread of the disease at different temperature levels.
Mathematical Model for Optimal Control of Soil-Transmitted Helminth Infection
(Hindawi, 2020-08-01) Lambura, Aristide G.; Mwanga, Gasper G.; Luboobi, Livingstone; Kuznetsov, Dmitry
In this paper, we study the dynamics of soil-transmitted helminth infection. We formulate and analyse a deterministic compartmental model using nonlinear differential equations. The basic reproduction number is obtained and both disease-free and endemic equilibrium points are shown to be asymptotically stable under given threshold conditions. The model may exhibit backward bifurcation for some parameter values, and the sensitivity indices of the basic reproduction number with respect to the parameters are determined. We extend the model to include control measures for eradication of the infection from the community. Pontryagian’s maximum principle is used to formulate the optimal control problem using three control strategies, namely, health education through provision of educational materials, educational messages to improve the awareness of the susceptible population, and treatment by mass drug administration that target the entire population(preschool- and school-aged children) and sanitation through provision of clean water and personal hygiene. Numerical simulations were done using MATLAB and graphical results are displayed. The cost effectiveness of the control measures were done using incremental cost-effective ratio, and results reveal that the combination of health education and sanitation is the best strategy to combat the helminth infection. Therefore, in order to completely eradicate soil-transmitted helminths, we advise investment efforts on health education and sanitation controls.
Mathematical model for the effects of treatment and vaccination controls on the dynamics of rotavirus disease with reference to Uganda
(SCIK Publishing Corporation, 2014) Namawejje, Hellen; Luboobi, Livingstone; Kuznetsov, Dmitry; Wobudeya, Eric
In this paper, while Rotavirus has been a recognised disease for a long time in developing countries like Uganda, the control of this endemic disease is still a challenge. We formulated a mathematical model for the dynamics of Rotavirus disease with both treatment and vaccination. The equilibrium points are determined. The disease free equilibrium points are shown to be locally and globally asymptotically stable. We analyzed different reproduction numbers at different doses of vaccination with treatment. Numerical results indicate that rotavirus can be reduced when one or both interventions are implemented. The study recommends that children should always be treated and also complete all their doses of rotavirus vaccines so as to reduce severe infections.
A mathematical model for the MLND dynamics and Sensitivity analysis in a maize population
(Asian Journal of Mathematics and Applications, 2017) Aloyce, William; Kuznetsov, Dmitry; Luboobi, Livingstone s.
Maize Lethal Necrosis disease (MLND) is a viral disease that can cause fatal damage to the crop of maize plants. This is very common in East Africa countries and Democratic Republic of Congo (DRC). In this manuscript, a mathematical model has been developed to study and analyze the dynamics of the MLND in the maize crop population. The disease free (DFE) and endemic equilibrium (EE) points of the model has been com- puted and the basic reproduction number (R0) derived using the next generation matrix method. We performed sensitivity analysis by using parametric values from literature and estimated ones. We found that the rates of transmission, mo, o and mm are the most positively sensitive parameters. Numerical simulations were also performed to verify the an- alytical results.Thus, this research work recommends that deliberate strategic intervention should be targeted on the disease transmission rates which are signi cant for MLND trans- mission in order to eradicate the disease or reduce the intensity of the disease transmission in the maize population.
Mathematical Model for the Transmission Dynamics of Bovine Tuberculosis in Human and Livestock with Control Strategies
(International Journal of Advances in Scientific Research and Engineering (ijasre), 2019-10) Sabini, Theresia Shirima; Nyerere, Nkuba; Irunde, Jacob Ismail; Kuznetsov, Dmitry
A deterministic mathematical model for bovine tuberculosis (bovine TB) in humans and livestock is formulated and used to assess the effectiveness of dairy products inspection, human treatment and quarantine of infected livestock as the control of the disease transmission. The computed effective reproduction number shows that the disease can be cleared from the population if Re is less than unity and it persists if Re is greater than unity. It means that if Re<1 the disease-free equilibrium is asymptotically stable which means the disease can be cleared from the population and endemic when Re>1 which implies the disease persists in the population. Numerical analysis was carried out to investigate how the controls can help to minimize the spread of the disease.
Mathematical Modelling of Drug Abuse and it’s Effect in the Society
(Annals of Pure and Applied Mathematics, 2021-12) Matonya, Fikiri; Kuznetsov, Dmitry
Modeling and analysis of taeniasis and cysticercosis transmission dynamics in humans, pigs and cattle
(Springer Nature, 2021-03-19) Mwasunda, Joshua A.; Irunde, Jacob I.; Kajunguri, Damian; Kuznetsov, Dmitry
Taeniasis and cysticercosis pose a significant challenge to food safety and public health. Cysticercosis reduces the market value for pigs and cattle by making pork and beef unsafe for consumption. In this paper, a mathematical model for the transmission dynamics of taeniasis and cysticercosis in humans, pigs and cattle is formulated and analyzed. The analysis shows that both the disease free equilibrium (DFE) and the endemic equilibrium (EE) exist. To study the dynamics of the diseases, we derived the basic reproduction number R0 by next generation matrix method. When R0 < 1, the DFE is globally asymptotically stable whereas when R0 > 1 the EE is globally asymptotically stable. The normalized forward sensitivity index was used to determine sensitive parameters to the diseases. Humans’ recruitment rate, probability of humans’ infection with taeniasis and the defecation rate of taenia eggs by humans with taeniasis are the most positive sensitive parameters to diseases’ transmission whereas the human natural death rate is the most negative sensitive parameter. However, it is biologically unethical and not practical to increase human natural mortality rate for disease control. In this case, other parameters with negative sensitivity indices such as death rate of taenia eggs and proportions of unconsumed infected beef and pork can be considered for disease control. Generally, to control the diseases, more efforts should be made directed to reducing the number of humans who have taeniasis and defecate in the open environment. Also meat inspection and indoor keeping of cattle and pigs should be emphasized.
Modeling cryptosporidiosis in humans and cattle: Deterministic and stochastic approaches
(Elsevier, 2023-02-25) Luhanda, Faraja; Irunde, Jacob; Kuznetsov, Dmitry
Cryptosporidiosis is a zoonotic disease caused by Cryptosporidium. The disease poses a public and veterinary health problem worldwide. A deterministic model and its corresponding continuous time Markov chain (CTMC) stochastic model are developed and analyzed to investigate cryptosporidiosis transmission dynamics in humans and cattle. The basic reproduction number for the deterministic model and stochastic threshold for the CTMC stochastic model are computed by the next generation matrix method and multitype branching process, respectively. The normalized forward sensitivity index method is used to determine the sensitivity index for each parameter in . Per capita birth rate of cattle, the rate of cattle to acquire cryptosporidiosis infection from the environment and the rate at which infected cattle shed Cryptosporidium oocysts in the environment play an important role in the persistence of the disease whereas Cryptosporidium oocysts natural death rate, cattle recovery rate and cattle natural death rate are most negative sensitive parameters in the dynamics of cryptosporidiosis. Numerical results for CTMC stochastic model show that the likelihood of cryptosporidiosis extinction is high when it arises from an infected human. However, there is a major outbreak if cryptosporidiosis emerges either from infected cattle or from Cryptosporidium oocysts in the environment or when it emerges from all three infectious compartments. Therefore to control the disease, control measures should focus on maintaining personal and cattle farm hygiene and decontaminating the environment to destroy Cryptosporidium oocysts.
Modeling malaria sensitive and resistant strains with superinfection
(Communications in Mathematical Biology and Neuroscience, 2019-03-08) Bimbiga, Ivy; Irunde, Jacob Ismail; Kuznetsov, Dmitry
In this paper, a deterministic model to examine the dynamics of malaria with sensitive and resistant strains in the presence of superinfection is formulated and presented. The basic reproduction is computed using next-generation method and sensitivity index for each parameter with respect to reproduction number R0 is derived. The rate at which human beings are infected by malaria resistant strain and mosquitoes’ mortality rate are the most sensitive parameters to malaria transmission dynamics. Though analysis shows that the rates at which humans are infected by resistant strain and mosquitoes’ natural mortality rate are more sensitive to the disease transmission dynamics, still mosquitoes’ biting rate plays an important role in the transmission dynamics of malaria. To control malaria infection, interventions which increase mosquitoes’ natural mortality, and decrease their biting rate are recommended.
Modeling the Dynamics of Rabies Transmission with Vaccination and Stability Analysis
(Science Publishing Group, 2015-10-09) Ega, Tesfaye Tadesse; Luboobi, Livingstone; Kuznetsov, Dmitry
In this paper we formulate a deterministic mathematical model for the transmission dynamics of rabies in human and animal within and around Addis Ababa, Ethiopia. Our model involves vaccination program for dog population. The basic reproduction number and effective reproduction numbers are computed and the results are entirely depending on the parameters of dog population, which shows the responsibility of dog population for human and livestock infection. For a specified set of values of parameters as deduced from the data provided by Ethiopian Public Health Institute of Addis Ababa, the basic reproduction number 0 R and the effective reproduction number e R works out to be 2 and 1.6 respectively, which indicates the disease will be endemic. The numerical simulation of reproduction ratio shows that the combination of vaccination, culling of stray dogs and controlling annual crop of new born puppies are the best method to control rabies transmission within and around Adds Ababa. The disease - free equilibrium 0 e is computed. When the effective reproduction number 1 e R < it is proved to be globally asymptotically stable in the feasible region F . When 1 e R > there exists one endemic equilibrium point which is locally asymptotically stable.
Modeling the Effects of Helminth Infection on the Transmission Dynamics of Mycobacterium tuberculosis under Optimal Control Strategies.
(Hindawi, 2020-11-18) Lambura, Aristide; Mwanga, Gasper; Luboobi, Livingstone; Kuznetsov, Dmitry
A deterministic mathematical model for the transmission and control of cointeraction of helminths and tuberculosis is presented, to examine the impact of helminth on tuberculosis and the effect of control strategies. The equilibrium point is established, and the effective reproduction number is computed. The disease-free equilibrium point is confirmed to be asymptotically stable whenever the effective reproduction number is less than the unit. The analysis of the effective reproduction number indicates that an increase in the helminth cases increases the tuberculosis cases, suggesting that the control of helminth infection has a positive impact on controlling the dynamics of tuberculosis. The possibility of bifurcation is investigated using the Center Manifold Theorem. Sensitivity analysis is performed to determine the effect of every parameter on the spread of the two diseases. The model is extended to incorporate control measures, and Pontryagin's Maximum Principle is applied to derive the necessary conditions for optimal control. The optimal control problem is solved numerically by the iterative scheme by considering vaccination of infants for Mtb, treatment of individuals with active tuberculosis, mass drug administration with regular antihelminthic drugs, and sanitation control strategies. The results show that a combination of educational campaign, treatment of individuals with active tuberculosis, mass drug administration, and sanitation is the most effective strategy to control helminth-Mtb coinfection. Thus, to effectively control the helminth-Mtb coinfection, we suggest to public health stakeholders to apply intervention strategies that are aimed at controlling helminth infection and the combination of vaccination of infants and treatment of individuals with active tuberculosis.
Modeling the effects of temperature and heterogeneous biting exposure on chikungunya virus disease dynamics
(Elsevier, 2022-07-06) Lusekelo, Eva; Helikumi, Mlyashimbi; Kuznetsov, Dmitry; Mushayabasa, Steady
Within the last decades, chikungunya virus (CHIKV), a mosquito-borne arboviral disease transmitted by Aedes species mosquitoes has been a growing public health burden. Approximately 1.3 billion inhabitants in 94 countries are estimated to be at risk of chikungunya virus infection. Prior studies suggest that temperature and heterogeneous biting exposure are some of the key determinant factors in transmission dynamics of vector-borne diseases. In order to direct preparedness for future outbreaks, it is imperative to evaluate the effects of heterogeneous biting exposure and temperature variations on transmission dynamics of CHIKV. In this paper, a mathematical model that incorporates heterogeneous biting exposure and temperature effects has been developed and analyzed. The basic reproduction number, an important metric for infectious disease models has been determined. Data from literature has been used to calibrate the model and observed CHIKV data for Kadmat primary health centre, India (2 July to 7 September 2007) has been used to validate the model. Results from the study suggest that disease prevention strategies which are effective at stopping transmission of CHIKV more than 80% of the time, will be highly effective minimizing disease burden during outbreaks. The proposed model can be used to inform policy makers on effective ways of managing CHIKV during outbreaks.
Modeling the Transmission Dynamics of Bovine Tuberculosis
(Hindawi, 2020-03-30) Sabini, Theresia Shirima; Irunde, Jacob Ismail; Kuznetsov, Dmitry
Bovine tuberculosis (bTB) is a bacterial and zoonotic disease which is transmitted through consumption of unpasteurized milk and uncooked meat and inhalation of aerosols. In this paper, a deterministic mathematical model is formulated to study the transmission dynamics of bTB in humans and animals. The basic reproduction number R0 is computed to determine the behavior of the disease. Stability analysis shows that there is a possibility for disease-free equilibrium and endemic equilibrium to coexist when R0 = 1. To determine parameters which drive the dynamics of bTB, we performed sensitivity analysis. The analysis shows that the rate at which dairy products are produced, the rate of transmission of bTB from animal to animal, and the rates at which human acquires bTB from infectious dairy products and animals drive the transmission of bTB. However, the disease decreases as the rate of consumption of dairy products decreases. To control bTB, education campaign, inspection of dairy products, treatment of infected humans, and quarantine of infected animals are recommended.
Modeling Transmission Dynamics of Northern Corn Leaf Blight Disease with Seasonal Weather Variations
(Journal of Mathematics and Informatics, 2019-01-31) Kapange, Fred; Irunde, Jacob Ismail; Kuznetsov, Dmitry
Northern Corn Leaf Blight (NCLB) disease is a fungal foliar disease caused by Exserohilum turcicum. Moderate temperature and high relative humidity are climatical conditions which favor the development of NCLB disease. A deterministic model for transmission dynamics of NCLB disease with seasonal weather variations is developed and vigorously analyzed. The basic reproduction number in the absence and presence of the impact of temperature and relative humidity is computed and the sensitivity analysis performed to determine the parameters’ relationship with basic reproduction number . The analysis shows that transmission rate from pathogen to susceptible maize plants, the pathogen’s shedding rate from infected maize plants to the environment and maize plants disease induced death rate are more sensitive to NCLB disease dynamics and they play an important role in its transmission. On the other hands when there are high or low temperature and low humidity, sensitive negative parameters increases as the results NCLB disease development decreases. However, increasing parameters such as natural death rate of the pathogen, the natural death rate for maize plants and maize plants’ disease induced death rate which have negative indices will reduce new infections. Moderate temperature and high relative humidity influence NCLB disease development.
Modelling the Control of the Impact of Fall Armyworm (Spodoptera frugiperda) Infestations on Maize Production
(Hindawi, 2021-02-18) Daudi, Salamida; Luboobi, Livingstone; Kgosimore, Moatlhodi; Kuznetsov, Dmitry
In this paper, we propose and analyze a stage-structured mathematical model for modelling the control of the impact of Fall Armyworm infestations on maize production. Preliminary analysis of the model in the vegetative and reproductive stages revealed that the two systems had a unique and positively bounded solution for all time . Numerical analysis of the model in both stages under two different cases was also considered: Case 1: different number of the adult moths in the field assumed at and Case 2: the existence of exogenous factors that lead to the immigration of adult moths in the field at time . The results indicate that the destruction of maize biomass which is accompanied by a decrease in maize plants to an average of 160 and 142 in the vegetative and reproductive stages, respectively, was observed to be higher in Case 2 than in Case 1 due to subsequent increase in egg production and density of the caterpillars in first few (10) days after immigration. This severe effect on maize plants caused by the unprecedented number of the pests influenced the extension of the model in both stages to include controls such as pesticides and harvesting. The results further show that the pest was significantly suppressed, resulting in an increase in maize plants to an average of 467 and 443 in vegetative and reproductive stages, respectively.