Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control

A mathematical model of HIV transmission is built and studied in this paper. The system’s equilibrium is calculated. A next-generation matrix is used to calculate the reproduction number. The novel method is used to examine the developed model’s bifurcation and equilibrium stability. The stability analysis result shows that the disease-free equilibrium is locally asymptotically stable if 0 < R 0 < 1 but unstable if R 0 > 1 . However, the endemic equilibrium is locally and globally asymptotically stable if R 0 > 1 and unstable otherwise. The sensitivity analysis shows that the most sensitive parameter that contributes to increasing of the reproduction number is the transmission rate β 2 of HIV transmission from HIV individuals to susceptible individuals and the parameter that contributes to the decreasing of the reproduction number is identified as progression rate η of HIV-infected individuals to AIDS individuals. Furthermore, it is observed that as we change η from 0.1 to 1 , the reproduction number value decreases from 1.205 to 1.189, where the constant value of β 2 = 0.1 . On the other hand, as we change the value of β 2 from 0.1 to 1 , the value of the reproduction number increases from 0.205 to 1.347, where the constant value of η = 0.1 . Further, the developed model is extended to the optimal control model of HIV/AIDS transmission, and the cost-effectiveness of the control strategy is analyzed. Pontraygin’s Maximum Principle (PMP) is applied in the construction of the Hamiltonian function. Moreover, the optimal system is solved using forward and backward Runge–Kutta fourth-order methods. The numerical simulation depicts the number of newly infected HIV individuals and the number of individuals at the AIDS stage reduced as a result of taking control measures. The cost-effectiveness study demonstrates that when combined and used, the preventative and treatment control measures are effective. MATLAB is used to run numerical simulations.

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Cost-effectiveness of influenza control measures: a dynamic transmission model-based analysis

Epidemiology and Infection ◽

10.1017/s0950268813000423 ◽

2013 ◽

Vol 141 (12) ◽

pp. 2581-2594 ◽

Cited By ~ 7

Author(s):

S.-C. CHEN ◽

C.-M. LIAO

Keyword(s):

Cost Effectiveness ◽

Natural Ventilation ◽

Seasonal Influenza ◽

School Setting ◽

Control Measures ◽

Transmission Model ◽

Model Based ◽

Dynamic Transmission Model ◽

Influenza Transmission ◽

The Cost

SUMMARYWe investigated the cost-effectiveness of different influenza control strategies in a school setting in Taiwan. A susceptible-exposure-infected-recovery (SEIR) model was used to simulate influenza transmission and we used a basic reproduction number (R0)–asymptomatic proportion (θ) control scheme to develop a cost-effectiveness model. Based on our dynamic transmission model and economic evaluation, this study indicated that the optimal cost-effective strategy for all modelling scenarios was a combination of natural ventilation and respiratory masking. The estimated costs were US$10/year per person in winter for one kindergarten student. The cost for hand washing was estimated to be US$32/year per person, which was much lower than that of isolation (US$55/year per person) and vaccination (US$86/year per person) in containing seasonal influenza. Transmission model-based, cost-effectiveness analysis can be a useful tool for providing insight into the impacts of economic factors and health benefits on certain strategies for controlling seasonal influenza.

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Analisis kestabilan dan kontrol optimal model matematika penyebaran penyakit Ebola dengan variabel kontrol berupa karantina

Jambura Journal of Biomathematics (JJBM) ◽

10.34312/jjbm.v2i1.10258 ◽

2021 ◽

Vol 2 (1) ◽

pp. 29-41

Author(s):

Erzalina Ayu Satya Megananda ◽

Cicik Alfiniyah ◽

Miswanto Miswanto

Keyword(s):

Optimal Control ◽

Human Population ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Optimal Model ◽

The Family ◽

The Stability ◽

The Cost ◽

Two Equilibria

Ebola disease is an infectious disease caused by a virus from the genus Ebolavirus and the family Filoviridae. Ebola disease is one of the most deadly diseases for human. The purpose of the thesis is to analyze the stability of the equilibrium point and to apply the optimal control of quarantine on a mathematical model of the spread of ebola. Model without control has two equilibria, non-endemic equilibrium and endemic equilibrium. The existence of endemic equilibrium and local stability depends on the basic reproduction number (R0). The non-endemic equilibrium is asymptotically stable if R0 1 and endemic equilibrium tend to asymptotically stable if R0 1. The problem of optimal control is solved by Pontryagin’s Maximum Principle. From the numerical simulation, the result shows that control is effective enough to minimize the number of infected human population and to minimize the cost of its control.

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Biological and Chemical Control of Mosquito Population by Optimal Control Approach

Games ◽

10.3390/g11040062 ◽

2020 ◽

Vol 11 (4) ◽

pp. 62

Author(s):

Juddy Heliana Arias-Castro ◽

Hector Jairo Martinez-Romero ◽

Olga Vasilieva

Keyword(s):

Biological Control ◽

Optimal Control ◽

Cost Effectiveness ◽

Negative Impact ◽

Control Measures ◽

External Control ◽

Modeling Framework ◽

Control Intervention ◽

The Cost ◽

Predacious Species

This paper focuses on the design and analysis of short-term control intervention measures seeking to suppress local populations of Aedes aegypti mosquitoes, the major transmitters of dengue and other vector-borne infections. Besides traditional measures involving the spraying of larvicides and/or insecticides, we include biological control based on the deliberate introduction of predacious species feeding on the aquatic stages of mosquitoes. From the methodological standpoint, our study relies on application of the optimal control modeling framework in combination with the cost-effectiveness analysis. This approach not only enables the design of optimal strategies for external control intervention but also allows for assessment of their performance in terms of the cost-benefit relationship. By examining numerous scenarios derived from combinations of chemical and biological control measures, we try to find out whether the presence of predacious species at the mosquito breeding sites may (partially) replace the common practices of larvicide/insecticide spraying and thus reduce their negative impact on non-target organisms. As a result, we identify two strategies exhibiting the best metrics of cost-effectiveness and provide some useful insights for their possible implementation in practical settings.

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The Cost-Effectiveness Analysis and Optimal Strategy of the Tobacco Control

Computational and Mathematical Methods in Medicine ◽

10.1155/2019/8189270 ◽

2019 ◽

Vol 2019 ◽

pp. 1-15

Author(s):

Liuyong Pang ◽

Sanhong Liu ◽

Xinan Zhang ◽

Tianhai Tian

Keyword(s):

Smoking Cessation ◽

Cost Effectiveness ◽

Basic Reproduction Number ◽

Media Coverage ◽

Reproduction Number ◽

Control Measures ◽

Smoking Cessation Treatment ◽

Cessation Treatment ◽

The Cost ◽

The Basic Reproduction Number

This paper aims at investigating how the media coverage and smoking cessation treatment should be implemented, for a certain period, to reduce the numbers of smokers and patients caused by smoking while minimizing the total cost. To this end, we first propose a new mathematical model without any control strategies to investigate the dynamic behaviors of smoking. Furthermore, we calculate the basic reproduction number R0 and discuss the global asymptotic stabilities of the equilibria. Then, from the estimated parameter values, we know that the basic reproduction number R0 is more than 1, which reveals that smoking is one of the enduring problems of the society. Hence, we introduce two control measures (media coverage and smoking cessation treatment) into the model. Finally, in order to investigate their effects in smoking control and provide an analytical method for the strategic decision-makers, we apply a concrete example to calculate the incremental cost-effectiveness ratios and analyze the cost-effectiveness of all possible combinations of the two control measures. The results indicate that the combination of media coverage and smoking cessation treatment is the most cost-effective strategy for tobacco control.

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503. The Misappropriation of COVID-19 Related TV Ad Expenditures

Open Forum Infectious Diseases ◽

10.1093/ofid/ofaa439.696 ◽

2020 ◽

Vol 7 (Supplement_1) ◽

pp. S317-S317

Author(s):

Kartavya J Vyas

Keyword(s):

Cost Effectiveness ◽

Performance Metrics ◽

Control Measures ◽

Mortality Data ◽

Preventive Interventions ◽

Social Distancing ◽

Epidemic Curve ◽

Incidence And Mortality ◽

The Cost ◽

The U.S

Abstract Background With nearly three-fourths of the U.S. population isolated in their homes between early March and the end of May, almost all of whom regularly watch television (TV), it was no surprise that companies began to purchase airtime on major television networks to advertise (ad) their brands and showcase their empathy with the populace. But how would the coronavirus disease 2019 (COVID-19) epidemic curve have changed had these same dollars been allocated to proven preventive interventions? Methods Performance and activity metrics on all COVID-19 related TV ads that have aired in the U.S. between February 26th and June 7th, 2020, were provided by iSpot.tv, Inc., including expenditures. COVID-19 incidence and mortality data were collected from the Centers for Disease Control and Prevention (CDC). Descriptive statistics were performed to calculate total TV ad expenditures and other performance metrics across industry categories. Leveraging a previously published stochastic agent-based model that was used to assess the cost-effectiveness of non-pharmaceutical interventions to control COVID-19, the number of cases that would have been prevented had these same dollars been used for preventive interventions was calculated using cost-effectiveness ratios (CERs), the cost divided by cases prevented. Results A total of 1,513 companies purchased TV airtime during the study period, totaling approximately 1.1 million airings, 215.5 billion impressions, and $2.7 billion in expenditures; most of the expenditures were spent by the restaurant (15.9%), electronics and communications (15.4%), and vehicle (13.7%) industries. The CERs for PPE and social distancing measures were $13,856 and $29,552, respectively; therefore, had all of these TV ad dollars instead been allocated to PPE or social distancing measures, approximately 194,908 and 91,386 cases of COVID-19 may have been prevented by the end of the study period, respectively. Figure 2. COVID-19 cases prevented had TV ad expenditures been reallocated for interventions. Conclusion Americans were inundated with COVID-19 related TV ads during the early months of the pandemic and companies are now showing some signs to relent. In times of disaster, however, it is paramount that the private sector go beyond showcasing their empathy and truly become socially responsible by allocating their funds to proven prevention and control measures. Disclosures All Authors: No reported disclosures

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Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination

Scientific Reports ◽

10.1038/s41598-021-95913-8 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Md Abdul Kuddus ◽

M. Mohiuddin ◽

Azizur Rahman

Keyword(s):

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Equilibrium Points ◽

Rank Correlation ◽

Dynamical Analysis ◽

Model Parameters ◽

Progression Rate ◽

Double Dose ◽

The Basic Reproduction Number

AbstractAlthough the availability of the measles vaccine, it is still epidemic in many countries globally, including Bangladesh. Eradication of measles needs to keep the basic reproduction number less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}<1)$$ ( i . e . R 0 < 1 ) . This paper investigates a modified (SVEIR) measles compartmental model with double dose vaccination in Bangladesh to simulate the measles prevalence. We perform a dynamical analysis of the resulting system and find that the model contains two equilibrium points: a disease-free equilibrium and an endemic equilibrium. The disease will be died out if the basic reproduction number is less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{ R}}_{0}<1)$$ ( i . e . R 0 < 1 ) , and if greater than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}>1)$$ ( i . e . R 0 > 1 ) epidemic occurs. While using the Routh-Hurwitz criteria, the equilibria are found to be locally asymptotically stable under the former condition on $${\mathrm{R}}_{0}$$ R 0 . The partial rank correlation coefficients (PRCCs), a global sensitivity analysis method is used to compute $${\mathrm{R}}_{0}$$ R 0 and measles prevalence $$\left({\mathrm{I}}^{*}\right)$$ I ∗ with respect to the estimated and fitted model parameters. We found that the transmission rate $$(\upbeta )$$ ( β ) had the most significant influence on measles prevalence. Numerical simulations were carried out to commissions our analytical outcomes. These findings show that how progression rate, transmission rate and double dose vaccination rate affect the dynamics of measles prevalence. The information that we generate from this study may help government and public health professionals in making strategies to deal with the omissions of a measles outbreak and thus control and prevent an epidemic in Bangladesh.

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Estimated transmissibility and impact of SARS-CoV-2 lineage B.1.1.7 in England

Science ◽

10.1126/science.abg3055 ◽

2021 ◽

Vol 372 (6538) ◽

pp. eabg3055 ◽

Cited By ~ 82

Author(s):

Nicholas G. Davies ◽

Sam Abbott ◽

Rosanna C. Barnard ◽

Christopher I. Jarvis ◽

Adam J. Kucharski ◽

...

Keyword(s):

United States ◽

Severe Acute Respiratory Syndrome ◽

Reproduction Number ◽

The United States ◽

Control Measures ◽

Transmission Model ◽

Educational Institutions ◽

Stringent Control ◽

Dynamic Transmission Model ◽

Credible Intervals

A severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) variant, VOC 202012/01 (lineage B.1.1.7), emerged in southeast England in September 2020 and is rapidly spreading toward fixation. Using a variety of statistical and dynamic modeling approaches, we estimate that this variant has a 43 to 90% (range of 95% credible intervals, 38 to 130%) higher reproduction number than preexisting variants. A fitted two-strain dynamic transmission model shows that VOC 202012/01 will lead to large resurgences of COVID-19 cases. Without stringent control measures, including limited closure of educational institutions and a greatly accelerated vaccine rollout, COVID-19 hospitalizations and deaths across England in the first 6 months of 2021 were projected to exceed those in 2020. VOC 202012/01 has spread globally and exhibits a similar transmission increase (59 to 74%) in Denmark, Switzerland, and the United States.

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Fractional-Order Modelling and Optimal Control of Cholera Transmission

Fractal and Fractional ◽

10.3390/fractalfract5040261 ◽

2021 ◽

Vol 5 (4) ◽

pp. 261

Author(s):

Silvério Rosa ◽

Delfim F. M. Torres

Keyword(s):

Mathematical Model ◽

Optimal Control ◽

Sensitivity Analysis ◽

Fractional Order ◽

Control Measures ◽

Time Interval ◽

Variable Order ◽

Effectiveness Analysis ◽

The Cost ◽

Derivative Order

A Caputo-type fractional-order mathematical model for “metapopulation cholera transmission” was recently proposed in [Chaos Solitons Fractals 117 (2018), 37–49]. A sensitivity analysis of that model is done here to show the accuracy relevance of parameter estimation. Then, a fractional optimal control (FOC) problem is formulated and numerically solved. A cost-effectiveness analysis is performed to assess the relevance of studied control measures. Moreover, such analysis allows us to assess the cost and effectiveness of the control measures during intervention. We conclude that the FOC system is more effective only in part of the time interval. For this reason, we propose a system where the derivative order varies along the time interval, being fractional or classical when more advantageous. Such variable-order fractional model, that we call a FractInt system, shows to be the most effective in the control of the disease.

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Simple MSEIR Model for Measles Transmission

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v12i330090 ◽

2019 ◽

pp. 1-11

Author(s):

Mojeeb Al-Rahman EL-Nor Osman ◽

Appiagyei Ebenezer ◽

Isaac Kwasi Adu

Keyword(s):

Mathematical Model ◽

Basic Reproduction Number ◽

Recovery Rate ◽

Birth Rate ◽

Reproduction Number ◽

Progression Rate ◽

Asymptotically Stable ◽

The Stability ◽

Disease Free ◽

The Basic Reproduction Number

In this paper, an Immunity-Susceptible-Exposed-Infectious-Recovery (MSEIR) mathematical model was used to study the dynamics of measles transmission. We discussed that there exist a disease-free and an endemic equilibria. We also discussed the stability of both disease-free and endemic equilibria. The basic reproduction number is obtained. If , then the measles will spread and persist in the population. If , then the disease will die out. The disease was locally asymptotically stable if and unstable if . ALSO, WE PROVED THE GLOBAL STABILITY FOR THE DISEASE-FREE EQUILIBRIUM USING LASSALLE'S INVARIANCE PRINCIPLE OF Lyaponuv function. Furthermore, the endemic equilibrium was locally asymptotically stable if , under certain conditions. Numerical simulations were conducted to confirm our analytic results. Our findings were that, increasing the birth rate of humans, decreasing the progression rate, increasing the recovery rate and reducing the infectious rate can be useful in controlling and combating the measles.

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The Role of Migration in Maintaining the Transmission of Avian Influenza in Waterfowl: A Multisite Multispecies Transmission Model along East Asian-Australian Flyway

Canadian Journal of Infectious Diseases and Medical Microbiology ◽

10.1155/2018/3420535 ◽

2018 ◽

Vol 2018 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Akira Endo ◽

Hiroshi Nishiura

Keyword(s):

Avian Influenza ◽

Reproduction Number ◽

Transmission Rate ◽

Temporal Variations ◽

Transmission Model ◽

Global Dynamics ◽

Long Distance ◽

Transmission Potential ◽

Influenza Transmission

Background. Migratory waterfowl annually migrate over the continents along the routes known as flyways, serving as carriers of avian influenza virus across distant locations. Prevalence of influenza varies with species, and there are also geographical and temporal variations. However, the role of long-distance migration in multispecies transmission dynamics has yet to be understood. We constructed a mathematical model to capture the global dynamics of avian influenza, identifying species and locations that contribute to sustaining transmission.Methods. We devised a multisite, multispecies SIS (susceptible-infectious-susceptible) model, and estimated transmission rates within and between species in each geographical location from prevalence data. Parameters were directly sampled from posterior distribution under Bayesian inference framework. We then analyzed contribution of each species in each location to the global patterns of influenza transmission.Results. Transmission and migration parameters were estimated by Bayesian posterior sampling. The basic reproduction number was estimated at 1.1, slightly above the endemic threshold. Mallard was found to be the most important host with the highest transmission potential, and high- and middle-latitude regions appeared to act as hotspots of influenza transmission. The local reproduction number suggested that the prevalence of avian influenza in the Oceania region is dependent on the inflow of infected birds from other regions.Conclusion. Mallard exhibited the highest transmission rate among the species explored. Migration was suggested to be a key factor of the global prevalence of avian influenza, as transmission is locally sustainable only in the northern hemisphere, and the virus could be extinct in the Oceania region without migration.

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