Dynamical analysis, optimal control and spatial pattern in an influenza model with adaptive immunity in two stratified population

<abstract>Consistently, influenza has become a major cause of illness and mortality worldwide and it has posed a serious threat to global public health particularly among the immuno-compromised people all around the world. The development of medication to control influenza has become a major challenge now. This work proposes and analyzes a structured model based on two geographical areas, in order to study the spread of influenza. The overall underlying population is separated into two sub populations: urban and rural. This geographical distinction is required as the immunity levels are significantly higher in rural areas as compared to urban areas. Hence, this paper is a novel attempt to proposes a linear and non-linear mathematical model with adaptive immunity and compare the host immune response to disease. For both the models, disease-free equilibrium points are obtained which are locally as well as globally stable if the reproduction number is less than 1 (<italic>R</italic>01 < 1 & <italic>R</italic>02 < 1) and the endemic point is stable if the reproduction number is greater then 1 (<italic>R</italic>01 > 1 & <italic>R</italic>02 > 1). Next, we have incorporated two treatments in the model that constitute the effectiveness of antidots and vaccination in restraining viral creation and slow down the production of new infections and analyzed an optimal control problem. Further, we have also proposed a spatial model involving diffusion and obtained the local stability for both the models. By the use of local stability, we have derived the Turing instability condition. Finally, all the theoretical results are verified with numerical simulation using MATLAB.</abstract>

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Fractional optimal control of COVID-19 pandemic model with generalized Mittag-Leffler function

Advances in Difference Equations ◽

10.1186/s13662-021-03546-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Amir Khan ◽

Rahat Zarin ◽

Usa Wannasingha Humphries ◽

Ali Akgül ◽

Anwar Saeed ◽

...

Keyword(s):

Optimal Control ◽

Existence And Uniqueness ◽

Local Stability ◽

Reproduction Number ◽

Equilibrium Points ◽

Control Analysis ◽

Fractional Operator ◽

Uniqueness Of The Solution ◽

Alternative Type ◽

Alternative Type Theorems

AbstractIn this paper, we consider a fractional COVID-19 epidemic model with a convex incidence rate. The Atangana–Baleanu fractional operator in the Caputo sense is taken into account. We establish the equilibrium points, basic reproduction number, and local stability at both the equilibrium points. The existence and uniqueness of the solution are proved by using Banach and Leray–Schauder alternative type theorems. For the fractional numerical simulations, we use the Toufik–Atangana scheme. Optimal control analysis is carried out to minimize the infection and maximize the susceptible people.

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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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Utilisasi Kelambu Berinsektisida Pada Daerah Endemis Tinggi Malaria di Indonesia: Analisis Data Riset Kesehatan Dasar (Riskesdas) 2018

Buletin Penelitian Kesehatan ◽

10.22435/bpk.v49i1.3428 ◽

2021 ◽

Vol 49 (1) ◽

pp. 9-20

Author(s):

Asep Hermawan

Keyword(s):

Logistic Regression ◽

Rural Areas ◽

Urban Areas ◽

Household Head ◽

Public Health Problem ◽

Bed Nets ◽

Global Public Health ◽

Cross Sectional ◽

Endemic Areas ◽

Insecticide Treated Bed Nets

Malaria is still a global public health problem, especially in the tropical countries including Indonesia. The use of insecticide-treated bed nets (ITN’s) is an effective way to reduce the prevalence of malaria. However, the proportion of households that use ITN’s in low to high malaria edemic areas in Indonesia is still low (15.8%). The purpose of the analysis is to asses the determinants affecting the use of ITN’s in high endemic areas in Indonesia in 2018. The analysis uses the Basic Health Research (Riskesdas) 2018 data whose design is a cross-sectional study. The population is residents in 28 high malaria endemic districts/ cities with the number of samples interviewed were 33,001 people. The dependent variable was the habit of using ITN’s, while the independent variable is sociodemographic factors (relationship with household head, age group, education , occupations, and residential/ urban or rural areas) and the other ways to prevent mosquito bites. Data were analyzed using logistic regression test. The logistic regression analysis showed that the population with the characteristics of living in urban areas (aOR 2.55, 95% CI 2.38-2.74), parents (aOR 1.29, 95% CI 1.02-1.64), farmers (aOR 1.69, 95% CI 1.49-1.92) and completed Junior High (aOR 1.61, 95% CI 1.35-1.91), have the opportunity to use insecticide-treated bed nets, while the method of preventing mosquito bites others, most of them are protective of this habit. A new canal initiative is needed to increase people's knowledge about the importance of using insecticide-treated bed nets. Keyword: insecticide-treated bed nets, high malaria endemic areas Abstrak Malaria masih menjadi masalah kesehatan masyarakat global, terutama di daerah tropis termasuk Indonesia. Penggunaan kelambu berinsektisida merupakan cara efektif untuk mengurangi prevalensi malaria. Namun, proporsi rumah tangga yang menggunakan kelambu berinsektisida di daerah edemis rendah sampai tinggi di Indonesia masih rendah (15,8%). Tujuan analisis ini adalah untuk mengetahui determinan yang berpengaruh terhadap penggunaan kelambu berinsektisida di daerah endemis tinggi malaria di Indonesia pada 2018. Analisis ini menggunakan data Riset Kesehatan Dasar (Riskesdas) 2018 yang desainnya adalah studi potong lintang. Populasi pada studi ini adalah penduduk di 28 kabupaten/ kota dengan katagori endemis malaria tinggi dengan jumlah sampel yang diwawancara sebanyak 33.001 orang. Variabel dependen adalah kebiasaan penggunaan kelambu berinsektisida, sedangkan variabel independen adalah faktor sosiodemografi (hubungan dengan KRT, kelompok usia, tingkat pendidikan, jenis pekerjaan, dan wilayah tempat tinggal/ perkotaan atau pedesaan) dan cara pencegahan gigitan nyamuk. Data dianalisis menggunakan uji logistic regression. Hasil analisis logistic regression menunjukkan bahwa penduduk dengam dengan karakteristik tinggal di perkotaan (aOR 2,55, 95%CI2,38-2,74), orang tua (aOR1,29, 95%CI 1,02-1,64), petani (aOR1,69, 95%CI 1,49-1,92) dan tamat SLTP/MTS (aOR 1,61, 95%CI 1,35-1,91), berpeluang menggunakan kelambua berinsektisida, sedangkan cara pencegahan gigitan nyamuk lainnya, sebagian besar bersifat protektif terhadap kebiasaan ini. Perlu inisiatif kanal baru untuk meningkatkan pengetahuan penduduk tentang pentingnya utilisasi kelambu berinsektisida. Kata kunci: kelambu berinsektisida, daerah endemis malaria tinggi

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Mathematical Modeling and Optimal Control Analysis of COVID-19 in Ethiopia

10.1101/2020.07.23.20160473 ◽

2020 ◽

Cited By ~ 1

Author(s):

Haileyesus Tessema Alemneh ◽

Getachew Teshome Telahun

Keyword(s):

Optimal Control ◽

Necessary Conditions ◽

Reproduction Number ◽

Control Strategies ◽

Equilibrium Points ◽

Control Model ◽

Control Analysis ◽

Basic Model ◽

Short Period ◽

Disease Free

In this paper we developed a deterministic mathematical model of the pandemic COVID-19 transmission in Ethiopia, which allows transmission by exposed humans. We proposed an SEIR model using system of ordinary differential equations. First the major qualitative analysis, like the disease free equilibruim point, endemic equilibruim point, basic reproduction number, stability analysis of equilibrium points and sensitivity analysis was rigorously analysed. Second, we introduced time dependent controls to the basic model and extended to an optimal control model of the disease. We then analysed using Pontryagins Maximum Principle to derive necessary conditions for the optimal control of the pandemic. The numerical simulation indicated that, an integrated strategy effective in controling the epidemic and the gvernment must apply all control strategies in combating COVID-19 at short period of time.

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Spatiotemporal spread of chikungunya virus in Sarawak, Malaysia

Transactions of the Royal Society of Tropical Medicine and Hygiene ◽

10.1093/trstmh/trab053 ◽

2021 ◽

Author(s):

Sarat Dass ◽

Romano Ngui ◽

Balvinder Singh Gill ◽

Yoke Fun Chan ◽

Wan Yusoff Wan Sulaiman ◽

...

Keyword(s):

Spatial Data ◽

Rural Areas ◽

Urban Areas ◽

Chikungunya Virus ◽

Reproduction Number ◽

Control Measures ◽

River Networks ◽

Scan Statistic ◽

R Software ◽

Time Scan

Abstract Background We studied the spatiotemporal spread of a chikungunya virus (CHIKV) outbreak in Sarawak state, Malaysia, during 2009–2010. Methods The residential addresses of 3054 notified CHIKV cases in 2009–2010 were georeferenced onto a base map of Sarawak with spatial data of rivers and roads using R software. The spatiotemporal spread was determined and clusters were detected using the space-time scan statistic with SaTScan. Results Overall CHIKV incidence was 127 per 100 000 population (range, 0–1125 within districts). The average speed of spread was 70.1 km/wk, with a peak of 228 cases/wk and the basic reproduction number (R0) was 3.1. The highest age-specific incidence rate was 228 per 100 000 in adults aged 50–54 y. Significantly more cases (79.4%) lived in rural areas compared with the general population (46.2%, p<0.0001). Five CHIKV clusters were detected. Likely spread was mostly by road, but a fifth of rural cases were spread by river travel. Conclusions CHIKV initially spread quickly in rural areas mainly via roads, with lesser involvement of urban areas. Delayed spread occurred via river networks to more isolated areas in the rural interior. Understanding the patterns and timings of arboviral outbreak spread may allow targeted vector control measures at key transport hubs or in large transport vehicles.

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ANALISIS KESTABILAN MODEL PENYEBARAN PENYAKIT TUBERKULOSIS DENGAN LAJU INFEKSI TERSATURASI (STABILITY ANALYSIS OF TUBERCOLOSIS SPREAD DISEASES MODEL CONSIDERING SATURATED INFECTION RATE)

Jurnal Matematika Thales ◽

10.22146/jmt.51490 ◽

2020 ◽

Vol 2 (1) ◽

Author(s):

Handika Lintang Saputra ◽

Sutimin Sutimin ◽

Sutrisno Sutrisno

Keyword(s):

Equilibrium Point ◽

Local Stability ◽

Stability Property ◽

Reproduction Number ◽

Equilibrium Points ◽

Dynamical Behavior ◽

Asymptotically Stable ◽

Simulation Results ◽

Tuberculosis Disease ◽

The Basic Reproduction Number

This paper deals with the analysis of tuberculosis disease spread model with saturated infection rate and the treatment effect. We analyze the dynamical behavior of the model to observe the stability peroperty of the model’s equilibrium points. The Routh-Hurwitz Theorem is used to analyze the local stability peroperty of the free disease equilibrium point whereas Transcritical Bifurcation principle is used to analyze the local stability property of the endemic equilibrium pont. The result show that the local stability property of the equilibrium points is depending on the basic reproduction number value calculated by the next generation matrix (NGM). When the basic reproduction number is less than 1, the free disease equilibrium point is locally asymptotically stable, and when it is greater than 1, the endemic equilibrium point is locally asymptotically stable. Numeric simulation results were presented to describe the evolution of the dynamical behavior and to understand the treatment effectiveness for the tuberculosis disease of the population. From the simulation results, it was derived that the treatment in the infected subpopulation had a better result than the one in latent.

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Kestabilan Model Epidemi Seir dengan Matriks Hurwitz

Jurnal Ilmiah Poli Rekayasa ◽

10.30630/jipr.11.2.76 ◽

2016 ◽

Vol 11 (2) ◽

pp. 74

Author(s):

Roni Tri Putra ◽

Sukatik - ◽

Sri Nita

Keyword(s):

Basic Reproduction Number ◽

Epidemic Model ◽

Local Stability ◽

Reproduction Number ◽

Equilibrium Points ◽

Asymptotically Stable ◽

Stability Of Equilibrium ◽

Disease Free ◽

The Basic Reproduction Number

In this paper, it will be studied local stability of equilibrium points of a SEIR epidemic model with infectious force in latent, infected and immune period. From the model it will be found investigated the existence and its stability of points its equilibrium by Hurwitz matrices. The local stability of equilibrium points is depending on the value of the basic reproduction number If the disease free equilibrium is local asymptotically stable.

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The local stability of a modified multi-strain SIR model for emerging viral strains

PLoS ONE ◽

10.1371/journal.pone.0243408 ◽

2020 ◽

Vol 15 (12) ◽

pp. e0243408

Author(s):

Miguel Fudolig ◽

Reka Howard

Keyword(s):

Local Stability ◽

Reproduction Number ◽

Equilibrium Points ◽

Sir Model ◽

Original Strain ◽

Viral Strain ◽

Sir Epidemic Model ◽

Sir Epidemic ◽

Cross Immunity ◽

Disease Free

We study a novel multi-strain SIR epidemic model with selective immunity by vaccination. A newer strain is made to emerge in the population when a preexisting strain has reached equilbrium. We assume that this newer strain does not exhibit cross-immunity with the original strain, hence those who are vaccinated and recovered from the original strain become susceptible to the newer strain. Recent events involving the COVID-19 virus shows that it is possible for a viral strain to emerge from a population at a time when the influenza virus, a well-known virus with a vaccine readily available, is active in a population. We solved for four different equilibrium points and investigated the conditions for existence and local stability. The reproduction number was also determined for the epidemiological model and found to be consistent with the local stability condition for the disease-free equilibrium.

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Optimal control application to the epidemiology of HBV and HCV co-infection

International Journal of Biomathematics ◽

10.1142/s1793524521501011 ◽

2021 ◽

Author(s):

Muhammad Naeem Jan ◽

Gul Zaman ◽

Nigar Ali ◽

Imtiaz Ahmad ◽

Zahir Shah

Keyword(s):

Optimal Control ◽

Global Stability ◽

Local Stability ◽

Equilibrium Points ◽

Optimality System ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Before And After ◽

Control Versus ◽

Order Four

It is very important to note that a mathematical model plays a key role in different infectious diseases. Here, we study the dynamical behaviors of both hepatitis B virus (HBV) and hepatitis C virus (HCV) with their co-infection. Actually, the purpose of this work is to show how the bi-therapy is effective and include an inhibitor for HCV infection with some treatments, which are frequently used against HBV. Local stability, global stability and its prevention from the community are studied. Mathematical models and optimality system of nonlinear DE are solved numerically by RK4. We use linearization, Lyapunov function and Pontryagin’s maximum principle for local stability, global stability and optimal control, respectively. Stability curves and basic reproductive number are plotted with and without control versus different values of parameters. This study shows that the infection will spread without control and can cover with treatment. The intensity of HBV/HCV co-infection is studied before and after optimal treatment. This represents a short drop after treatment. First, we formulate the model then find its equilibrium points for both. The models possess four distinct equilibria: HBV and HCV free, and endemic. For the proposed problem dynamics, we show the local as well as the global stability of the HBV and HCV. With the help of optimal control theory, we increase uninfected individuals and decrease the infected individuals. Three time-dependent variables are also used, namely, vaccination, treatment and isolation. Finally, optimal control is classified into optimality system, which we can solve with Runge–Kutta-order four method for different values of parameters. Finally, we will conclude the results for implementation to minimize the infected individuals.

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Optimal Control Analysis of Pneumonia and Meningitis Coinfection

Computational and Mathematical Methods in Medicine ◽

10.1155/2019/2658971 ◽

2019 ◽

Vol 2019 ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

Getachew Teshome Tilahun

Keyword(s):

Optimal Control ◽

Basic Reproduction Number ◽

Deterministic Model ◽

Reproduction Number ◽

Equilibrium Points ◽

Optimality System ◽

Control Analysis ◽

Prevention Effort ◽

Treatment Effort ◽

Basic Model

In this paper, we proposed a deterministic model of pneumonia-meningitis coinfection. We used a system of seven ordinary differential equations. Firstly, the qualitative behaviours of the model such as positivity of the solution, existence of the solution, the equilibrium points, basic reproduction number, analysis of equilibrium points, and sensitivity analysis are studied. The disease-free equilibrium is locally asymptotically stable if the basic reproduction number is kept less than unity, and conditions for global stability are established. Then, the basic model is extended to optimal control by incorporating four control interventions, such as prevention of pneumonia as well as meningitis and also treatment of pneumonia and meningitis diseases. The optimality system is obtained by using Pontryagin’s maximum principle. For simulation of the optimality system, we proposed five strategies to check the effect of the controls. First, we consider prevention only for both diseases, and the result shows that applying prevention control has a great impact in bringing down the expansion of pneumonia, meningitis, and their coinfection in the specified period of time. The other strategies are prevention effort for pneumonia and treatment effort for meningitis, prevention effort for meningitis and treatment effort for pneumonia, treatment effort for both diseases, and using all interventions. We obtained that each of the listed strategies is effective in minimizing the expansion of pneumonia-only, meningitis-only, and coinfectious population in the specified period of time.

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