Incorporating heterogeneity into the transmission dynamics of a waterborne disease model

Collins, Obiora Cornelius; Govinder, K. S.

doi:10.1016/j.jtbi.2014.04.022

Cited by 27 publications

(28 citation statements)

References 27 publications

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“…Combining these assumptions and formulations, we obtain

\begin{matrix} true \\ right \overset{S}{true} (t) & = & left μ N (t) - S (t) \sum_{j = 1}^{n} b_{j} W_{j} (t) - μ S (t), \\ right \overset{I}{true} (t) & = & left S (t) \sum_{j = 1}^{n} b_{j} W_{j} (t) - (μ + γ) I (t), \\ right {\dot{W}}_{1} (t) & = & left ν_{1} I (t) - σ_{1} W_{1} (t), \\ right {\dot{W}}_{2} (t) & = & left ν_{2} I (t) - σ_{2} W_{2} (t), \\ right ⋮ & left ⋮ \\ right {\dot{W}}_{n} (t) & = & left ν_{n} I (t) - σ_{n} W_{n} (t), \\ right \overset{R}{true} (t) & = & left γ I (t) - μ R (t), \end{matrix}

where

σ_{j} = ξ_{j} - α_{j} > 0

, for

j = 1, 2, \dots, n

. Note, since our emphasis is to study optimal vaccination strategies for a community where individuals are exposed to multiple water sources, model is most suitable even though it is a simplified version of our previous model (Collins and Govinder []).…”

Section: Model Formulationmentioning

confidence: 99%

“…The final outbreak size denoted by Z of the

S I R

models together with some other related models is given by the relation

Z = 1 - exp (- {scriptR}_{0} Z) .

We show that the final outbreak size relation also holds for model . We consider the same approach used in Tien and Earn [], Ma and Earn [], and Collins and Govinder []. Proposition Let

μ = 0

and

{scriptR}_{0} > 1

.…”

Section: Model Analysismentioning

confidence: 99%

“…For the numerical simulation of the optimal system, we consider the published data in Table together with the estimated parameter values in Table . Note that these estimated parameters are computed from the Haiti cholera outbreak by using a similar approach to our previous work (Collins and Govinder []). The values of the cost factors used for the simulations are:

A_{1} = 6.00, B_{1} = 6.00, C_{1} = 10.00, C_{2} = 10.00

.…”

Section: Numerical Illustration Of the Optimal Systemmentioning

confidence: 99%

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Stability Analysis and Optimal Vaccination of a Waterborne Disease Model With Multiple Water Sources

Collins

Govinder

2016

Natural Resource Modeling

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Abstract. Waterborne diseases are among the major health problems facing the world today. This is especially true in developing countries where there is limited access to clean water. In such settings, even when multiple water sources exist, they tend to be contaminated. In this paper, we formulate a waterborne disease model where individuals are exposed to multiple contaminated water sources. The fundamental mathematical features of the model such as the basic reproduction number R 0 and final epidemic size are obtained and analyzed accordingly. The global stability analysis of the disease-free equilibrium is performed. The model is later extended by considering vaccination as a possible control intervention strategy. An optimal control problem is constructed to investigate the existence of an optimal control function that reduces the spread of the disease with minimum cost. We support our analytical predictions by carrying out numerical simulations using published and estimated data from the recent cholera outbreak in Haiti.

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“…Combining these assumptions and formulations, we obtain

\begin{matrix} true \\ right \overset{S}{true} (t) & = & left μ N (t) - S (t) \sum_{j = 1}^{n} b_{j} W_{j} (t) - μ S (t), \\ right \overset{I}{true} (t) & = & left S (t) \sum_{j = 1}^{n} b_{j} W_{j} (t) - (μ + γ) I (t), \\ right {\dot{W}}_{1} (t) & = & left ν_{1} I (t) - σ_{1} W_{1} (t), \\ right {\dot{W}}_{2} (t) & = & left ν_{2} I (t) - σ_{2} W_{2} (t), \\ right ⋮ & left ⋮ \\ right {\dot{W}}_{n} (t) & = & left ν_{n} I (t) - σ_{n} W_{n} (t), \\ right \overset{R}{true} (t) & = & left γ I (t) - μ R (t), \end{matrix}

where

σ_{j} = ξ_{j} - α_{j} > 0

, for

j = 1, 2, \dots, n

Section: Model Formulationmentioning

confidence: 99%

“…The final outbreak size denoted by Z of the

S I R

models together with some other related models is given by the relation

Z = 1 - exp (- {scriptR}_{0} Z) .

We show that the final outbreak size relation also holds for model . We consider the same approach used in Tien and Earn [], Ma and Earn [], and Collins and Govinder []. Proposition Let

μ = 0

and

{scriptR}_{0} > 1

.…”

Section: Model Analysismentioning

confidence: 99%

A_{1} = 6.00, B_{1} = 6.00, C_{1} = 10.00, C_{2} = 10.00

.…”

Section: Numerical Illustration Of the Optimal Systemmentioning

confidence: 99%

See 1 more Smart Citation

Stability Analysis and Optimal Vaccination of a Waterborne Disease Model With Multiple Water Sources

Collins

Govinder

2016

Natural Resource Modeling

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show abstract

“…High risk refers to susceptible individuals who have limited access to safe drinking water. As in [13,14] we assume that low risk susceptible individuals have low chances of acquiring the infection.…”

Section: Model Frameworkmentioning

confidence: 99%

Modeling the Effects of Spatial Heterogeneity and Seasonality on Guinea Worm Disease Transmission

Losio

Mushayabasa

2018

Journal of Applied Mathematics

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Guinea worm disease is one of the neglected tropical diseases that is on the verge of elimination. Currently the disease is endemic in four countries, namely, Ethiopia, Mali, Chad, and South Sudan. Prior studies have demonstrated that climate factors and limited access to safe drinking water have a significant impact on transmission and control of Guinea worm disease. In this paper, we present a new mathematical model to understand the transmission dynamics of Guinea worm disease in South Sudan. The model incorporates seasonal variations, educational campaigns, and spatial heterogeneity. Both qualitative and quantitative analysis of the model have been carried out. Utilizing Guinea worm disease surveillance data of South Sudan (2007-2013) we estimate the model parameters. Meanwhile, we perform an optimal control study to evaluate the implications of vector control on long-term Guinea worm infection dynamics. Our results demonstrate that vector control could play a significant role on Guinea worm disease eradication.

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“…A number of mathematical models have explored the dynamics of infectious disease, including waterborne diseases [1,[10][11][12][13][14][15][16][17][18][19][20][21]. These studies have contributed immensely to our understanding of the dynamics of these diseases as well as possible control strategies.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a waterborne disease model with socioeconomic classes

Collins

Robertson

Govinder

2015

Mathematical Biosciences

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Incorporating heterogeneity into the transmission dynamics of a waterborne disease model

Cited by 27 publications

References 27 publications

Stability Analysis and Optimal Vaccination of a Waterborne Disease Model With Multiple Water Sources

Stability Analysis and Optimal Vaccination of a Waterborne Disease Model With Multiple Water Sources

Modeling the Effects of Spatial Heterogeneity and Seasonality on Guinea Worm Disease Transmission

Analysis of a waterborne disease model with socioeconomic classes

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