- Research
- Open Access
Mathematical modeling of infectious disease transmission in macroalgae
- Artorn Nokkaew^{1},
- Charin Modchang^{2, 3},
- Somkid Amornsamankul^{3, 4},
- Yongwimon Lenbury^{3, 4},
- Busayamas Pimpunchat^{5} and
- Wannapong Triampo^{2, 3}Email author
https://doi.org/10.1186/s13662-017-1337-5
© The Author(s) 2017
- Received: 1 February 2017
- Accepted: 26 August 2017
- Published: 15 September 2017
Abstract
Understanding the infectious diseases outbreak of algae can provide significant knowledge for disease control intervention and/or prevention. We consider here a disease caused by highly pathogenic organisms that can result in the death of algae. Even though a great deal of understanding about diseases of algae has been reached, studies concerning effects of the outbreak at the population level are still rare. For this reason, we computationally model an outbreak in the algae reservoir or container systems consisting of several patches or clusters of algae being infected with a contagious infectious disease. We computationally investigate the systems as well as make some predictions via the deterministic SEIR epidemic model. We consider the factors that could affect the spread of the disease including the number of patches, the size of initial infected population, the distance between patches or spatial range, and the basic reproduction number (\(R_{0}\)). The results provide some information that may be beneficial to algae disease control, intervention or prevention.
Keywords
- outbreak
- epidemic
- infectious disease
- algae
- SEIR model
- mathematical modeling
1 Introduction
Algae are aquatic primitive multicellular photosynthetic plant species that play an essential role in aquatic ecosystems. They can come in many forms and colors. Algae can be generally characterized based on their photosynthetic pigments and combinations thereof: Cyanophyta, blue-green algae; Rhodophyta, red algae; Chrysophyceae, golden algae; Phaephyceae, brown algae; Chlorophyta, green algae [1]. They range in size from tiny, 1 micrometer in diameter, called microalgae, to giant kelps which can reach 60 meters in length, called macroalgae [2].
The macroalgae can be utilized as a crop [3]. Remarkably, it does not require any land or fertilizer. For farming purposes, it is not only used for direct consumption [4] in a number of ways such as human food, fish food, fertilizer, skin care, and biofuel, but also it can help to improve the environment. Like terrestrial forests, macroalgae forests (such as kelp forests) provide an extensive ecosystem for many organisms from the sea floor to the ocean surface. However, the increasing use of macroalgae as crop species for commercial purposes requires a good system and process for both cultivation and disease control.
Similar to any other living organisms, algae are plagued by diseases caused by fungi, bacteria or viruses. Here parasitism is mainly focused on. Parasitism is one of the common ecological interactions with algae. They represent a strong forcing factor for evolutionary and ecological processes, e.g., population dynamics, species successions, competition for resources, species diversification, and energy and gene flows [5]. Among their parasites, fungi are the most dominant ones [6]. Few attempts have been made to include parasites in the food web dynamics of aquatic systems [7]. As aquaculture continues to rise worldwide, pathogens of algae are becoming a significant economic burden. The filamentous green alga, Chaetomorpha media, from the western and eastern coasts of the Indian peninsula showed infection by an oomycetous fungus Pontisma lagenidioides. The infected cells appear brownish and the infection spreads from the tip downwards of the algal filament on incubation in seawater in the laboratory [8, 9].
In particular, infectious disease outbreak and control could become a huge problem for production management. Thus, in this work, a possible infectious disease outbreak or epidemic is our main focus. Even though the pathology aspect is quite well understood in the most part, how macroalgae respond to pathogens and how the disease can spread at the population level is not well known [10–12].
So far, to the best of our knowledge, there has been no publication on mathematical modeling studies about infectious disease outbreak or epidemics involving macroalgae. Since the epidemics could be extremely costly to farming, it is important to learn as much as possible how to prevent, control, or initiate an intervention, when it happens. Typically, there are many risk factors driving the emergence of the epidemic including population density, degree of transmission, degree of contagion, contact nature, water condition, and climate and so on [13, 14].
In this work, we apply a traditional susceptible-exposed-infectious-recovered (SEIR) model [13] to study the macroalgae system. Both time and spatial considerations were conducted. Computational results and analyses were given. Interpretation and connection between real world and model world were carefully done.
2 Methods
Parameter values used in the simulations
Parameter | Value |
---|---|
Basic reproduction number (\(R_{0}\)) | 1.4 or 2 |
Distance scaling factor (\(r_{0}\)) | 30 m |
Incubation period (1/σ) | 2 day |
Infectious period (1/γ) | 5 day |
Initial infectious individuals | 1-28% |
Total number of algae individuals | 900 |
Number of clusters | 2-6 |
Algae population of each cluster | 150-450 |
Area of each cluster | 1,000 m^{2} |
Distance between clusters | 30-600 m |
Time step | 10^{−3} day |
Susceptible algae \(S_{i}\) are infected at a rate of \(\sum_{j = 1}^{n} \beta_{ij}I_{j}\), where \(\beta_{ij}\) is the transmission rate from the jth patch to the ith patch. The summation is taken over all of the compartments that can spread the infection to theith patch. The infected individuals (\(E_{i}\)) incubate the infection for a mean duration of \(1/\sigma\). After passing through the exposed state, infected individuals become infectious (\(I_{i}\)) with the mean duration of the contagious stage of \(1/\gamma\).
It should be noted that the dynamics of an epidemic here is assumed to be much faster than the dynamics of natural birth and death (vital dynamics), therefore, birth and death are omitted in the model. In other words the time scale of the epidemic dynamics of algal system is much faster than that of vital dynamics. Or the birth rate and death rate of algae is considerably much less than the incubation and infectious period of the disease. As far as our current model is concerned, it would be more realistic if the life span of algae is relatively long enough compared to the infectious period. It was estimated that the life span of the large brown attached alga, Macrocystis pyrifera, a member of a widespread genus, could be several months [15].
3 Results and discussion
In this study, we quantitatively investigated the factors that cause the damage of algae farm due to an epidemic of an infectious disease. The factors studied in this investigation include \(R_{0}\), the number of patches (density of a planting patch), the distance between the patches and the number of initial infectious individuals.
Considering the effect of patch numbers and the time of the peak of the epidemic, unlike the percentage of the damage, the day of the maximum of the epidemic is sensitive to \(R_{0}\). Different behaviors were found for \(R_{0}\) of 1.4 and 2.0. For \(R_{0}\) of 1.4, the data of the peak varied. For \(R_{0}\) of 2.0, three patterns were found. For \(10 r_{0}\) and \(20 r_{0}\), the highest peak is slightly decreased with increase of the number of clusters (see Figure 3). For \(1r_{0}\) and \(2 r_{0}\), the period of the peak is slightly longer as the number of patches increases. For \(5 r_{0}\), the period of the peak is close to constant for \(R_{0}\) of 1.4 and 2.0. Note that the maximum number of infectious (at the peak) is different for different distances. Based on our system, a longer distance yields a lower maximum number. Computationally, we performed simulation and measured the following quantities, namely statistical hypothesis testing for investigation between the number of patches (\(c_{i}\), \(i=2,3,4,5\) and 6) on the loss or damage and the distance among patches such that \(1 r_{0}\), \(2 r_{0}\), \(5 r_{0}\), \(10 r_{0}\) and \(20 r_{0}\), respectively. We can test whether or not the number of patches and the distance between patches differ among groups and conduct the F-test at the \(\alpha = 0.05\) significant level. As a result, we rejected H0, and concluded that a difference existed between the various values of \(R_{0}\). There was no difference with regard to the number of patches.
4 Conclusions
In this work, we applied the SEIR epidemics model to study the infectious disease spreading in algae population. Our SEIR-based model suggests that the value of \(R_{0}\) plays a significant role in the epidemic dynamics of algae system. Our results are considerably consistent with the general theory. Due to the fact that \(R_{0}\) is the number of cases that one case generates on average over the course of its infectious period, the larger the value of \(R_{0}\), the more the infection can spread, and the harder it is to control the epidemic. When we compartmentalized the whole population into smaller compartments or clusters, it results in a lower algae population density and consequently a lower contact rate as well. However, in this work we used the transmission rule based on the modified gravity disease transmission model. It is apparent from the disease transmission rate that the distance between habitats plays a crucial role on the spread of the disease. Lastly, the number of clusters or habitats (for a given total population) is found to mitigate the epidemics when the number is increased.
Declarations
Acknowledgements
This research project is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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