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Reggi SD, Scarabel F, Vermiglio R. Approximating reproduction numbers: a general numerical method for age-structured models. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2024; 21:5360-5393. [PMID: 38872539 DOI: 10.3934/mbe.2024236] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2024]
Abstract
In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.
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Affiliation(s)
- Simone De Reggi
- Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100 Udine, Italy
- CDLab - Computational Dynamics Laboratory, University of Udine, Italy
| | - Francesca Scarabel
- Department of Applied Mathematics, University of Leeds, Woodhouse, Leeds LS2 9JT, United Kingdom
- CDLab - Computational Dynamics Laboratory, University of Udine, Italy
| | - Rossana Vermiglio
- Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100 Udine, Italy
- CDLab - Computational Dynamics Laboratory, University of Udine, Italy
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2
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Xie Y, Ahmad I, Ikpe TIS, Sofia EF, Seno H. What Influence Could the Acceptance of Visitors Cause on the Epidemic Dynamics of a Reinfectious Disease?: A Mathematical Model. Acta Biotheor 2024; 72:3. [PMID: 38402514 PMCID: PMC10894808 DOI: 10.1007/s10441-024-09478-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2023] [Accepted: 01/30/2024] [Indexed: 02/26/2024]
Abstract
The globalization in business and tourism becomes crucial more and more for the economical sustainability of local communities. In the presence of an epidemic outbreak, there must be such a decision on the policy by the host community as whether to accept visitors or not, the number of acceptable visitors, or the condition for acceptable visitors. Making use of an SIRI type of mathematical model, we consider the influence of visitors on the spread of a reinfectious disease in a community, especially assuming that a certain proportion of accepted visitors are immune. The reinfectivity of disease here means that the immunity gained by either vaccination or recovery is imperfect. With the mathematical results obtained by our analysis on the model for such an epidemic dynamics of resident and visitor populations, we find that the acceptance of visitors could have a significant influence on the disease's endemicity in the community, either suppressive or supportive.
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Affiliation(s)
- Ying Xie
- Department of Mathematical and Information Sciences, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai, 980-8579, Miyagi, Japan
| | - Ishfaq Ahmad
- Department of Mathematical and Information Sciences, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai, 980-8579, Miyagi, Japan
| | - ThankGod I S Ikpe
- Department of Mathematical and Information Sciences, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai, 980-8579, Miyagi, Japan
| | - Elza F Sofia
- Department of Mathematical and Information Sciences, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai, 980-8579, Miyagi, Japan
| | - Hiromi Seno
- Department of Mathematical and Information Sciences, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai, 980-8579, Miyagi, Japan.
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3
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Mfangnia CNT, Tonnang HEZ, Tsanou B, Herren J. Mathematical modelling of the interactive dynamics of wild and Microsporidia MB-infected mosquitoes. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:15167-15200. [PMID: 37679176 DOI: 10.3934/mbe.2023679] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/09/2023]
Abstract
A recent discovery highlighted that mosquitoes infected with Microsporidia MB are unable to transmit the Plasmodium to humans. Microsporidia MB is a symbiont transmitted vertically and horizontally in the mosquito population, and these transmission routes are known to favor the persistence of the parasite in the mosquito population. Despite the dual transmission, data from field experiments reveal a low prevalence of MB-infected mosquitoes in nature. This study proposes a compartmental model to understand the prevalence of MB-infected mosquitoes. The dynamic of the model is obtained through the computation of the basic reproduction number and the analysis of the stability of the MB-free and coexistence equilibria. The model shows that, in spite of the high vertical transmission efficiency of Microsporidia MB, there can still be a low prevalence of MB-infected mosquitoes. Numerical analysis of the model shows that male-to-female horizontal transmission contributes more than female-to-male horizontal transmission to the spread of MB-infected mosquitoes. Moreover, the female-to-male horizontal transmission contributes to the spread of the symbiont only if there are multiple mating occurrences for male mosquitoes. Furthermore, when fixing the efficiencies of vertical transmission, the parameters having the greater influence on the ratio of MB-positive to wild mosquitoes are identified. In addition, by assuming a similar impact of the temperature on wild and MB-infected mosquitoes, our model shows the seasonal fluctuation of MB-infected mosquitoes. This study serves as a reference for further studies, on the release strategies of MB-infected mosquitoes, to avoid overestimating the MB-infection spread.
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Affiliation(s)
- Charlène N T Mfangnia
- Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, P.O. Box: 67, Cameroon
- International Centre of Insect Physiology and Ecology (icipe), Nairobi, P.O. Box: 30772, Kenya
| | - Henri E Z Tonnang
- International Centre of Insect Physiology and Ecology (icipe), Nairobi, P.O. Box: 30772, Kenya
| | - Berge Tsanou
- Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, P.O. Box: 67, Cameroon
- Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
| | - Jeremy Herren
- International Centre of Insect Physiology and Ecology (icipe), Nairobi, P.O. Box: 30772, Kenya
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Harrington PD, Cantrell DL, Lewis MA. Next-generation matrices for marine metapopulations: The case of sea lice on salmon farms. Ecol Evol 2023; 13:e10027. [PMID: 37122768 PMCID: PMC10133530 DOI: 10.1002/ece3.10027] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/26/2022] [Revised: 03/27/2023] [Accepted: 03/31/2023] [Indexed: 05/02/2023] Open
Abstract
Classifying habitat patches as sources or sinks and determining metapopulation persistence requires coupling connectivity between habitat patches with local demographic rates. While methods to calculate sources, sinks, and metapopulation persistence exist for discrete-time models, there is no method that is consistent across modeling frameworks. In this paper, we show how next-generation matrices, originally popularized in epidemiology to calculate new infections after one generation, can be used in an ecological context to calculate sources and sinks as well as metapopulation persistence in marine metapopulations. To demonstrate the utility of the method, we construct a next-generation matrix for a network of sea lice populations on salmon farms in the Broughton Archipelago, BC, an intensive salmon farming region on the west coast of Canada where certain salmon farms are currently being removed under an agreement between local First Nations and the provincial government. The column sums of the next-generation matrix can determine if a habitat patch is a source or a sink and the spectral radius of the next-generation matrix can determine the persistence of the metapopulation. With respect to salmon farms in the Broughton Archipelago, we identify the salmon farms which are acting as the largest sources of sea lice and show that in this region the most productive sea lice populations are also the most connected. The farms which are the largest sources of sea lice have not yet been removed from the Broughton Archipelago, and warming temperatures could lead to increased sea louse growth. Calculating sources, sinks, and persistence in marine metapopulations using the next-generation matrix is biologically intuitive, mathematically equivalent to previous methods, and consistent across different modeling frameworks.
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Affiliation(s)
- Peter D. Harrington
- Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonAlbertaCanada
- Department of MathematicsUniversity of British ColumbiaVancouverBritish ColumbiaCanada
| | - Danielle L. Cantrell
- California Department of Fish and WildlifeMarine Region's Fisheries Analytics ProjectMontereyCaliforniaUSA
| | - Mark A. Lewis
- Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonAlbertaCanada
- Department of Biological SciencesUniversity of AlbertaEdmontonAlbertaCanada
- Department of Mathematics and StatisticsUniversity of VictoriaVictoriaBritish ColumbiaCanada
- Department of BiologyUniversity of VictoriaVictoriaBritish ColumbiaCanada
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5
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SEIR-Metapopulation model of potential spread of West Nile virus. Ecol Modell 2022. [DOI: 10.1016/j.ecolmodel.2022.110213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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6
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Ghakanyuy BM, Teboh-Ewungkem MI, Schneider KA, Ngwa GA. Investigating the impact of multiple feeding attempts on mosquito dynamics via mathematical models. Math Biosci 2022; 350:108832. [PMID: 35718220 DOI: 10.1016/j.mbs.2022.108832] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2021] [Revised: 04/07/2022] [Accepted: 05/09/2022] [Indexed: 11/27/2022]
Abstract
A deterministic differential equation model for the dynamics of terrestrial forms of mosquito populations is studied. The model assesses the impact of multiple probing attempts by mosquitoes that quest for blood within human populations by including a waiting class for mosquitoes that failed a blood feeding attempt. The equations are derived based on the idea that the reproductive cycle of the mosquito can be viewed as a set of alternating egg laying and blood feeding outcomes realized on a directed path called the gonotrophic cycle pathway. There exists a threshold parameter, the basic offspring number for mosquitoes, whose nature is affected by the way we interpret the transitions involving the different classes on the gonotrophic cycle path. The trivial steady state for the system, which always exists, can be globally asymptomatically stable whenever the threshold parameter is less than unity. The non-trivial steady state, when it exists, is stable for a range of values of the threshold parameter but can also be driven to instability via a Hopf bifurcation. The model's output reveals that the waiting class mosquitoes do contribute positively to sustain mosquito populations as well as increase their interactions with humans via increased frequency and initial amplitude of oscillations. We conclude that to understand human-mosquito interactions, it is informative to consider multiple probing attempts; known to occur when mosquitoes quest for blood meals within human populations.
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Affiliation(s)
- Bime M Ghakanyuy
- Department of Mathematics, University of Buea, P.O. Box 63, Buea, Cameroon
| | | | - Kristan A Schneider
- Department of Applied Computer and Bio-Sciences, University of Applied Sciences, Mittweida, Technikumplatz 17, 09648 Mittweida, Germany
| | - Gideon A Ngwa
- Department of Mathematics, University of Buea, P.O. Box 63, Buea, Cameroon.
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Ito H, Yamamoto T, Morita S. The effect of men who have sex with men (MSM) on the spread of sexually transmitted infections. Theor Biol Med Model 2021; 18:18. [PMID: 34635123 PMCID: PMC8504019 DOI: 10.1186/s12976-021-00148-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2021] [Accepted: 08/30/2021] [Indexed: 11/10/2022] Open
Abstract
Sexually transmitted infections (STIs) have remained a worldwide public health threat. It is difficult to control the spread of STIs, not only because of heterogeneous sexual transmission between men and women but also because of the complicated effects of sexual transmission among men who have sex with men (MSM) and mother-to-child transmission. Many studies point to the existence of a ‘bisexual bridge’, where STIs spread from the MSM network via bisexual connections. However, it is unclear how the MSM network affects heterosexual networks as well as mother-to-child transmission. To analyse the effect of MSM on the spread of STIs, we divided the population into four subpopulations: (i) women, (ii) men who have sex with women only (MSW), (iii) men who have sex with both men and women (MSMW), (iv) men who have sex with men exclusively (MSME). We calculated the type-reproduction numbers of these four subpopulations, and our analysis determined what preventive measures may be effective. Our analysis shows the impact of bisexual bridge on the spread of STIs does not outweigh their population size. Since MSM and mother-to-child transmission rates do not have a strong synergistic effect when combined, complementary prevention measures are needed. The methodologies and findings we have provided here will contribute greatly to the future development of public health.
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Affiliation(s)
- Hiromu Ito
- Department of International Health and Medical Anthropology, Institute of Tropical Medicine, Nagasaki University, Nagasaki, Nagasaki, 852-8523, Japan
| | - Taro Yamamoto
- Department of International Health and Medical Anthropology, Institute of Tropical Medicine, Nagasaki University, Nagasaki, Nagasaki, 852-8523, Japan
| | - Satoru Morita
- Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, Shizuoka, 432-8561, Japan. .,Department of Environment and Energy Systems, Graduate School of Science and Technology, Shizuoka University, Hamamatsu, Shizuoka, 432-8561, Japan.
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8
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Bloomquist A, Vaidya NK. Modelling the risk of HIV infection for drug abusers. JOURNAL OF BIOLOGICAL DYNAMICS 2021; 15:S81-S104. [PMID: 33164703 DOI: 10.1080/17513758.2020.1842921] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2020] [Accepted: 10/19/2020] [Indexed: 06/11/2023]
Abstract
Drugs of abuse, such as opiates, are one of the leading causes for transmission of HIV in many parts of the world. Drug abusers often face a higher risk of acquiring HIV because target cell (CD4+ T-cell) receptor expression differs in response to morphine, a metabolite of common opiates. In this study, we use a viral dynamics model that incorporates the T-cell expression difference to formulate the probability of infection among drug abusers. We quantify how the risk of infection is exacerbated in morphine conditioning, depending on the timings of morphine intake and virus exposure. With in-depth understanding of the viral dynamics and the increased risk for these individuals, we further evaluate how preventive therapies, including pre- and post-exposure prophylaxis, affect the infection risk in drug abusers. These results are useful to devise ideal treatment protocols to combat the several obstacles those under drugs of abuse face.
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Affiliation(s)
- Angelica Bloomquist
- Department of Mathematics and Statistics, San Diego State University, San Diego, CA, USA
- Computational Science Research Center, San Diego State University, San Diego, CA, USA
- Viral Information Institute, San Diego State University, San Diego, CA, USA
| | - Naveen K Vaidya
- Department of Mathematics and Statistics, San Diego State University, San Diego, CA, USA
- Computational Science Research Center, San Diego State University, San Diego, CA, USA
- Viral Information Institute, San Diego State University, San Diego, CA, USA
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9
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Botelho C, Dzevela Kong J, Ali Ber Lucien M, Shuai Z, Wang H. A mathematical model for Vibrio-phage interactions. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:2688-2712. [PMID: 33892567 DOI: 10.3934/mbe.2021137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
A cholera model has been formulated to incorporate the interaction of bacteria and phage. It is shown that there may exist three equilibria: one disease free and two endemic equilibria. Threshold parameters have been derived to characterize stability of these equilibria. Sensitivity analysis and disease control strategies have been employed to characterize the impact of bacteria-phage interaction on cholera dynamics.
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Affiliation(s)
- Christopher Botelho
- Department of Mathematics, University of Central Florida, Orlando, Florida, 32816, USA
| | - Jude Dzevela Kong
- Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada
- The Canadian Center for Disease Modelling, York University, Toronto, Ontario, M3J 1P3, Canada
| | - Mentor Ali Ber Lucien
- Laboratoire National de Santé Publique, Ministère de la Santé Publique et de la Population d'Haiti, Port-au-Prince, HT6120, Haiti
| | - Zhisheng Shuai
- Department of Mathematics, University of Central Florida, Orlando, Florida, 32816, USA
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
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10
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Wang X, Zhao XQ. Target reproduction numbers for reaction-diffusion population models. J Math Biol 2020; 81:625-647. [PMID: 32737586 DOI: 10.1007/s00285-020-01523-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2019] [Revised: 02/10/2020] [Indexed: 10/23/2022]
Abstract
A very important population threshold quantity is the target reproduction number, which is a measure of control effort required for a target prevention, intervention or control. This concept, as a generalization of type reproduction number, was first introduced in Shuai et al. (J Math Biol 67:1067-1082, 2013) for nonnegative matrices with immediate applications to compartmental population models of ordinary differential equations. The current paper is devoted to the study of all target reproduction numbers for reaction-diffusion population models with compartmental structure. It turns out that the target reproduction number can be regarded as the basic reproduction number of a modified system, where the state of newborn individuals is limited to the target control set and the offspring from the non-target set is regarded as a part of the transition. In other words, the target reproduction number can be interpreted as the expected number of offspring in a specific target set that a primary newborn individual of the same set would produce during its lifetime. We also characterize the target reproduction number so that it can be easily computed numerically for reaction-diffusion models. At the end, we demonstrate our theoretical observations using two examples.
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Affiliation(s)
- Xueying Wang
- Department of Mathematics and Statistics, Washington State University, Pullman, WA, 99164, USA
| | - Xiao-Qiang Zhao
- Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada.
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11
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Seno H. An SIS model for the epidemic dynamics with two phases of the human day-to-day activity. J Math Biol 2020. [PMID: 32270285 DOI: 10.1007/s00285-020-01491-0/figures/13] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/21/2023]
Abstract
An SIS model is analyzed to consider the contribution of community structure to the risk of the spread of a transmissible disease. We focus on the human day-to-day activity introduced by commuting to a central place for the social activity. We assume that the community is classified into two subpopulations: commuter and non-commuter, of which the commuter has two phases of the day-to-day activity: private and social. Further we take account of the combination of contact patterns in two phases, making use of mass-action and ratio-dependent types for the infection force. We investigate the dependence of the basic reproduction number on the commuter ratio and the daily expected duration at the social phase as essential factors characterizing the community structure, and show that the dependence is significantly affected by the combination of contact patterns, and that the difference in the commuter ratio could make the risk of the spread of a transmissible disease significantly different.
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Affiliation(s)
- Hiromi Seno
- Department of Computer and Mathematical Sciences, Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, Japan.
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12
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Seno H. An SIS model for the epidemic dynamics with two phases of the human day-to-day activity. J Math Biol 2020; 80:2109-2140. [PMID: 32270285 PMCID: PMC7139907 DOI: 10.1007/s00285-020-01491-0] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2019] [Revised: 03/13/2020] [Indexed: 01/11/2023]
Abstract
An SIS model is analyzed to consider the contribution of community structure to the risk of the spread of a transmissible disease. We focus on the human day-to-day activity introduced by commuting to a central place for the social activity. We assume that the community is classified into two subpopulations: commuter and non-commuter, of which the commuter has two phases of the day-to-day activity: private and social. Further we take account of the combination of contact patterns in two phases, making use of mass-action and ratio-dependent types for the infection force. We investigate the dependence of the basic reproduction number on the commuter ratio and the daily expected duration at the social phase as essential factors characterizing the community structure, and show that the dependence is significantly affected by the combination of contact patterns, and that the difference in the commuter ratio could make the risk of the spread of a transmissible disease significantly different.
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Affiliation(s)
- Hiromi Seno
- Department of Computer and Mathematical Sciences, Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, Japan.
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13
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LUPICA ANTONELLA, GUMEL ABBAB, PALUMBO ANNUNZIATA. THE COMPUTATION OF REPRODUCTION NUMBERS FOR THE ENVIRONMENT-HOST-ENVIRONMENT CHOLERA TRANSMISSION DYNAMICS. J BIOL SYST 2020. [DOI: 10.1142/s021833902040001x] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
This study presents a new model for the environment-host-environment transmission dynamics of V. cholerae in a community with an interconnected aquatic pond–river water network. For the case when the human host is the sole target of anti-cholera control and the volume of water in the pond is maximum, the disease-free equilibrium of the model is shown to be globally asymptotically stable whenever a certain epidemiological threshold, known as the basic reproduction number [Formula: see text], is less than unity. The epidemiological implication of this result is that cholera can be eliminated from the community if the control strategies implemented can bring (and maintain) [Formula: see text] to a value less than unity. Four scenarios, that represent different interpretations of the role of the V. cholerea pathogen within the environment, were studied. The corresponding basic reproduction numbers were shown to exhibit the same threshold property with respect to the value unity (i.e., if one is less (equal, greater) than unity, then the three others are also less (equal, greater) than unity. Further, it was shown that for the case where anti-cholera control is focused on the human host population, the associated type reproduction number of the model (corresponding to each of the four transmission scenarios considered) is unique. The implication of this result is that the estimate of the effort needed for disease elimination (i.e., the required herd immunity threshold) is unique, regardless of which of the four transmission scenarios is considered. However, when any of the other two bacterial population types in the aquatic environment (i.e., bacterial in the pond or river) is the focus of the control efforts, this study shows that the associated type reproduction number is not unique. Extensive numerical simulations of the model, using a realistic set of parameters from the published literature, show that the community-wide implementation of a strategy that focus on improved water quality, sanitation, and hygiene (known as WASH-only strategy), using the current estimated coverage of 50% and efficacy of 60%, is unable to lead to the elimination of the disease. Such elimination is attainable if the coverage and efficacy are increased (e.g., to 80% and 90%, respectively). Further, elimination can be achieved using a strategy that focuses on oral rehydration therapy and the use of antibiotics to treat the infected humans (i.e., treatment-only strategy) for moderate effectiveness and coverage levels. The combined hybrid WASH-treatment strategy provides far better population-level impact vis a vis disease elimination. This study ranks the three interventions in the following order of population-level effectiveness: combined WASH-treatment, followed by treatment-only and then WASH-only strategy.
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Affiliation(s)
- ANTONELLA LUPICA
- Department of Mathematics and Computer Sciences, University of Catania, V.le A. Doria 6, 95125 Catania, Italy
- Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, V.le F. D’Alcontres 31, 98166 Messina, Italy
| | - ABBA B. GUMEL
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona, USA
- Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
| | - ANNUNZIATA PALUMBO
- Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, V.le F. D’Alcontres 31, 98166 Messina, Italy
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14
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Harrington PD, Lewis MA. A Next-Generation Approach to Calculate Source-Sink Dynamics in Marine Metapopulations. Bull Math Biol 2020; 82:9. [PMID: 31932972 DOI: 10.1007/s11538-019-00674-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2019] [Accepted: 12/02/2019] [Indexed: 11/26/2022]
Abstract
In marine systems, adult populations confined to isolated habitat patches can be connected by larval dispersal. Source-sink theory provides effective tools to quantify the effect of specific habitat patches on the dynamics of connected populations. In this paper, we construct the next-generation matrix for a marine metapopulation and demonstrate how it can be used to calculate the source-sink dynamics of habitat patches. We investigate the effect of environmental variables on the source-sink dynamics and demonstrate how the next-generation matrix can provide useful biological insight into transient as well as asymptotic dynamics of the model.
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Affiliation(s)
- Peter D Harrington
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada.
| | - Mark A Lewis
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
- Department of Biological Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
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15
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A Population Dynamics Model of Mosquito-Borne Disease Transmission, Focusing on Mosquitoes' Biased Distribution and Mosquito Repellent Use. Bull Math Biol 2019; 81:4977-5008. [PMID: 31595380 DOI: 10.1007/s11538-019-00666-1] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Accepted: 09/24/2019] [Indexed: 01/03/2023]
Abstract
We present an improved mathematical model of population dynamics of mosquito-borne disease transmission. Our model considers the effect of mosquito repellent use and the mosquito's behavior or attraction to the infected human, which cause mosquitoes' biased distribution around the human population. Our analysis of the model clearly shows the existence of thresholds for mosquito repellent efficacy and its utilization rate in the human population with respect to the elimination of mosquito-borne diseases. Further, the results imply that the suppression of mosquito-borne diseases becomes more difficult when the mosquitoes' distribution is biased to a greater extent around the human population.
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16
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Dansu EJ, Seno H. A model for epidemic dynamics in a community with visitor subpopulation. J Theor Biol 2019; 478:115-127. [PMID: 31228488 PMCID: PMC7094103 DOI: 10.1016/j.jtbi.2019.06.020] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2019] [Revised: 06/16/2019] [Accepted: 06/19/2019] [Indexed: 11/04/2022]
Abstract
With a model consisting of SIR and SIS models, we affirm claims in previous works. We derive different basic reproduction numbers looking at varying perspectives. We discuss the biological meanings of these basic reproduction numbers. All the basic reproduction numbers coincide with respect to the critical condition. Relevant public health policies are proposed based on our findings.
With a five dimensional system of ordinary differential equations based on the SIR and SIS models, we consider the dynamics of epidemics in a community which consists of residents and short-stay visitors. Taking different viewpoints to consider public health policies to control the disease, we derive different basic reproduction numbers and clarify their common/different mathematical natures so as to understand their meanings in the dynamics of the epidemic. From our analyses, the short-stay visitor subpopulation could become significant in determining the fate of diseases in the community. Furthermore, our arguments demonstrate that it is necessary to choose one variant of basic reproduction number in order to formulate appropriate public health policies.
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Affiliation(s)
- Emmanuel J Dansu
- Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai 980-8579, Japan.
| | - Hiromi Seno
- Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 6-3-09, Aoba-ku, Sendai 980-8579, Japan
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