# American Institute of Mathematical Sciences

August 2018, 15(4): 841-862. doi: 10.3934/mbe.2018038

## Continuous and pulsed epidemiological models for onchocerciasis with implications for eradication strategy

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA 2 Mathematics Department, Seattle University, 901 12th Ave, Seattle, WA 98122-1090, USA 3 Department of Mathematics, Indiana University of Pennsylvania, Indiana, PA 15705, USA 4 Department of Mathematics, New York City College of Technology -CUNY, Brooklyn, NY 11201, USA

* Corresponding author: Glenn Ledder

Received  March 2017 Accepted  January 03, 2018 Published  March 2018

Fund Project: This work was supported by NSF grant DMS 1239280

Onchocerciasis is an endemic disease in parts of sub-Saharan Africa. Complex mathematical models are being used to assess the likely efficacy of efforts to eradicate the disease; however, their predictions have not always been borne out in practice. In this paper, we represent the immunological aspects of the disease with a single empirical parameter in order to reduce the model complexity. Asymptotic approximation allows us to reduce the vector-borne epidemiological model to a model of an infectious disease with nonlinear incidence. We then consider two versions, one with continuous treatment and a more realistic one where treatment occurs only at intervals. Thorough mathematical analysis of these models yields equilibrium solutions for the continuous case, periodic solutions for the pulsed case, and conditions for the existence of endemic disease equilibria in both cases, thereby leading to simple model criteria for eradication. The analytical results and numerical experiments show that the continuous treatment version is an excellent approximation for the pulsed version and that the current onchocerciasis eradication strategy is inadequate for regions where the incidence is highest and unacceptably slow even when the long-term behavior is the disease-free state.

Citation: Glenn Ledder, Donna Sylvester, Rachelle R. Bouchat, Johann A. Thiel. Continuous and pulsed epidemiological models for onchocerciasis with implications for eradication strategy. Mathematical Biosciences & Engineering, 2018, 15 (4) : 841-862. doi: 10.3934/mbe.2018038
##### References:
 [1] African programme for onchocerciasis control: Progress report, 2014-2015, Relevé Épidémiologique Hebdomadaire / Section D'hygiène Du Secrétariat De La Société Des Nations = Weekly Epidemiological Record / Health Section Of The Secretariat Of The League Of Nations, 90 (2015), 661–674. URL: http://www.who.int/iris/handle/10665/254534. [2] M. Basáñez and M. Boussinesq, Population biology of human onchocerciasis, Philosophical Transactions: Biological Sciences, 354 (1999), 809-826. doi: 10.1098/rstb.1999.0433. [3] L. E. Coffeng, W. A. Stolk, H. G. M. Zouré, J. L. Veerman and K. B. Agblewonu, African programme for onchocerciasis control 1995-2015: Model-estimated health impact and cost, PLoS Neglected Tropical Diseases, 7 (2013), e2032. doi: 10.1371/journal.pntd.0002032. [4] Y. Dadzie, M. Neira and D. Hopkins, Final report of the conference on the eradicability of onchocerciasis, Filaria Journal, 2 (2003), 22-134. doi: 10.1186/1475-2883-2-2. [5] J. B. Davies, Description of a computer model of forest onchocerciasis transmission and its application to field scenarios of vector control and chemotherapy, Ann Trop Med Parasitol, 87 (1992), 41–63, URL: https://www.ncbi.nlm.nih.gov/pubmed/8346991. doi: 10.1080/00034983.1993.11812738. [6] K. Dietz, The population dynamics of onchocerciasis, Population Dynamics of Infectious Diseases (ed. R. M. Anderson), Chapman and Hall, (1982), 209–241. doi: 10.1007/978-1-4899-2901-3_7. [7] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [8] A. Hopkins and B. A. Boatin, Onchocerciasis, Water and Sanitation-Related Diseases and the Environment: Challenges, Interventions, and Preventive Measures (ed J.M.H. Selendy), John Wiley & Sons, Inc, (2011), 133-149. doi: 10.1002/9781118148594.ch11. [9] E. I. Jury, A simplified stability criterion for linear discrete systems, Proceedings of the IRE, 50 (1962), 1493-1500. doi: 10.1109/JRPROC.1962.288193. [10] M. N. Katabarwa, A. Eyamba, P. Nwane, P. Enyong, S. Yaya, J. Baldagai, T. K. Madi, A. Yougouda, G. O. Andze and F. O. Richards, Seventeen years of annual distribution of ivermectin has not interrupted onchocerciasis transmission in North Region, Cameroon, The American Journal Of Tropical Medicine And Hygiene [serial online], 85 (2011), 1041-1049. doi: 10.4269/ajtmh.2011.11-0333. [11] G. Ledder, Forest defoliation scenarios, Math Biosci. Eng., 4 (2007), 15-28. doi: 10.3934/mbe.2007.4.15. [12] G. Ledder, Mathematics for the Life Sciences: Calculus, Modeling, Probability, and Dynamical Systems, Springer-Verlag, 2013. doi: 10.1007/978-1-4614-7276-6. [13] A. P. Plaisier, E. S. Alley, B. A. Boatin, G. J. van Oortmarssen and H. Remme, Irreversible effects of ivermectin on adult parasites in onchocerciasis patients in the Onchocerciasis Control Programme in West Africa, Journal of Infectious Disease, 172 (1995), 204–210, URL: https://academic.oup.com/jid/article-abstract/172/1/204/853322. doi: 10.1093/infdis/172.1.204. [14] A. P. Plaisier, G. J. van Oortmarssen, J. D. F. Habbema, J. Remme and E. S. Alley ONCHOSIM: a model and computer simulation program for the transmission and control of onchocerciasis, Comput Methods Programs Biomed, 31 (1990), 43–56. URL: https://www.ncbi.nlm.nih.gov/pubmed/2311368. doi: 10.1016/0169-2607(90)90030-D. [15] B. Ranganathan, Onchocerciasis: An overview, Amrita Journal of Medicine, 8 (2012), 1-9. [16] J. Remme, O. Ba, K. Y. Dadzie and M. Karam, A force-of-infection model for onchocerciasis and its applications in the epidemiological evaluation of the Onchocerciasis Control Programme in the Volta River basin area, Bulletin of the World Health Organization, 64 (1986), 667–681, URL: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2490951/. [17] F. Richards, N. Rizzo and A. Domínguez, One hundred years after its discovery in Guatemala by Rodolfo Robles, Onchocerca volvulus transmission has been eliminated from the Central Endemic Zone, The American Journal Of Tropical Medicine And Hygiene [serial online], 93 (2015), 1295-1304. doi: 10.4269/ajtmh.15-0364. [18] W. A. Stolk, M. Walker, L. E. Coffeng, M. G. Basáñez and S. J. de Vlas, Required duration of mass ivermectin treatment for onchocerciasis elimination in Africa: a comparative modeling analysis, Parasites and Vectors, 8 (2015), 552-567. doi: 10.1186/s13071-015-1159-9. [19] H. C. Turner, W. Walker, T. S. Churcher and M. Basáñez, Modeling the impact of ivermectin on river blindness and its burden of morbidity and mortality in African Savannah: EpiOncho projections, Parasites and Vectors, 7 (2014), 241-255. doi: 10.1186/1756-3305-7-241. [20] F. Weldegebreal, G. Medhin, Z. Weldegebriel and M. Legesse, Knowledge, attitude and practice of community drug distributors' about onchocerciasis and community directed treatment with ivermectin in Quara district, North Western Ethiopia, BMC Research Notes, 9 (2016), 1-8. doi: 10.1186/s13104-016-2010-x.

show all references

##### References:
 [1] African programme for onchocerciasis control: Progress report, 2014-2015, Relevé Épidémiologique Hebdomadaire / Section D'hygiène Du Secrétariat De La Société Des Nations = Weekly Epidemiological Record / Health Section Of The Secretariat Of The League Of Nations, 90 (2015), 661–674. URL: http://www.who.int/iris/handle/10665/254534. [2] M. Basáñez and M. Boussinesq, Population biology of human onchocerciasis, Philosophical Transactions: Biological Sciences, 354 (1999), 809-826. doi: 10.1098/rstb.1999.0433. [3] L. E. Coffeng, W. A. Stolk, H. G. M. Zouré, J. L. Veerman and K. B. Agblewonu, African programme for onchocerciasis control 1995-2015: Model-estimated health impact and cost, PLoS Neglected Tropical Diseases, 7 (2013), e2032. doi: 10.1371/journal.pntd.0002032. [4] Y. Dadzie, M. Neira and D. Hopkins, Final report of the conference on the eradicability of onchocerciasis, Filaria Journal, 2 (2003), 22-134. doi: 10.1186/1475-2883-2-2. [5] J. B. Davies, Description of a computer model of forest onchocerciasis transmission and its application to field scenarios of vector control and chemotherapy, Ann Trop Med Parasitol, 87 (1992), 41–63, URL: https://www.ncbi.nlm.nih.gov/pubmed/8346991. doi: 10.1080/00034983.1993.11812738. [6] K. Dietz, The population dynamics of onchocerciasis, Population Dynamics of Infectious Diseases (ed. R. M. Anderson), Chapman and Hall, (1982), 209–241. doi: 10.1007/978-1-4899-2901-3_7. [7] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [8] A. Hopkins and B. A. Boatin, Onchocerciasis, Water and Sanitation-Related Diseases and the Environment: Challenges, Interventions, and Preventive Measures (ed J.M.H. Selendy), John Wiley & Sons, Inc, (2011), 133-149. doi: 10.1002/9781118148594.ch11. [9] E. I. Jury, A simplified stability criterion for linear discrete systems, Proceedings of the IRE, 50 (1962), 1493-1500. doi: 10.1109/JRPROC.1962.288193. [10] M. N. Katabarwa, A. Eyamba, P. Nwane, P. Enyong, S. Yaya, J. Baldagai, T. K. Madi, A. Yougouda, G. O. Andze and F. O. Richards, Seventeen years of annual distribution of ivermectin has not interrupted onchocerciasis transmission in North Region, Cameroon, The American Journal Of Tropical Medicine And Hygiene [serial online], 85 (2011), 1041-1049. doi: 10.4269/ajtmh.2011.11-0333. [11] G. Ledder, Forest defoliation scenarios, Math Biosci. Eng., 4 (2007), 15-28. doi: 10.3934/mbe.2007.4.15. [12] G. Ledder, Mathematics for the Life Sciences: Calculus, Modeling, Probability, and Dynamical Systems, Springer-Verlag, 2013. doi: 10.1007/978-1-4614-7276-6. [13] A. P. Plaisier, E. S. Alley, B. A. Boatin, G. J. van Oortmarssen and H. Remme, Irreversible effects of ivermectin on adult parasites in onchocerciasis patients in the Onchocerciasis Control Programme in West Africa, Journal of Infectious Disease, 172 (1995), 204–210, URL: https://academic.oup.com/jid/article-abstract/172/1/204/853322. doi: 10.1093/infdis/172.1.204. [14] A. P. Plaisier, G. J. van Oortmarssen, J. D. F. Habbema, J. Remme and E. S. Alley ONCHOSIM: a model and computer simulation program for the transmission and control of onchocerciasis, Comput Methods Programs Biomed, 31 (1990), 43–56. URL: https://www.ncbi.nlm.nih.gov/pubmed/2311368. doi: 10.1016/0169-2607(90)90030-D. [15] B. Ranganathan, Onchocerciasis: An overview, Amrita Journal of Medicine, 8 (2012), 1-9. [16] J. Remme, O. Ba, K. Y. Dadzie and M. Karam, A force-of-infection model for onchocerciasis and its applications in the epidemiological evaluation of the Onchocerciasis Control Programme in the Volta River basin area, Bulletin of the World Health Organization, 64 (1986), 667–681, URL: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2490951/. [17] F. Richards, N. Rizzo and A. Domínguez, One hundred years after its discovery in Guatemala by Rodolfo Robles, Onchocerca volvulus transmission has been eliminated from the Central Endemic Zone, The American Journal Of Tropical Medicine And Hygiene [serial online], 93 (2015), 1295-1304. doi: 10.4269/ajtmh.15-0364. [18] W. A. Stolk, M. Walker, L. E. Coffeng, M. G. Basáñez and S. J. de Vlas, Required duration of mass ivermectin treatment for onchocerciasis elimination in Africa: a comparative modeling analysis, Parasites and Vectors, 8 (2015), 552-567. doi: 10.1186/s13071-015-1159-9. [19] H. C. Turner, W. Walker, T. S. Churcher and M. Basáñez, Modeling the impact of ivermectin on river blindness and its burden of morbidity and mortality in African Savannah: EpiOncho projections, Parasites and Vectors, 7 (2014), 241-255. doi: 10.1186/1756-3305-7-241. [20] F. Weldegebreal, G. Medhin, Z. Weldegebriel and M. Legesse, Knowledge, attitude and practice of community drug distributors' about onchocerciasis and community directed treatment with ivermectin in Quara district, North Western Ethiopia, BMC Research Notes, 9 (2016), 1-8. doi: 10.1186/s13104-016-2010-x.
The O. volvulus life cycle. Microfilariae produced in the human host are transmitted to the black fly of the genus Simulium via a bite. Within the black fly, the larvae pass through larval stages L1-L3. At larval stage L3, they are transmitted to a human host via a bite, where they pass through the final larval stages L3-L5 and become adults [8]
The SEIPMS-UV epidemiological model. Fly bites transfer the disease from infected flies ($V$) to susceptible humans ($S$) and from infective humans ($I$, $P$, and $M$) to uninfected flies ($U$), with the transfer from medicated humans ($M$) decreased by a factor $\nu$. Exposed humans ($E$) become infective, with the fraction $p$ counting as premedicated ($P$) while waiting for treatment and the remaining fraction $q = 1-p$ becoming unmedicated infectives ($I$). Premedicated humans become medicated when they receive treatment. All three infective classes can become susceptible by clearing all the adult worms. Birth and death rates for humans are equal, with all births into the susceptible class; similarly, birth and death rates for flies are equal with all births into the uninfected class
Simulation of the introduction of a small population of human infectives into a previously unexposed population, showing $E/N$, $V/F$, $I/N$, and $S/N$ in Figure 3a, from bottom up, and $V/F$, $I/N$ in Figure 3b, from bottom up, using $a = 0.9$, $b = 3.0$, $\eta = 0.1$, $\epsilon = 0$, $p = 0$, with solid for $\delta = 0.01$ and dash-dot for the asymptotic simplification
Periodic solutions for the exposed ($x$, dashed) and total infective ($y = h+i$, solid) classes, with treatment intervals of 2 years (top), 1 year (middle), and 6 months (bottom), using $a = 0.9$, $b = 3.5$, $\eta = 0.1$, $\epsilon = 0$, $\nu p = 0.6$
Time average infective populations, with $a = 0.9$, $b = 3$, $\eta = 0.1$, $\epsilon = 0$, $\nu p = 0.6$. Humans: top 2; Flies: bottom 2; Pulsed: solid; Continuous: dashed
Simulations of various treatment scenarios, with $a = 0.9$, $\eta = 0.1$, $\epsilon = 0$
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