November  2019, 24(11): 6297-6323. doi: 10.3934/dcdsb.2019140

Effects of migration on vector-borne diseases with forward and backward stage progression

Department of Mathematics & Center for Computational and Applied Mathematics, California State University, Fullerton, Fullerton, CA 92831, USA

Received  August 2018 Revised  February 2019 Published  July 2019

Is it possible to break the host-vector chain of transmission when there is an influx of infectious hosts into a naïve population and competent vector? To address this question, a class of vector-borne disease models with an arbitrary number of infectious stages that account for immigration of infective individuals is formulated. The proposed model accounts for forward and backward progression, capturing the mitigation and aggravation to and from any stages of the infection, respectively. The model has a rich dynamic, which depends on the patterns of infected immigrant influx into the host population and connectivity of the transfer between infectious classes. We provide conditions under which the answer of the initial question is positive.

Citation: Derdei Mahamat Bichara. Effects of migration on vector-borne diseases with forward and backward stage progression. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6297-6323. doi: 10.3934/dcdsb.2019140
References:
[1]

N. BameS. BowongJ. MbangG. Sallet and J. Tewa, Global stability for seis models with n latent classes, Math. Biosci. Eng., 5 (2008), 20-33. doi: 10.3934/mbe.2008.5.20. Google Scholar

[2]

E. D. Barnett and P. F. Walker, Role of immigrants and migrants in emerging infectious diseases, Medical Clinics of North America, 92 (2008), 1447-1458. doi: 10.1016/j.mcna.2008.07.001. Google Scholar

[3]

M. Q. BenedictR. S. LevineW. A. Hawley and L. P. Lounibos, Spread of the tiger: Global risk of invasion by the mosquito aedes albopictus, Vector-borne and Zoonotic Diseases, 7 (2007), 76-85. Google Scholar

[4]

D. BicharaA. Iggidr and L. Smith, Multi-stage vector-borne zoonoses models: A global analysis, Bulletin of Mathematical Biology, 80 (2018), 1810-1848. doi: 10.1007/s11538-018-0435-1. Google Scholar

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B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, 184. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0619-4. Google Scholar

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F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. Google Scholar

[7]

C. Castillo-Chavez and H. R. Thieme, Asymptotically Autonomous Epidemic Models, in Mathematical Population Dynamics: Analysis of Heterogeneity, Volume One: Theory of Epidemics, O. Arino, A. D.E., and M. Kimmel, eds., Wuerz, 1995.Google Scholar

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Centers for Disease Control and Prevention, Illnesses from mosquito, tick, and flea bites increasing in the us, Centers for Disease Control and Prevention, https://www.cdc.gov/media/releases/2018/p0501-vs-vector-borne.html, (2018).Google Scholar

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G. ChowellP. Diaz-DuenasJ. MillerA. Alcazar-VelazcoJ. HymanP. Fenimore and C. Castillo-Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data, Mathematical biosciences, 208 (2007), 571-589. doi: 10.1016/j.mbs.2006.11.011. Google Scholar

[10]

G. Cruz-PachecoL. Esteva and C. Vargas, Control measures for chagas disease, Mathematical biosciences, 237 (2012), 49-60. doi: 10.1016/j.mbs.2012.03.005. Google Scholar

[11]

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Math. Biosci., 150 (1998), 131-151. doi: 10.1016/S0025-5564(98)10003-2. Google Scholar

[12]

E. A. GouldP. GallianX. De Lamballerie and R. N. Charrel, First cases of autochthonous dengue fever and chikungunya fever in france: From bad dream to reality!, Clinical Microbiology and Infection, 16 (2010), 1702-1704. doi: 10.1111/j.1469-0691.2010.03386.x. Google Scholar

[13]

M. Grandadam, V. Caro, S. Plumet, J.-M. Thiberge, Y. Souares, A.-B. Failloux, H. J. Tolou, M. Budelot, D. Cosserat, I. Leparc-Goffart and P. Desprès, Chikungunya virus, southeastern france, Emerging Infectious Diseases, 17 (2011), p. 910.Google Scholar

[14]

H. Guo and M. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525. doi: 10.3934/mbe.2006.3.513. Google Scholar

[15]

H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. Google Scholar

[16]

H. GuoM. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM Journal on applied mathematics, 72 (2012), 261-279. doi: 10.1137/110827028. Google Scholar

[17]

J. M. HymanJ. Li and E. Stanley, The differential infectivity and staged progression models for the transmission of HIV., Math. Biosci., 155 (1999), 77-109. doi: 10.1016/S0025-5564(98)10057-3. Google Scholar

[18]

M. Isaäcson, Airport malaria: A review, Bulletin of the World Health Organization, 67 (1989), p. 737.Google Scholar

[19]

T. L. Johnson, E. L. Landguth and E. F. Stone, Modeling relapsing disease dynamics in a host-vector community, PLoS Negl Trop Dis, 10 (2016), p. e0004428. doi: 10.1371/journal.pntd.0004428. Google Scholar

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K. E. JonesN. G. PatelM. A. LevyA. StoreygardD. BalkJ. L. Gittleman and P. Daszak, Global trends in emerging infectious diseases, Nature, 451 (2008), 990-993. doi: 10.1038/nature06536. Google Scholar

[21]

A. M. Kilpatrick and S. E. Randolph, Drivers, dynamics, and control of emerging vector-borne zoonotic diseases, The Lancet, 380 (2012), 1946-1955. doi: 10.1016/S0140-6736(12)61151-9. Google Scholar

[22]

M. Li and J. S. Muldowney, On r.a. smith's automonmous convergence theorem, Rocky Mountain J. Math., 25 (1995), 365-379. doi: 10.1216/rmjm/1181072289. Google Scholar

[23]

M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5. Google Scholar

[24]

C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16. doi: 10.1016/S0025-5564(02)00149-9. Google Scholar

[25]

C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Differ. Equations, 16 (2004), 139-166. doi: 10.1023/B:JODY.0000041283.66784.3e. Google Scholar

[26]

C. J. Mitchell, Geographic spread of aedes albopictus and potential for involvement in arbovirus cycles in the mediterranean basin, Journal of Vector Ecology, 20 (1985), 44-58. Google Scholar

[27]

J. Moon, Counting Labeled Trees, Canadian Math, Monographs, 1970. Google Scholar

[28]

C. PalmerE. LandguthE. Stone and T. Johnson, The dynamics of vector-borne relapsing diseases, Math. Biosci., 297 (2018), 32-42. doi: 10.1016/j.mbs.2018.01.001. Google Scholar

[29]

M. Paty, C. Six, F. Charlet, G. Heuzé, A. Cochet, A. Wiegandt, J. Chappert, D. Dejour-Salamanca, A. Guinard, P. Soler, V. Servas, M. Vivier-Darrigol, M. Ledrans, M. Debruyne, O. Schaal, C. Jeannin, B. Helynck, I. Leparc-Goffart and B. Coignard, Large number of imported chikungunya cases in mainland france, 2014: a challenge for surveillance and response, Eurosurveillance, 19 (2014), 20856. doi: 10.2807/1560-7917.ES2014.19.28.20856. Google Scholar

[30]

P. Poletti, G. Messeri, M. Ajelli, R. Vallorani, C. Rizzo and S. Merler, Transmission potential of chikungunya virus and control measures: The case of italy, PLoS One, 6 (2011), e18860. doi: 10.1371/journal.pone.0018860. Google Scholar

[31]

G. RezzaL. NicolettiR. AngeliniR. RomiA. FinarelliM. PanningP. CordioliC. FortunaS. BorosF. MaguranoG. SilviP. AngeliniM. DottoriM. CiufoliniG. Majori and A. Cassone, Infection with hikungunya virus in italy: An outbreak in a temperate region, The Lancet, 370 (2007), 1840-1846. doi: 10.1016/S0140-6736(07)61779-6. Google Scholar

[32]

D. Rogers and S. Randolph, Climate change and vector-borne diseases, Advances in parasitology, 62 (2006), 345-381. doi: 10.1016/S0065-308X(05)62010-6. Google Scholar

[33]

D. A. Shroyer, AEDES ALBOPICTUS and arboviruses: A concise review of the literaturei, Journal of the American Mosquito Control Association, (1986).Google Scholar

[34]

F. SimonH. Savini and P. Parola, Chikungunya: A paradigm of emergence and globalization of vector-borne diseases, Medical Clinics of North America, 92 (2008), 1323-1343. doi: 10.1016/j.mcna.2008.07.008. Google Scholar

[35]

T. TabataM. PetittH. Puerta-GuardoD. MichlmayrC. WangJ. Fang-HooverE. Harris and L. Pereira, Zika virus targets different primary human placental cells, suggesting two routes for vertical transmission, Cell Host & Microbe, 20 (2016), 155-166. doi: 10.1016/j.chom.2016.07.002. Google Scholar

[36]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. Google Scholar

[37]

J. TumwiineJ. Mugisha and L. Luboobi, A host-vector model for malaria with infective immigrants, Journal of Mathematical Analysis and Applications, 361 (2010), 139-149. doi: 10.1016/j.jmaa.2009.09.005. Google Scholar

[38]

A. Vega-Rua, K. Zouache, V. Caro, L. Diancourt, P. Delaunay, M. Grandadam and A.-B. Failloux, High efficiency of temperate aedes albopictus to transmit chikungunya and dengue viruses in the southeast of france, PLoS One, 8 (2013), e59716. doi: 10.1371/journal.pone.0059716. Google Scholar

[39]

M. Vidyasagar, Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability., IEEE Trans. Autom. Control, 25 (1980), 773-779. doi: 10.1109/TAC.1980.1102422. Google Scholar

[40]

N. WauquierBe cquartD. NkogheC. PadillaA. Ndjoyi-Mbiguino and E. M. Leroy, The acute phase of chikungunya virus infection in humans is associated with strong innate immunity and t cd8 cell activation, Journal of Infectious Diseases, 204 (2011), 115-123. doi: 10.1093/infdis/jiq006. Google Scholar

[41]

A. Wilder-Smith, M. Quam, O. Sessions, J. Rocklov, J. Liu-Helmersson, L. Franco and K. Khan, The 2012 dengue outbreak in madeira: Exploring the origins, Euro Surveill, 19 (2014), 20718. doi: 10.2807/1560-7917.ES2014.19.8.20718. Google Scholar

[42]

World Health Organization et al., The World Health Report: 2004: Changing History, 2004.Google Scholar

[43]

—, A Global Brief on Vector-Borne Diseases, 2014.Google Scholar

show all references

References:
[1]

N. BameS. BowongJ. MbangG. Sallet and J. Tewa, Global stability for seis models with n latent classes, Math. Biosci. Eng., 5 (2008), 20-33. doi: 10.3934/mbe.2008.5.20. Google Scholar

[2]

E. D. Barnett and P. F. Walker, Role of immigrants and migrants in emerging infectious diseases, Medical Clinics of North America, 92 (2008), 1447-1458. doi: 10.1016/j.mcna.2008.07.001. Google Scholar

[3]

M. Q. BenedictR. S. LevineW. A. Hawley and L. P. Lounibos, Spread of the tiger: Global risk of invasion by the mosquito aedes albopictus, Vector-borne and Zoonotic Diseases, 7 (2007), 76-85. Google Scholar

[4]

D. BicharaA. Iggidr and L. Smith, Multi-stage vector-borne zoonoses models: A global analysis, Bulletin of Mathematical Biology, 80 (2018), 1810-1848. doi: 10.1007/s11538-018-0435-1. Google Scholar

[5]

B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, 184. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0619-4. Google Scholar

[6]

F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. Google Scholar

[7]

C. Castillo-Chavez and H. R. Thieme, Asymptotically Autonomous Epidemic Models, in Mathematical Population Dynamics: Analysis of Heterogeneity, Volume One: Theory of Epidemics, O. Arino, A. D.E., and M. Kimmel, eds., Wuerz, 1995.Google Scholar

[8]

Centers for Disease Control and Prevention, Illnesses from mosquito, tick, and flea bites increasing in the us, Centers for Disease Control and Prevention, https://www.cdc.gov/media/releases/2018/p0501-vs-vector-borne.html, (2018).Google Scholar

[9]

G. ChowellP. Diaz-DuenasJ. MillerA. Alcazar-VelazcoJ. HymanP. Fenimore and C. Castillo-Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data, Mathematical biosciences, 208 (2007), 571-589. doi: 10.1016/j.mbs.2006.11.011. Google Scholar

[10]

G. Cruz-PachecoL. Esteva and C. Vargas, Control measures for chagas disease, Mathematical biosciences, 237 (2012), 49-60. doi: 10.1016/j.mbs.2012.03.005. Google Scholar

[11]

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Math. Biosci., 150 (1998), 131-151. doi: 10.1016/S0025-5564(98)10003-2. Google Scholar

[12]

E. A. GouldP. GallianX. De Lamballerie and R. N. Charrel, First cases of autochthonous dengue fever and chikungunya fever in france: From bad dream to reality!, Clinical Microbiology and Infection, 16 (2010), 1702-1704. doi: 10.1111/j.1469-0691.2010.03386.x. Google Scholar

[13]

M. Grandadam, V. Caro, S. Plumet, J.-M. Thiberge, Y. Souares, A.-B. Failloux, H. J. Tolou, M. Budelot, D. Cosserat, I. Leparc-Goffart and P. Desprès, Chikungunya virus, southeastern france, Emerging Infectious Diseases, 17 (2011), p. 910.Google Scholar

[14]

H. Guo and M. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525. doi: 10.3934/mbe.2006.3.513. Google Scholar

[15]

H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. Google Scholar

[16]

H. GuoM. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM Journal on applied mathematics, 72 (2012), 261-279. doi: 10.1137/110827028. Google Scholar

[17]

J. M. HymanJ. Li and E. Stanley, The differential infectivity and staged progression models for the transmission of HIV., Math. Biosci., 155 (1999), 77-109. doi: 10.1016/S0025-5564(98)10057-3. Google Scholar

[18]

M. Isaäcson, Airport malaria: A review, Bulletin of the World Health Organization, 67 (1989), p. 737.Google Scholar

[19]

T. L. Johnson, E. L. Landguth and E. F. Stone, Modeling relapsing disease dynamics in a host-vector community, PLoS Negl Trop Dis, 10 (2016), p. e0004428. doi: 10.1371/journal.pntd.0004428. Google Scholar

[20]

K. E. JonesN. G. PatelM. A. LevyA. StoreygardD. BalkJ. L. Gittleman and P. Daszak, Global trends in emerging infectious diseases, Nature, 451 (2008), 990-993. doi: 10.1038/nature06536. Google Scholar

[21]

A. M. Kilpatrick and S. E. Randolph, Drivers, dynamics, and control of emerging vector-borne zoonotic diseases, The Lancet, 380 (2012), 1946-1955. doi: 10.1016/S0140-6736(12)61151-9. Google Scholar

[22]

M. Li and J. S. Muldowney, On r.a. smith's automonmous convergence theorem, Rocky Mountain J. Math., 25 (1995), 365-379. doi: 10.1216/rmjm/1181072289. Google Scholar

[23]

M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5. Google Scholar

[24]

C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16. doi: 10.1016/S0025-5564(02)00149-9. Google Scholar

[25]

C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Differ. Equations, 16 (2004), 139-166. doi: 10.1023/B:JODY.0000041283.66784.3e. Google Scholar

[26]

C. J. Mitchell, Geographic spread of aedes albopictus and potential for involvement in arbovirus cycles in the mediterranean basin, Journal of Vector Ecology, 20 (1985), 44-58. Google Scholar

[27]

J. Moon, Counting Labeled Trees, Canadian Math, Monographs, 1970. Google Scholar

[28]

C. PalmerE. LandguthE. Stone and T. Johnson, The dynamics of vector-borne relapsing diseases, Math. Biosci., 297 (2018), 32-42. doi: 10.1016/j.mbs.2018.01.001. Google Scholar

[29]

M. Paty, C. Six, F. Charlet, G. Heuzé, A. Cochet, A. Wiegandt, J. Chappert, D. Dejour-Salamanca, A. Guinard, P. Soler, V. Servas, M. Vivier-Darrigol, M. Ledrans, M. Debruyne, O. Schaal, C. Jeannin, B. Helynck, I. Leparc-Goffart and B. Coignard, Large number of imported chikungunya cases in mainland france, 2014: a challenge for surveillance and response, Eurosurveillance, 19 (2014), 20856. doi: 10.2807/1560-7917.ES2014.19.28.20856. Google Scholar

[30]

P. Poletti, G. Messeri, M. Ajelli, R. Vallorani, C. Rizzo and S. Merler, Transmission potential of chikungunya virus and control measures: The case of italy, PLoS One, 6 (2011), e18860. doi: 10.1371/journal.pone.0018860. Google Scholar

[31]

G. RezzaL. NicolettiR. AngeliniR. RomiA. FinarelliM. PanningP. CordioliC. FortunaS. BorosF. MaguranoG. SilviP. AngeliniM. DottoriM. CiufoliniG. Majori and A. Cassone, Infection with hikungunya virus in italy: An outbreak in a temperate region, The Lancet, 370 (2007), 1840-1846. doi: 10.1016/S0140-6736(07)61779-6. Google Scholar

[32]

D. Rogers and S. Randolph, Climate change and vector-borne diseases, Advances in parasitology, 62 (2006), 345-381. doi: 10.1016/S0065-308X(05)62010-6. Google Scholar

[33]

D. A. Shroyer, AEDES ALBOPICTUS and arboviruses: A concise review of the literaturei, Journal of the American Mosquito Control Association, (1986).Google Scholar

[34]

F. SimonH. Savini and P. Parola, Chikungunya: A paradigm of emergence and globalization of vector-borne diseases, Medical Clinics of North America, 92 (2008), 1323-1343. doi: 10.1016/j.mcna.2008.07.008. Google Scholar

[35]

T. TabataM. PetittH. Puerta-GuardoD. MichlmayrC. WangJ. Fang-HooverE. Harris and L. Pereira, Zika virus targets different primary human placental cells, suggesting two routes for vertical transmission, Cell Host & Microbe, 20 (2016), 155-166. doi: 10.1016/j.chom.2016.07.002. Google Scholar

[36]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. Google Scholar

[37]

J. TumwiineJ. Mugisha and L. Luboobi, A host-vector model for malaria with infective immigrants, Journal of Mathematical Analysis and Applications, 361 (2010), 139-149. doi: 10.1016/j.jmaa.2009.09.005. Google Scholar

[38]

A. Vega-Rua, K. Zouache, V. Caro, L. Diancourt, P. Delaunay, M. Grandadam and A.-B. Failloux, High efficiency of temperate aedes albopictus to transmit chikungunya and dengue viruses in the southeast of france, PLoS One, 8 (2013), e59716. doi: 10.1371/journal.pone.0059716. Google Scholar

[39]

M. Vidyasagar, Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability., IEEE Trans. Autom. Control, 25 (1980), 773-779. doi: 10.1109/TAC.1980.1102422. Google Scholar

[40]

N. WauquierBe cquartD. NkogheC. PadillaA. Ndjoyi-Mbiguino and E. M. Leroy, The acute phase of chikungunya virus infection in humans is associated with strong innate immunity and t cd8 cell activation, Journal of Infectious Diseases, 204 (2011), 115-123. doi: 10.1093/infdis/jiq006. Google Scholar

[41]

A. Wilder-Smith, M. Quam, O. Sessions, J. Rocklov, J. Liu-Helmersson, L. Franco and K. Khan, The 2012 dengue outbreak in madeira: Exploring the origins, Euro Surveill, 19 (2014), 20718. doi: 10.2807/1560-7917.ES2014.19.8.20718. Google Scholar

[42]

World Health Organization et al., The World Health Report: 2004: Changing History, 2004.Google Scholar

[43]

—, A Global Brief on Vector-Borne Diseases, 2014.Google Scholar

Figure 1.  Flow diagram of Model 1. Note that, to unclutter the figure, we did not display the arrows that represent the recruitments for $ I_2 $, $ I_3 $ and $ I_4 $. Similarly, the arrows representing the death, $ \mu_i $, and recovery rates, $ \eta_i, $ in all host classes are not displayed
Figure 2.  Effects of host-vector transmission on the dynamics of Model (2) with $ n = 4 $. The proportions of infectious influx are $ p_1 = 0.2 $, $ p_3 = 0.1 $, $ p_4 = 0 $ and $ p_{5} = 0.3 $. The transfer matrix $ M $ is such as $ \gamma_{13} = \gamma_{24} = 0.1 $, $ \gamma_{14} = 0.2 $, $ \delta_{21} = 0.01 $, $ \delta_{31} = 0.02 $, $ \delta_{41} = 0.001 $, $ \delta_{32} = 0.03 $, $ \delta_{42} = 0.01 $ and $ \delta_{43} = 0.03 $
Figure 3.  Dynamics of infected hosts and vectors when the hypotheses of Theorem 3.1, Item 3 are satisfied. The proportions of infectious influx are $ p_0 = 0 $, $ {\bf p} = (0, 0, p_3, p_4)^T = (0, 0, 0.2, 0.0001)^T $ and $ p_{5} = 0.3 $. The transfer matrix $ M $ is such as $ \gamma_{13} = 0.1 $, $ \gamma_{14} = \gamma_{24} = 0 $, $ \delta_{21} = 0.01 $, $ \delta_{31} = \delta_{32} = \delta_{42} = 0 $, $ \delta_{41} = 0.035 $, and $ \delta_{43} = 0.03 $
Figure 4.  Dynamics of infected hosts and vectors when the hypotheses of Theorem 3.1, Item 4 are satisfied. The proportions of infectious influx are $ p_0 = 0 $, $ {\bf p} = (0, 0, p_3, p_4)^T = (0, 0, 0.2, 0.0001)^T $ and $ p_{5} = 0.3 $. The transfer matrix $ M $ is such as $ \gamma_{13} = 0.1 $, $ \gamma_{14} = \gamma_{24} = 0 $, $ \delta_{21} = 0.01 $, $ \delta_{31} = \delta_{32} = \delta_{41} = \delta_{42} = 0 $, and $ \delta_{43} = 0.03 $. With these parameters, $ {\mathcal N}_0^2 = 0.3237\leq1 $. As expected, the vector population will be disease-free (Figure 4(b) and Figure 4(d)) and the infectious hosts are generated only through influx of infectious immigrants at stage 3 and 4 (Figure 4(a) and Figure 4(c))
Figure 5.  Dynamics of infected hosts and vectors when the hypotheses of Theorem 3.1, Item 4 are satisfied. The proportions of infectious influx are $ p_0 = 0 $, $ {\bf p} = (0, 0, p_3, p_4)^T = (0, 0, 0.2, 0.0001)^T $ and $ p_{5} = 0.3 $. The transfer matrix $ M $ is such as $ \gamma_{13} = 0.1 $, $ \gamma_{14} = \gamma_{24} = 0 $, $ \delta_{21} = 0.01 $, $ \delta_{31} = \delta_{32} = \delta_{41} = \delta_{42} = 0 $, and $ \delta_{43} = 0.03 $. Using the values $ a = 0.9 $ and $ \beta_{vh} = 0.9 $, $ {\mathcal N}_0^2 = 1.6051>1 $, and thus the trajectories of the system converge towards an interior equilibrium
Table 1.  Description of the parameters used in System (1)
Parameters Description
$ \pi_h $ Recruitment of the host
$ \pi_v $ Recruitment of vectors
$ p_0 $ Proportion of latent immigrants
$ p_i $ Proportion of infectious immigrants at stage $ i $
$ a $ Biting rate
$ \mu_h $ Host's natural death rate
$ \beta_{v, h} $ Host's infectiousness by mosquitoes per biting
$ \beta_{i} $ Vector's infectiousness by host at stage $ i $ per biting
$ \nu_h $ Host's rate at which the exposed individuals become infectious
$ \eta_i $ Per capita recovery rate of an infected host at stage $ i $
$ \gamma_{ij} $ Host's per capita progression rate from stage $ i $ to $ j $
$ \delta_{ij} $ Host's per capita regression rate from stage $ i $ to $ j $
$ \mu_v $ Vectors' natural mortality rate
$ \delta_v $ Vectors' control-induced mortality rate
$ \nu_v $ Rate at which the exposed vectors become infectious
Parameters Description
$ \pi_h $ Recruitment of the host
$ \pi_v $ Recruitment of vectors
$ p_0 $ Proportion of latent immigrants
$ p_i $ Proportion of infectious immigrants at stage $ i $
$ a $ Biting rate
$ \mu_h $ Host's natural death rate
$ \beta_{v, h} $ Host's infectiousness by mosquitoes per biting
$ \beta_{i} $ Vector's infectiousness by host at stage $ i $ per biting
$ \nu_h $ Host's rate at which the exposed individuals become infectious
$ \eta_i $ Per capita recovery rate of an infected host at stage $ i $
$ \gamma_{ij} $ Host's per capita progression rate from stage $ i $ to $ j $
$ \delta_{ij} $ Host's per capita regression rate from stage $ i $ to $ j $
$ \mu_v $ Vectors' natural mortality rate
$ \delta_v $ Vectors' control-induced mortality rate
$ \nu_v $ Rate at which the exposed vectors become infectious
[1]

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