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Doctoral Degrees (Applied Mathematics)

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Browsing Doctoral Degrees (Applied Mathematics) by SDG "SDG3"

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    Modeling the effectiveness of control measures from the within-host dynamics to the population dynamics of sleeping sickness.
    (2024) Mulalo, Makhuvha.; Mambili-Mamboundou, Hermane.
    This study addresses the persistent health challenge of sleeping sickness in sub-Saharan Africa by comprehensively exploring its transmission dynamics. The primary focus is on developing, studying, and analyzing mathematical models spanning from within-host dynamics to multiscale interactions of sleeping sickness. The thesis begins by investigating the dynamics within a single sleeping sickness-infected human host. The model considers the interplay between parasite types, macrophages, and cytokines within the human host. Findings suggest that the immune response play a pivotal part in regulating parasite growth, and the absence of switching enables effective immune system intervention. Two distinct optimal control problems emerge, emphasizing the need to identify the disease stage for effective drug application, thereby preventing parasite persistence and reducing drug toxicity. The study’s second phase focuses on developing an epidemiological model of sleeping sickness transmission among humans, non-human hosts, and tsetse flies. Sensitivity analysis reveals that parameters such as biting rate and death rate are particularly influential. The examination of bifurcation concerning the effective reproductive number (Re) indicates the presence of a backward bifurcation within the system which signifies that the traditional condition of Re < 1 is no longer adequate for achieving effective disease elimination. Disease dynamics are strongly scale dependent, so we propose a multi-temporal scale model that incorporates within-host and between-host disease dynamics. Results are compared with the single-scaled models. Furthermore, the numerical solutions for susceptible humans in both the between-host and multiscale models demonstrated a similar pattern: a rapid reduction in susceptible human populations within the first 50 days in the absence of intervention. The vector profiles in both the multiscale and between-host models showed similar trends, with intervention initially applied at maximum intensity and rapidly reduced within the first 50 days of use. However, a notable distinction arose regarding treatment profiles: whereas within-host results indicated that drugs needed to be administered at maximum dosage to reduce the burden of infection effectively, the opposite was observed in the between-host model. In conclusion, this thesis offers valuable insights into the disease transmission dynamics of sleeping sickness, highlighting the significance of comprehending both within host and between-host interactions. Essentially, our models proved to be more mathematically, numerically, and biologically tractable compared to most standard models used for projecting sleeping sickness. Iqoqa. Lolu cwaningo lubuka izingqinamba zezempilo ezingaphezi zesifo sokulala e-Afrika emaphakathi nasezansi ngokuhlola okuphelele ukuthelelana kwaso okwehlukahlukile. Kugxilwe kakhulu ekuthuthukiseni, ukucwaninga, kanye nokuhlaziya amamodeli ezibalo kuhlanganisa kusukela phakathi ekwehlukeni komsingathi kuya ekuhlumaneni okuvamile kwesifo sokulala. Ucwaningo luqala ngokuphenya ukwehlukahluka phakathi komsingathi wesifo sokulala oyedwa othelelekile. Imodeli ibuka ukuxhumana phakathi kwezinhlobo zento ephila ngokudla emzimbeni wenye into, izigaba ezinkulu, kanye nokwehlukana kwezinhlayiya phakathi komsingathi owumuntu. Okutholakele kuphakamisa ukuthi ukuphendula kwamasosha omzimba kubamba iqhaza elibalulekile ekwengameleni ukukhula kwezinto eziphila ngokudla emzimbeni wenye into, kanye nokungabi bikho kokushintsha okuvumela ukungenelela kwendlela yamasosha omzimba eyimpumelelo. Izinkinga ezimbili ezilungile ezingefani zokwengamela zaqubuka, okugcizelela Isidingo sokukhomba isigaba sesifo ukuze kubukwe ukusebenza kwemithi yokwelapha, ngenxa yalokho kuvikele ukusabalala kanye nokwehlisa ubungozi bemithi yokwelapha. Isigaba sesibili socwaningo sigxile ekwakheni imodeli yokukhombela, ukuvela kanye nokusabalala kokuthelelana ngezifo nokuthelelana ngesifo sokulala phakathi kwabantu, abasingathi abangebona abantu, kanye nezinambuzane zetsetse. Ukuhlaziya kokuzwela kuveza ukuthi imikhawulo efana nezinga lokuluma kanye nezinga lokufa kujwayele ukuba nomthelela. Ukuhlola okuhlukanisa kabili mayelana nokusebenza kokwanda kwenombolo i-R_e kuveza ubukhona bokuhlukanisa kabili okubheke emuva phakathi kwenqubo ekhomba ukuthi isimo sendabuko se-R_e<1 asisakufanele ukufinyelela ekwehliseni isifo. Ukwehlukahluka kwesifo kuncike kakhulu esikalini, ngaleyo ndlela siphakamisa imodeli yesikali engaphindaphindeka isikhashana ehlanganisa maphakathi nomsingathi kanye naphakathi kokwehluka kwesifo kubasingathi. Imiphumela iqhathaniswa namamodeli esikali esisodwa. Okunye futhi, izixazululo zezinombolo zabantu abasebungozini kukho kokubili phakathi abasingathi kanye namamodeli ezikali eziningana zaveza indlela efanayo; ukwehla ngamandla komphakathi wabantu abangabasingathi esinsukwini zokuqala ezingama-50 uma kungekho ukungenelela. Izinhlaka zobungako obunezakhi ezimbili ezizimele kuko kokubili izikali eziningana kanye namamodeli aphakathi nabasingathi kwaveza izinkomba ezifanayo, ngokungenelela okuqale kwasetshenziswa kwase kwehliswa futhi kakhulu ezinsukwini ezingama-50 zisetshenziswa. Yize noma kunjalo, umehluko oqaphelekayo wavela mayelana nezinhlaka zokwelashwa: kanti imiphumela ephakathi kwabasingathi yaveza ukuthi imithi yokwelashwa idinga ukusetshenziswa ngesikali esikhulukazi ukwehlisa umthwalo wokutheleleka ngempumelelo okuphikisanayo kwabonakala kwimodeli ephakathi nabasingathi. Isiphetho, lolu cwaningo lunikeka imibono ebalulekile ekuhlukeni kokutheleleka ngesifo sokulala, lugqamisa ukubaluleka kokuqonda kokubili imaphakathi lomsingathi kanye nokuxhumana kwaphakathi nomsingathi. Okubalulekile, amamodeli ethu aqinisekisa ukusebenza ngezibalo, izinombolo, kanye nokuphilayo okuphathekayo uma kuqhathaniswa namamodeli ajwayelekile asetshenziswa ukukhomba isifo sokulala.