Mathematics

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https://hdl.handle.net/20.500.14540/131

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Browsing Mathematics by Advisor "Prof.Dr. Kedar Nath Uprety"

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    Dynamics of Transmission and Control of Measles
    (Institute of Sciecne and Technology, 2024-08) Pokharel, Anjana; Prof.Dr. Kedar Nath Uprety
    The emergence and re-emergence of infectious diseases have become a global problem. Measles is a highly contagious human viral disease whose outbreaks frequently occur in many countries, including Nepal, despite the availability of vaccines partly due to the lack of compliance with vaccination. While the National Immunization Program is in place in Nepal, the frequent occurrence of measles in Nepal remains a major cause of child morbidity and mortality. Mathematical modeling for infectious diseases aids in forecasting and comprehending the dynamics of such diseases, facilitating the deployment of effective public health interventions and the allocation of resources. In this work, we developed a novel transmission dynamics model in the form of system of nonlinear ordinary differential equations to evaluate the effects of monitored vaccination programs on individuals who have skipped the regular vaccination program, aiming to control and eliminate measles properly. Our model was validated by Nepal's yearly incidence case data from 2000 to 2019. We calculated the vaccinated reproduction number, Rv, using the Next Generation Matrix method. We also computed the effective reproduction number of measles in Nepal. We performed model analyses to establish the global asymptotic stability of the disease-free equilibrium point for Rv<1 and the uniform persistence of the disease for Rv>1. Moreover, we performed model simulations to identify monitored vaccination strategies for successfully controlling measles in Nepal. Additionally, using the model, we analyzed the long-term dynamics of the epidemic. Our model demonstrates that the monitored vaccination programs can help control the potential resurgence of the disease. Due to the nationwide lockdown enforced by the government of Nepal during the COVID-19 pandemic, the scheduled immunization program was disrupted. As a result, amid the ongoing lockdown, measles outbreaks, including fatal cases, have been reported in several districts of Nepal. Moreover, measles cases in adult groups, albeit small in number, indicate that the previously neglected adult group may need to be brought into vaccine coverage to achieve the WHO’s goal of measles eradication around the world. To examine the role of measles-infected adult groups and to evaluate combined adult-child vaccination programs for eradication, we develop a further extended transmission dynamics model describing measles cases in adults and children. We validated the new model using measles outbreak cases in Nepal from November 24, 2022, to March 10, 2023. Detailed analyses of our model provide the vaccination reproduction number, conditions for measles eradication or persistence, and the role of contact network size. Furthermore, our results highlight that while children are the primary targets for measles outbreaks, a small infection in adults may act as a reservoir for measles, posing obstacles to eradication. Moreover, our model predicts that while impactful controls can be achieved by children-focused vaccines, a combined adult-child vaccination program may help ensure the eradication of the disease.
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    Modeling and Analysis of Dynamics of Malaria Transmission with Control Measures: Imported Cases
    (Institute of Science & Technology, 2024-08) Gautam, Ramesh; Prof.Dr. Kedar Nath Uprety
    Many countries, including low and middle-income countries like Nepal, are facing many challenges in pursuing malaria elimination. Despite progress in reducing the malaria burden significantly, these countries still struggle with low levels of malaria transmission, making complete elimination difficult. Mathematical modeling of diseases like malaria provides essential insights into disease dynamics. These insights help develop effective public health strategies for allocating resources efficiently, and support evidence-based policies to reduce endemic diseases like malaria. In the global effort to eliminate malaria, human mobility and the relapse of Plasmodium Vivax and Plasmodium Ovale malaria pose significant challenges. In this thesis, first, we develop a mathematical model of malaria transmission, integrating the cross-border mobility of migrant workers from low-endemic countries like Nepal to high-endemic countries like India. The model describes how migrant workers become infectious abroad and bring malaria back to their home country as imported cases. Despite complicated features with eight-dimensional nonlinear non-homogeneous systems, we were able to derive three disease-free equilibria and three epidemic thresholds, R0, R1, and R2, which establish their local stability. In addition, we established the theorems for global stability and uniform persistence. Our model simulations show that among Insecticide Treated Nets (ITN), Indoor Residual Spraying (IRS), Border Screening and Isolation (BSI), and Migration Reduction (MR), MR is the most effective strategy at low mosquito biting rates, whereas ITN is the most effective at high mosquito biting rates for malaria control and elimination. Second, we conducted a thorough bifurcation analysis of our model to examine whether migration can cause backward bifurcation phenomena, demonstrating bistability for threshold values less than one. Along with theoretical derivations, we developed MATLAB code to obtain different types of bifurcation diagrams associated with various modes of mobility. Our backward bifurcation analysis revealed three major results considering three different mobility conditions based on policies implemented at home and abroad: (a) If the mobility of migrants is completely restricted, the home country becomes free from malaria when the threshold R0 < 1 and the disease-induced death rate falls below some threshold. (b) If the mobility of migrants continues with complete protection from malaria transmission abroad, then both the home country and migrants abroad are free from malaria when the threshold R1 < 1, provided the mobility of infectious migrants is below the certain threshold. However, a backward bifurcation occurs if the mobility exceeds the threshold. (c) If there is mobility of migrants without protection abroad, the home country can only reduce the malaria burden with local control strategies and by reducing the mobility rate of migrants below some levels. However, elimination is only possible if the abroad region is free from malaria. Third, we developed a model incorporating delay in relapses to address the role of Plasmodium vivax and Plasmodium Ovale malaria relapses in malaria elimination programs in low-endemic countries like Nepal. Our model analyses and simulations predict that in the absence of imported cases, with less than 50% initial relapse rate of malaria and less than 14% subsequent relapses within five months, malaria can potentially be eliminated by 2025. However, the initial relapse rate above 28% and the subsequent relapses above 25% stand as the obstacle to eliminating malaria by 2025. Also, shortening the relapse interval to two months under an initial relapse rate below 50% enables malaria elimination by 2024, while extending it to six months will cause a delay in elimination beyond 2025. Furthermore, periodic outbreaks are observed via Hopf bifurcation when the reproduction number exceeds unity. Our study has made two major policy recommendations to ensure the successful elimination of malaria in Nepal by 2026: (1) Implementing a comprehensive awareness program to protect migrants from malaria transmission abroad, coupled with rigorous border screening and isolation of infectious migrants, to maintain the mobility rate of infectious migrants below the threshold. (2) Establishing and enforcing a comprehensive radical cure treatment protocol, along with a structured follow-up program, to keep relapse proportions below a critical level. नेपालजस्ता न्यून र मध्यम आय भएका देश लगायत धेरै देशले औलो उन्मूलनका लागि धेरै चुनौतीहरूको सामना गरिरहेका छन् । औलोको बोझलाई उल्लेखनीय रूपमा घटाउनमा प्रगति भएता पनि, यी देशहरू अझै पनि औलो प्रसारणको न्यून स्तरसँग संघर्ष गरिरहेका छन् र पूर्ण रूपमा उन्मूलन गर्न गाह्रो भइरहेको छ । औलो जस्ता रोगहरूको गणितीय मोडेलिङले रोगको गतिशीलतामा आवश्यक अन्तरदृष्टि प्रदान गर्दछ । यी अन्तर्दृष्टिहरूले प्रभावकारी रूपमा स्रोतहरू विनियोजन गर्न प्रभावकारी सार्वजनिक स्वास्थ्य रणनीतिहरू विकास गर्न मद्दत गर्दछ, र औलो जस्ता लामोसमयसम्म रहने रोगहरू कम गर्न र उन्मूलन गर्नका लागी प्रमाण-आधारित नीति निर्माण गर्नका लागी सहयोग गर्दछ। औलो उन्मूलन गर्ने विश्वव्यापी प्रयासमा, मानव गतिशीलता र प्लाज्मोडियम भाइभ्याक्स र ओभेल मलेरियाको पुनरावृत्तिले महत्त्वपूर्ण चुनौतीहरू खडा गरेको छ । यस थीसिसमा, सर्वप्रथम, हामी नेपाल जस्ता न्यून-स्थानीय देशहरूबाट भारत जस्ता उच्च-स्थानीय देशहरूमा आप्रवासी कामदारहरूको सीमापार गतिशीलतालाई एकीकृत गर्दै औलो प्रसारणको गणितीय मोडेल विकास गर्छौं । मोडेलले प्रवासी कामदारहरू विदेशमा कसरी संक्रामक हुन्छन् र औलो लाई आयातित केसहरूको रूपमा आफ्नो देशमा फिर्ता ल्याउँदछ भनेर वर्णन गर्दछ । आठ-आयामी ननलिनयर ननहोमोजीनियस प्रणालीहरूसँगको जटिलताको बावजुद पनि, हामीले तीन औलो मुक्त सन्तुलन र तीन महामारी मापक थ्रेसहोल्डहरू, R0, R1, र R2 प्राप्त गर्न सक्षम भयौं, जसले तिनीहरूको स्थानीय स्थिरता स्थापना गर्दछ । थप रूपमा, हामीले विश्वव्यापी स्थिरता र समान दृढताका लागि प्रमेयहरू स्थापना गरेका छौं । हाम्रो मोडेल सिमुलेशनहरूले देखाउँदछ कि कीटनाशक-उपचारित जालहरू (ITN), भित्री अवशिष्ट स्प्रेइ (IRS), सीमा जाँच र अलगाव (BSI), र माइग्रेसन रिडक्सन (MR), हरूमा MR कम लामखुट्टे टोक्ने दरहरूमा सबैभन्दा प्रभावकारी रणनीति र ITN औलो नियन्त्रण र उन्मूलनको लागि उच्च लामखुट्टेले टोक्ने दरमा सबैभन्दा प्रभावकारी पुष्टी भय। दोस्रो, हामीले माइग्रेसनले गर्दा महामारी मापक थ्रेसहोल्ड एक भन्दा कम भयर मात्रै औलो मुक्त हुन नसक्ने र थ्रेसहोल्ड मानहरूको लागि बिस्टेबिलिटी प्रदर्शन गर्दै पछाडिको विभाजन घटना निम्त्याउन सक्छ कि भनेर जाँच्नको लागि मोडेलको विस्तृत विभाजन विश्लेषण गरीयको छ । सैद्धान्तिक व्युत्पन्नहरूको साथमा, हामीले गतिशीलताका विभिन्न मोडहरूसँग सम्बन्धित विभिन्न प्रकारका विभाजन रेखाचित्रहरू प्राप्त गर्न MATLAB कोड विकास गर्यौं । हाम्रो पछाडी विभाजन विश्लेषणले स्वदेश र विदेशमा लागू हुनसक्ने नीतिहरु का कारण तीनवटा प्रमुख नतिजाहरू पत्ता लगाएको छ जुन तीन फरक गतिशीलता अवस्थाहरूमा आधारित छ: (क) यदि आप्रवासीहरूको गतिशीलता पूर्ण रूपमा प्रतिबन्धित गर्न सकियो भने, थ्रेसहोल्ड R0 <1 र रोग-प्रेरित मृत्यु दर केही थ्रेसहोल्ड भन्दा तल झर्दा गृह देश औलो बाट मुक्त हुन्छ । (ख) विदेशमा औलोको संक्रमणबाट पूर्ण सुरक्षाका साथ प्रवासीहरूको आवतजावत जारी छ भने, थ्रेसहोल्ड R1 <1 र संक्रामक आप्रवासीहरूको गतिशीलता निश्चित थ्रेसहोल्डभन्दा कम भएमा स्वदेश र विदेशमा बस्नेहरू दुवै औलोबाट मुक्त हुन्छन् । यद्यपि, यदि गतिशीलता थ्रेसहोल्ड भन्दा बढी छ भने पछाडि विभाजन हुन्छ र उन्मुलनको लागी थप प्रयास जरुरी पर्दछ । (ग) विदेशमा सुरक्षा बिना आप्रवासीहरूको गतिशीलता भएमा, स्वदेशले स्थानीय नियन्त्रण रणनीतिहरू र केही तहभन्दा तल आप्रवासीहरूको गतिशीलता दर घटाएर मात्र औलोको बोझ कम गर्न सक्छ । तर, विदेश क्षेत्र औलोमुक्त भए मात्रै उन्मूलन सम्भव हुन्छ । तेस्रो, हामीले पुन: आवृत्तिमा ढिलाइलाई समावेश गर्दै एक मोडल विकास गर्यौं जसले कम संक्रमण दर भएका देशहरू जस्तै नेपालमा औलो को उन्मूलन कार्यक्रमहरूमा प्लाज्मोडियम विवाक्स र ओभाले मलेरियाका पुन: आवृत्तिहरूको भूमिकालाई सम्बोधन गर्दछ । हाम्रो मोडल विश्लेषणहरू र सिमुलेशनहरूले भविष्यवाणी गर्छन् कि आयातित केसहरूको अभावमा, मलेरियाको प्रारम्भिक पुन: आवृत्ति दर ५०% भन्दा कम र पाँच महिनाभित्र दोस्रो पुन: आवृत्ति १४% भन्दा कम हुने अवस्थामा, २०२५ सम्ममा मलेरिया उन्मूलन गर्न सकिन्छ । तर, प्रारम्भिक पुन: आवृत्ति दर २८% भन्दा माथि र दोस्रो पुन: आवृत्ति २५% भन्दा माथि हुने अवस्थामा २०२५ सम्ममा मलेरियाको उन्मूलनमा बाधा पुग्नेछ । साथै, महामारी थ्रेसहोल्ड R0 एक भन्दा माथि पुगेमा हप्फ बिफरकेसनको माध्यमबाट समय-समयमा प्रकोपहरू देखिन्छन् । हाम्रो अध्ययनले २०२६ सम्ममा नेपालमा मलेरियाको सफल उन्मूलन सुनिश्चित गर्न दुई प्रमुख नीतिगत सिफारिसहरू गरेको छ: (१) विदेशमा मलेरिया संक्रमणबाट आप्रवासीहरूलाई जोगाउन व्यापक चेतना कार्यक्रम लागू गर्दै, संक्रमण भएका आप्रवासीहरूको गति दरलाई थ्रेसहोल्डभन्दा तल राख्न कडा सीमा स्क्रिनिङ र संक्रमित आप्रवासीहरूको अलगाव गर्न । (२) पुन: आवृत्ति अनुपातलाई एक महत्वपूर्ण स्तरभन्दा तल राख्न व्यापक उपचार विधि र संरचित अनुगमन कार्यक्रमको स्थापना र कार्यान्वयन गर्न ।