Chapter 3 Modelling Communicable Diseases

Chapter 3 Modelling Communicable Diseases [Spread Awareness not the Disease]

Jyoti Gupta 53 3.1: Introduction The diseases that can be transmitted directly or indirectly from one person to other are known as communicable diseases. These diseases are caused by various micro-organism such as bacteria, virus etc. They are well classified depending on their mode and method of transmission. The method of transmission can be classified as direct, indirect and zoonotic. Direct: The diseases which are communicated directly, without any intermediate agent, fall under this category. This includes o Airborne- Tuberculosis, Influenza o Blood/Body fluid borne (i.e. sexually transmitted diseases) HIV/AIDS, Hepatitis B, C o Water/Food borne Cholera, Hepatitis A, Typhoid, etc. Indirect: The diseases which need an intermediate agent such as mosquito, tick, flies or any other insect or animal, fall under this category. E.g. vector-borne diseases. Zoonotic The people handling or feeding infected animals are at a risk of catching their infections. E.g. bird flu, mad cow disease. In general, all infectious diseases are communicable, in one way or the other. In this chapter, the focus is HIV/AIDS and Tuberculosis. So, entire detail and modelling would be discussed for these two. 3.2 HIV/AIDS AIDS (Acquired Immuno Deficiency Syndrome), caused by HIV (Human Immuno-deficiency Virus), has become one of the world s most serious

Mathematical Modelling Control (In Indian Context) 54 health challenges. Out of 34 lakh global AIDS cases, 48 lakh cases are in Asia, and India has the highest (49%) number of people affected by this deadliest disease in the continent as revealed by UAIDS report (211). Due to lack of knowledge about AIDS people have fear in their minds against the AIDS patients. Also, due to fear of social boycott, people try to hide their disease status. After initial infection, the virus becomes less active in the body, although it is still present. During this period many people do not have any symptoms of HIV infection. This period is called latency phase. When person s immune system is severely damaged by the virus and has difficulty in fighting diseases, then at this stage the person is said to have AIDS. Thus, AIDS is the late stage of HIV infection. Before development of certain medications, people with HIV could progress to AIDS in just a few years. Currently, people can live much longer even decades with HIV before they develop AIDS. Scientists believe HIV came from a particular kind of chimpanzee in Western Africa. Humans probably came in contact with HIV when they hunted and ate infected animals. Human Immunodeficiency Virus (HIV) is lot like other viruses, including those that cause the common cold. But there is an important difference over the time, our immune system can clear most viruses out of our body, but not HIV. It belongs to an unusual group of viruses called retroviruses found in monkeys and apes, sheep and goats.

Jyoti Gupta 55 There are two main strains of HIV i.e. HIV-1 and HIV-2. HIV-1 has caused the majority (8%) of infections and AIDS cases and HIV-2 is contracted in Africa and in some parts of India. Once the virus enters the body, it attacks CD4+ type of white blood cells (WBCs) in blood and gradually kills them. These cells help us to fight against various infections. Once they are destroyed, our body s resistance goes down and the person suffers from lots of infections. Initially, an HIV infected person looks completely normal and healthy. As early as 3 - months after exposure to HIV, people can experience an acute illness, often described as the worst flu ever. This is called Acute Retroviral Syndrome (ARS) or primary HIV infection, and it is the body s natural response to HIV infection. During primary HIV infection, there are higher levels of virus circulating in the blood, which means the person can easily transmit the virus to others. It is important to remember that not everyone gets ARS when they become infected with HIV. The general symptoms may include fever, rashes, night-sweats, diarrhoea, mouth ulcers, sudden weight loss and other opportunistic illnesses. HIV do not survive outside the human body and therefore, can only be transmitted directly from person to person, either by sexual contact, exchange of blood or body fluids or from mother to child. HIV incidence and dynamics varies from country to country and from region to region, depending on various risk factors. Since this disease is driven

Mathematical Modelling Control (In Indian Context) 56 mainly by sexual transmission, the level and intensity of risk behaviour are the main determinants of the spread of the virus. A lot of efforts are being done in the direction of controlling the spread of HIV. World Health Organisation (WHO) had an International AIDS Conference with the theme stepping up the pace on July 2-25, 214. Here, they discussed various means of spreading awareness among population along with new treatment possibilities. Mathematical models can be very helpful in the present scenario in choosing the right directions of applying different control strategies. Mathematical models have been used since decades for the better understanding of HIV/AIDS dynamics and for applying control measures. From the very beginning Hymen (1988) used many models from simple to complex and addressed various risk factors. agelkerke (22) modelled the impact of various intervention methods and concluded that it is possible to have drugresistant HIV after some years. Srinivasa (23) presented a simple AIDS epidemic model and analysed it for India. Johnson (24) gave a note on basics of HIV/AIDS modelling. Srinivasa and Kakehashi (24) applied convolution to understand the prevalence. Hsieh and Chen (24) highly structured the population into various classes on the basis of their activity type, gender etc. Srinivasa et al. (29) studied impact of HIV/AIDS epidemic in India. aresh et al. (29) divided the infected class into two parts on the basis of their activity levels. Hove-Musekwa and yabadza (29) included screening while modelling.

Jyoti Gupta 57 In this section 3.2, a mathematical model is proposed for predicting the epidemiological status of disease in long term and to find the right parameters to be worked upon for controlling the spread of the disease. 3.2.1 Mathematical Model Here, we divide the entire population into six compartments. At time t, there are S susceptible or HIV negatives, H n (L n ) HIV positive high (low) sexual activity group who are not aware that they are infected, H a (L a ) HIV positive high (low) sexual activity group who are aware that they are infected, A is population with AIDS. The population dynamics among these compartments is shown in Fig.3.2.1 H n S L n H a A L n Fig 3.2.1 Population Dynamics among various compartments The state variables and all the parameters are listed below: S: umber of susceptible (i.e. HIV negative) H n (L n ): umber of unaware HIV positive persons with high (low) sexual activity

Mathematical Modelling Control (In Indian Context) 58 H a (L a ): umber of highly (less) active persons who are now aware that they are HIV positive A: umber of people who has developed AIDS B: Recruitment rate in susceptible class μ: atural death rate δ: Disease induced death rate β 1 (β 2 /β 3 ): Probability of transmission of infection from an infective H n (L n /H a ) to a susceptible per contact per unit time c 1 (c 2 /c 3 ): umber of contacts made by a person from H n (L n /H a ) class σ 1 (σ 2 ): Rate of unaware infective H n (L n ) to become aware by screening. α: Rate with which all type of infective develops AIDS. Here, we have assumed that when a person from L n class moves to L a class, it does not spread infection any more. Taking all the assumptions into account, the model takes the form as follows: SH SL SH dt ds n n a B β c 1 1 β c β c μs 2 2 3 3 dhn SHn SHa β1c 1 β3c3 μ α σ1h dt dl dt n dh dt dl dt a a SLn β2c2 μ α σ2 L 2 1 n n σ H μ α H σ L μ α L da α H n L n H a L a μ δ A dt a a n n (3.2.1.1)

Jyoti Gupta 59 with S Hn Ln Ha La A Adding all the above equations, we have d S H n L n H a L a A B μ S H L H L A n n a a δa dt or S H L H L A B μs H L H L A n n a a n n a a or B μ B lim sup n n a a t μ Then S H L H L A So, the feasible region for the system is B S, Hn, Ln, Ha, La, A : S Hn Ln Ha La A, Λ μ S, Hn, Ln, Ha, La, A Let, n, n, a, a, E S H L H L A be the equilibrium point of the system (3.2.1.1) of equation. ow, we calculate basic reproduction number, R, as follows: Since, the recruitment term B can never be zero and population cannot vanish, therefore, there is no trivial equilibrium point like S, Hn, Ln, Ha, La, A,,,,,. So, lete S, Hn, Ln, Ha, La, A S,,,,,. Then system (3.2.1.1) of equations at this point gives B S. μ So, we can see that there is a disease free equilibrium at B E S, Hn, Ln, Ha, La, A,,,,, μ

Mathematical Modelling Control (In Indian Context) 6. Let X H, L, H, L, A, S n n a a T dx Therefore, X F( X) - V( X) dt where F( X ) = SH β c β c SLn βc 2 2 n 1 1 3 3 SH a and V( X ) = μ α σ1 Hn μ α σ2 Ln σ1h n μ α Ha σ2ln μ α La α Hn Ln Ha La μ δ A SH SL SH n n a B β1c 1 β2c2 β3c3 μs The derivatives DF( E) and DV( E) at disease free esquilibrium point E, are partitioned as F DF( E) = and DV( E) = V J J 1 2 where F and V are 3 3 matrices given by

Jyoti Gupta 61 F = β1c 1B β3c3b μ μ β2c2b μ μ α σ1 V = μ α σ2 - σ1 μ α FV β1c 1B β3c3b β3c3b μ μ α σ1 μ μ α σ1 μ α μ μ α β c B = μ μ α σ2 1 2 2 Therefore, basic reproduction number R 1 ρ FV = spectral radius of FV 1 1 1 3 3 1 2 2 μ α σ μ α μ α σ β c μ α β c σ βc R max, 1 2 R μ α σ μ α β c μ α β c σ 1 1 3 3 1 1 (3.2.1.2)

Mathematical Modelling Control (In Indian Context) 62 3.2.2 Stability of Disease Free Equilibrium The disease free equilibrium is stable if all the eigen values of the Jacobian matrix of the system (3.2.1.1) have negative real parts. For this, the Jacobian B of the system (3.2.1.1) at E,,,,, μ takes the form J β1c 1B β2c2b β3c3b μ μ μ μ β1c 1B β3c3b μ α σ1 μ μ β2c2b μ α σ2 μ σ1 μ α σ2 μ α α α α α μ δ Here, trace( J) 6μ 4α σ σ δ 1 1 2 2 1 2. β c β c B μ Clearly, trace (J) <. ow, for det (J) to be >, we proceed as follows: Expanding det (J), we get β1c 1B μ α σ μ 2 2 det( J) μ μ δ μ α μ α σ2 1 1 β c B μ β3c3b μ σ μ α

Jyoti Gupta 63 β c B det( J) μ μ δ μ α μ α σ μ 2 2 β1c 1B β c Bσ μ α μ α σ1 μ μ 2 3 3 1 For det (J) >, after some algebraic simplification, we get the relation as 2 β c μ α σ μ α β c σ μ α σ μ α μ α σ 1 1 1 3 3 1 1 1 μ α μ α σ β c μ α β c σ 1 1 3 3 1 1 1 R 1 (by (3.2.1.2)) This implies that the disease free equilibrium is locally asymptotically stable if R 1 otherwise unstable. 3.2.3 Endemic Equilibrium * * * * * * Let the endemic equilibrium point be E S, H, L, H, L, A,,,,, with e n n a a * μ α δ B μr μ α δ αδ R 1 (3.2.3.1) H * n 2 μ α R 1 1 R μ α σ μ α σ 1 2 (3.2.3.2) μ δμ α BR 1 μ α σ μr μ α δ αδ R 1 L μ α σ H * * n 1 n 2 H * 1 * a n σ μ α H (3.2.3.3) (3.2.3.4)

Mathematical Modelling Control (In Indian Context) 64 L * 2 * a n σ μ α L (3.2.3.5) A * BαR 1 1 μr μ α δ αδ R (3.2.3.6) with S * B. (3.2.3.7) μr 3.2.3.1 Local Stability of Endemic Equilibrium * * * * * * For this, the Jacobian of the system (3.2.1.1) at e, n, n, a, a, the form E S H L H L A takes

Jyoti Gupta 65 1 β c S β c S β3c3s 1 β1c 1S β3c3s β1c 1H n β3c3ha μ α σ1 β c S σ1 μ α σ2 μ α α α α α μ δ 1 1 2 2 β1c 1H n β2c2ln β3c3ha μ J 2 2 E μ α σ2

Mathematical Modelling Control (In Indian Context) 66 where all the state variables are at endemic equilibrium point given by relations (3.2.3.1) (3.2.3.7). Here, 1 S trace( JE ) β1c 1Hn β2c2ln β3c3ha β1c 1 β2c2 6μ 4α σ1 σ2 δ Clearly, trace (J E ) <. ow for det (J E ) to be >, we proceed as follows: Expanding det (J E ) and placing it to be greater than zero, we get * * * * β2c2l n RS RS Hn 1 * * * μ α σ1 R * 1 μ μ Substituting values of state variables at endemic equilibrium point and then after some algebraic simplifications we reach the relation Rα μ α σ2 μ δ R R 1 1 μ α σ μ α δ β c 2 2 2 (3.2.3.1.1) * * * * * * This implies that the endemic equilibrium point e, n, n, a, a, locally asymptotically stable if the relation (3.2.3.1.1) holds. E S H L H L A is 3.2.3.2 Global Stability of Endemic Equilibrium Consider the Lyapunov function 1 2 * * * H n * * H a V m1h n Hn Hn log m * 2Ha Ha Ha log * 2 Hn Ha m m m 2 2 2 * 4 * * Ln Ln La La A A 2 2 2 3 5

Jyoti Gupta 67 * d H H dh Ha H a dha * dl V m1 m2 m3ln Ln dt H dt H dt dt * * n n n n n a dl m4 L L m5 A A dt * a * a a da dt Using (3.2.1.1) and simplifying we get * 2 * * * * 2 a a 3 V μ δ A A m μ α L L m μ δ A A * Hn H n * * * * * m4β3c3ha Hn Hn Ln Ln Ha Ha La La H n 5 * H a Ha m μ α Hence, V is negative. Also note that V if and only if H a 2, H H, L L, H H, L L and A A. Therefore the * * * * * * n n n n a a a a largest compact invariant set in S H L H L A e,,,,, Λ : V is the n n a a singleton sete, where E e is endemic equilibrium. Therefore, LaSalle s invariant principle implies that E e is globally asymptotically stable in Lyapunov sense. 3.2.4 Sensitivity Analysis ow we evaluate the sensitivity indices of R to all the different parameters it depends on. These indices guide us to find the right parameter(s) responsible for disease spread and need to be taken care of. We use following parameter values: β 1 = β 2 = β 3 =.47, σ 1 = σ 2 =.15, μ =.2, δ = 1, α =.1, c 1 = 3, c 2 =.4, c 3 =.6. Most of the values are taken from previous works done in Indian context. Others are derived on the basis

Mathematical Modelling Control (In Indian Context) 68 of statistics released by some government organisations. The analysis results are given in table 3.2.1. Table 3.2.1 Sensitivity Indices of R Parameter Sign Value β 1 +.9756 β 3 +.244 c 1 +.9756 c 3 +.244 µ -.1522 α -.7611 σ 1 - -.867 These results show that we need to target high activity group. The effective index of rate, with which they spread infection, turns out to be β1c1.9518. Also, the rate of progression from infectious class to AIDS class is an important parameter having negative impact on disease spread as generally humans, after developing AIDS, rarely spread infection any more. 3.2.5 umerical Simulation Here, in order to find future trends of the disease we simulated the data with a sample population size of = 23. The results obtained are shown in Fig. 3.2.2.

Population Jyoti Gupta 69 Fig. 3.2.2 Population Dynamics in Different compartments These simulation results in fig 3.2.2 show that people in high activity group are more liable for catching HIV infection as well as more responsible for spreading it. 45 4 35 3 25 2 15 1 5 1 4 7 1 13 16 19 22 25 28 31 34 37 4 43 46 49 52 Years Susceptible Hn Ha Fig. 3.2.3 Population trend in High Activity Group ot aware verses Aware

Population Mathematical Modelling Control (In Indian Context) 7 Fig 3.2.3 shows that unaware population in high activity group contracts the infection very fast. When they become aware, they seem to be less prone for catching infection. 35 3 25 2 15 1 5 Susceptible Ln La 1 4 7 1 13 16 19 22 25 28 31 34 37 4 43 46 49 52 Years Fig. 3.2.4 Population trend in Low Activity Group ot aware verses Aware Fig 3.2.4 suggests that low activity group is not a matter of big concern. So, this group does not need as prompt response as compared to high activity group. 3.2.6 Results and Discussion In this study, we developed a mathematical model for HIV/AIDS in order to understand the population dynamics of the disease. We divided the infected class into four groups depending on their activity levels and screening status. For starting the qualitative analysis of the model, a relation for basic reproduction number R is established. Then, the existence of steady states and their stabilities are analysed. The analysis shows that the disease free equilibrium is locally asymptotically stable if R 1. For local stability of

Jyoti Gupta 71 endemic equilibrium we established a relation that should hold. For global stability of it, we defined a Lyapunov function and established the result accordingly. Sensitivity analysis is done using data for India. The results of sensitivity analysis show that we need to target the high activity group. If we make them aware about the disease and educate them for using some safety measures then it may reduce the spread of disease. The results of numerical simulation confirm the results of sensitivity analysis. The graphs predict the future dynamics of disease prevalence and guide us that we need to control high activity group s risky behaviour. 3.3 Tuberculosis Tuberculosis (TB) is a big public health problem in most developing countries including India. As per WHO TB-report 211, India accounts for one fourth (approximately 26%) of the global TB cases. Each year nearly 2,, people in India develop TB, of which around 8,7, are infectious cases. It is estimated that every day approximately 1, Indians die due to TB. Tuberculosis or TB (short form for Tubercle Bacillus) is an infectious bacterial disease caused by Mycobacterium Tuberculosis. TB is caused by various strains of mycobacteria most commonly by mycobacterium tuberculosis. It commonly affects the lungs (pulmonary) but can also affect other parts of the body (extra-pulmonary). It is transmitted from person to person through the air via droplets from the throat and lungs of people with active TB. These droplets come out when they cough or sneeze. TB generally develops slowly. In cases, when the bacteria infect the body and cause symptoms, it is called active TB otherwise it is called latent TB.

Mathematical Modelling Control (In Indian Context) 72 Depending on the type of TB (i.e. pulmonary and extra-pulmonary), the symptoms are different. The symptoms of active pulmonary TB are coughing, sometimes with sputum or blood, breathlessness, lack of appetite, weight loss, fever and night sweats. Sometimes, TB occurs outside lungs, which is known as extra-pulmonary TB. It is more common in people with weak immune system particularly with an HIV infection. An extra-pulmonary infection site may be lymph nodes, bones and joints, digestive system, bladder and reproduction system, nervous system and any other part of the body. Out of these all, lymph nodes are the most common site for extra-pulmonary TB. Once infected or infectious person may be cured with medication or may die from TB. Recovered individuals may relapse to disease or be re-infected. Many mathematical models are formulated and analysed using several techniques and mathematical tools. Most of these models are SEIR type. Castillo-Chavez and Feng (1997) modelled tuberculosis taking two-strain TB into account. Aparicio et al. (2) discussed the re-emergence of tuberculosis using SEIR model. Jung et at. (22) also discussed models for two-strain tuberculosis. Coljin et al. (26) analysed, reviewed and compared existing models showing that even small changes lead to some different types of useful interpretations. They presented their own model also using delay differential equations. Joshi and Bates (28) conducted a study to compare pulmonary and extra-pulmonary tuberculosis cases. Aparicio and

Jyoti Gupta 73 Castillo-Chavez (29) presented a rather complex models using homogeneous mixing and heterogeneous mixing epidemic models. Egbetada and Ibrahim (212) derived stability conditions for tuberculosis when it is in epidemic state. Although these models of tuberculosis look similar in structure yet there are important (may not be apparent), differences in the way the population is classified and the transmission process is represented, leading to some other important conclusions. In this section 3.3, we formulate a model with separating pulmonary and extra-pulmonary classes and with the help of sensitivity analysis we try to trace the parameters which are most responsible for the spread of disease. 3.3.1 Mathematical Model Here, we formulate a model based on SEIR model of infectious disease epidemiology. The entire population is divided into five compartments depending on disease status. These compartments are referred to as state variables. At time t, there are S susceptible, E exposed, I infectious (with pulmonary TB) and X with extra-pulmonary TB, and T treated. The population move from one compartment to the other in as given in fig 3.3.1.

Mathematical Modelling Control (In Indian Context) 74 Fig 3.3.1 Flow of Population from one compartment to the other The state variables and parameters used are listed below: S: umber of susceptibles E: umber of exposed (latently infected) I: umber of infectious (pulmonary TB) X: umber of individuals with extra-pulmonary TB (non- infectious) R: umber of treated B: Recruitment as susceptible per unit time ν 1 : Rate of progression of individuals from the exposed class to the infectious (pulmonary) class ν 2 : Rate of progression of individuals from the exposed class to the non-infectious (extra-pulmonary) class

Jyoti Gupta 75 γ 1 : treatment rate of latent individuals γ 2 : treatment rate of infectious (pulmonary) individuals γ 3 : treatment rate of non-infectious (extra-pulmonary) individuals δ 1 : Disease-induced death rate (pulmonary) δ 2 : Disease-induced death rate (extra-pulmonary) μ: atural death rate β: Probability that a susceptible becomes infected per contact per unit time β': Probability that a treated becomes infected per contact per unit time c: Contact rate The model takes the form as follows: ds dt SI B βc μs de SI TI dt di dt ' βc β c γ1 μ ν1 ν2 E ν E δ γ μ I 1 1 2 dx ν 2 E δ 2 γ 3 μ X dt dt dt TI ' γ1e γ2i γ3 X β c μt (3.3.1.1) Adding above equations, we have d S E I X T B μ S E I X T δ I δ X 1 2 dt

Mathematical Modelling Control (In Indian Context) 76 S E I X T B μs E I X T B μ Then lim sups E I X T t So, the feasible region for the system is B Λ S E I X T : S E I X T, S, E, I μ ow the basic reproduction number R will be calculated using the approach given in Drissche and Watmough (22). The basic reproduction number, R, is defined as the expected number of secondary cases produced by a single (typical) infection in a completely susceptible population. Let QS, E, I, X, T be the equilibrium point of the system (3.3.1.1). Since the recruitment term can never be zero and population can never extinct, therefore, there is no trivial equilibrium point like,,,,,,,, Q S E I X T. It is easy to see that the system has a disease free equilibrium at B Q S, E, I, X, T,,,,. μ Let Q E, I, X, T, S T dq Therefore, Q F( Q) - V( Q) dt

Jyoti Gupta 77 where F( Q) gives the rate of appearance of new infections in a compartment and V( Q) gives the transfer of individuals. And F( Q) SI βc TI βc ( γ1 ν1 ν2 μ) E ν1e ( γ2 μ δ1) I ν 2E ( γ3 μ δ2 ) X TI V( Q) = γ 1E γ2i γ3 X μt β c SI B βc μs Since Qis the disease free equilibrium therefore the derivatives DF( Q ) and DV( Q ) are partitioned as F DF( Q ) = and DV( Q ) = V J J 1 2 where F and V are 3 X 3 matrices given by

Mathematical Modelling Control (In Indian Context) 78 F = βcb μ and γ1 ν1 ν2 μ V = ν1 δ1 γ2 μ ν2 δ2 γ3 μ Therefore, FV 1 = βcbν1 βcb μ δ1 γ2 μ μ δ1 γ2 μ Therefore, basic reproduction number R 1 ρ FV = spectral radius of FV 1 R Bβcν1 μ δ γ μ γ ν ν μ 1 2 1 1 2 (3.3.1.2) 3.3.2 Stability of Disease Free Equilibrium The disease free equilibrium is stable if all the eigen values of the Jacobian matrix of the given system have negative real parts. In order to check this, the Jacobian of the system (3.3.1.1) at B Q S, E, I, X, T,,,, takes the form μ

Jyoti Gupta 79 Bβc μ μ Bβc γ1 ν1 ν2 μ μ J ν1 δ1 γ2 μ ν2 δ2 γ3 μ γ1 γ2 γ3 μ Here, trace (J) = δ δ γ γ γ ν ν μ 1 2 1 2 3 1 2 5 Clearly, trace (J) <. For det (J) to be >, we should have μ βcν1 B γ ν ν μ δ γ μ 1 1 2 2 2 (on solving determinant of J) or Bβcν 1 μ γ1 ν1 ν2 μδ2 γ2 μ 1 i.e. R 1. This implies that the disease free equilibrium is locally asymptotically stable if R 1 otherwise unstable. 3.3.3 Stability of Endemic Equilibrium Let the endemic equilibrium point be *, *, *, *, *,,,, Q S E I X T. e Here, the Jacobian takes the form

Mathematical Modelling Control (In Indian Context) 8 J Q e Iβc Sβc μ Iβc Sβc Tβc Iβc γ1 ν1 ν2 μ ν1 δ1 γ2 μ ν 2 δ2 γ3 μ Tβc Iβc γ1 γ 2 γ3 μ with S * * * * B I S βc I T βc 1, E *, μ I βc γ1 ν1 ν2 μ * * * * * * * * E ν * E ν * E γ I γ X γ I, X, T δ 1 γ2 μ δ2 γ3 μ Iβc μ 1 2 1 2 2 Here, Iβc Iβc trace( JQ e ) 2μ γ1 ν1 ν2 μ δ1 γ2 μ δ2 γ3 μ Clearly, trace (J(Q e )) <. For det (J(Q e )) >, after a long cumbersome calculation and assuming β β, we finally get that R 1. Thus, the endemic equilibrium is stable if R 1. 3.3.4 Sensitivity Analysis Here, we evaluate the sensitivity indices of R to all the different parameters it depends on. These indices tell us how crucial each parameter is to disease

Jyoti Gupta 81 spread and guide us to find the right parameter to be taken care of. Table 3.3.1 Values of Parameters for TB (In India) Model Parameter Value μ.8 δ 1.24 β 2. γ 1 2. γ 2 1.5 ν 1.57 ν 2.9 c 1 B 8 1 Table 3.3.2 Sensitivity Indices of R to the Parameters for the TB Model Parameter Sign Value μ - 1.85 δ 1 -.1373 β + 1 γ 1 -.9888 γ 2 -.8581 ν 1 +.9972 ν 2 -.45 c + 1 B + 1-1 Results in table 3.3.2 show that the infection rate is most responsible for the disease spread. It depends on probability of transmission of infection and contact rate. We do not have any control on probability but we can control the number of contacts made by an infectious individual. Rate of progression from infectious class to treated class has a positive impact on the disease scenario.

Mathematical Modelling Control (In Indian Context) 82 3.3.5 umerical simulation We simulated the tuberculosis model in the absence of treatment and then in the presence of treatment and compared them. The simulation results are shown in fig. 3.3.2 to fig. 3.3.4. Fig 3.3.2 Population dynamics of Tuberculosis in absence of treatment Fig 3.3.2 shows that the the infection spreads very fast among the population. The important thing to notice is that along with pulmonary cases, extrapulmonary cases are also increasing that makes the scenario worse.

Jyoti Gupta 83 Fig 3.3.3 Population dynamics in susceptible and exposed compartments in presence of treatment From fig 3.3.3 we see that susceptible moves to exposed class slowly as comapred to those shown in fig 3.3.2. Fig 3.3.4 Population dynamics in pulmonary (infectious), extrapulmonary and treated compartments in presence of treatment

Mathematical Modelling Control (In Indian Context) 84 Fig 3.3.4 shows that treatment has a positive impact on extrapulmonary class. Also, there is a significant difference in the number of pulmonary tuberculosis cases. 3.3.6 Result And Discussion In this study, we formulated a mathematical model for understanding the population dynamics of tuberculosis considering the pulmonary and extrapulmonary cases separately. For starting the qualitative analysis of the model, a relation for basic reproduction number R was established. Then the existence of steady states and their stabilities were analysed. The analysis showed that the disease free equilibrium was locally asymptotically stable if R 1. Also the endemic equilibrium was found to be globally asymptotically stable if R 1 provided it exists. Sensitivity analysis of the model was carried out using the data for India. The results of sensitivity analysis shows that overcrowding unsanitary conditions were the major causes in for the prevalence of this disease in India as it increases the contact rate and provides the bacteria the environment to survive longer. umerical simulation is carried out for the model in the absence of treatment and then in the presence of treatment. The results show that the treatment is helpful in controlling the disease spread. The results also tell us that we need to take special care of cases with pulmonary infection in terms of providing efficient treatment along with isolation for around two weeks sothat they recover fast and will not pass infection to others during their infectious stage.

Jyoti Gupta 85 3.4 HIV/AIDS with Tuberculosis Tuberculosis is the most common HIV-related opportunistic infection in the world, and caring for patients with both diseases together is a major public health challenge. India has about 1.8 million new cases of tuberculosis annually, accounting for a fifth of new cases in the world - a greater number than in any other country. Patients with latent Mycobacterium tuberculosis infection are at higher risk for progression into active TB if they are coinfected with HIV. In recent decades, the dramatic spread of the HIV epidemic in sub-saharan Africa has resulted in notification rates of TB increasing up to 1 - times. The incidence of TB is also increasing in other high HIV prevalence countries where the population with HIV infection and TB overlap. This is the underlying factor that suggests that TB control will not make much ahead way in HIV prevalent settings, unless HIV control is also achieved. The related spread of two or more infections has always been a cause of concern for human beings. HIV-TB co-infection is the largest cause among them. Tuberculosis (TB) is the most common opportunistic disease affecting HIV positive people and the leading cause of death in patients with AIDS. As per the World Health Organisation (WHO) estimation, one third of the world s population is infected with Mycobacterium Tuberculosis. All over the world, approximately 15% of TB patients have HIV co-infection. HIV patients are at higher risk for catching primary TB infection as well as reactivation of latent TB infection. HIV infection primarily affects those components of host immune system which are responsible for cell mediated immunity. These

Mathematical Modelling Control (In Indian Context) 86 cells help us fight against various infections. Once they are destroyed our body s resistance to fight infections goes down. Thus if a person with latent TB infection catches HIV infection, then the host s immunity reduces resulting in active tuberculosis. Moreover, the infection is poorly contained following reactivation, resulting in widespread dissemination causing extrapulmonary disease. The interaction between HIV and tuberculosis in patients who are co-infected is bidirectional. HIV infection accelerates the activation of tuberculosis and tuberculosis accelerates the HIV infection developing into AIDS. Also, the relative risk of death and development of other opportunistic infections is higher in HIV-TB co-infected patients as compared to those having only one disease out of two. Research work is already in progress for understanding the dynamics of this co-epidemic. Krischner (1999) modelled the CD4+ cell counts and viral load in the case of two infections together. Morris (SIAM) formed basic models and derived steady states for them. Escombe et al. (28) analysed how infectiousness of a co-infected person differs from the one having only TB. Sharomi et al. (28) formulated the mathematical model and also included the treatment factor in it. Bauer et al. (28) examined that how the latent TB patient moves to active TB class if it catches HIV-1 infection too. Pawlowski et al. (212) reviewed the available literature in order to highlight immunological events responsible for developing the one infection in the presence of the other. Shah and Gupta (213) developed a model for tuberculosis to explain the dynamics of disease taking pulmonary and extrapulmonary TB separately. Shah and Gupta (213) also developed a model to predict future trends of HIV/AIDS.

Jyoti Gupta 87 In this section 3.4, a model is formulated for HIV-TB co-infection using system of differential equations in order to understand the dynamics of disease spread. The model is analysed for all the parameters responsible for the disease spread in order to find the most sensitive parameters out of all. 3.4.1 Mathematical Model The entire population is divided into twelve compartments, namely (1) Susceptible (S) (i.e. no disease), (2) Latent TB o HIV (E 1 ), (3) Latent TB with HIV(E 2 ), (4) Active TB o HIV (I 1 ), (5) Active TB with HIV (I 2 ), (6) Only HIV (H), (7) TB treated o HIV (T 1 ), (8) TB treated with HIV (T 2 ), (9) AIDS o TB (A 1 ), (1) AIDS with Latent TB (A 2 ), (11) AIDS with Active TB (A 3 ), (12) AIDS Treated TB (A 4 ). Here, it is assumed that a person, when moves any of AIDS class, does not spread disease any more. The population dynamics among these compartments is shown in Fig 3.4.1.

Mathematical Modelling Control (In Indian Context) 88 E 2 LTB+HIV S E 1 LTB H HIV I 1 ATB I 2 ATB+HIV A 3 ATB+AID A 1 AIDS T 1 TTB A 4 TTB+AID A 2 T 2 LTB+AID TTB+HIV Fig 3.4.1 Flow of Population in Various Compartments The state variables and the parameters used in model formulation are as follows: S: umber of susceptible (i.e. no disease) E 1 : umber of persons with latent TB infection and no HIV

Jyoti Gupta 89 E 2 : umber of persons with latent TB infection and HIV positive H : umber of persons with HIV infection and no TB I 1 : umber of persons with active TB infection and no HIV I 2 : umber of persons with active TB infection and HIV positive T 1 : umber of persons with treated TB and no HIV T 2 : umber of persons with treated TB and HIV positive A 1 : umber of persons with AIDS and no TB A 2 : umber of persons with AIDS with latent TB infection A 3 : umber of persons with AIDS with active TB infection A 4 : umber of persons with AIDS with treated TB B: Recruitment rate in susceptible class μ: atural death rate δ A : AIDS induced death rate δ T : Tuberculosis induced death rate β 1 : Probability of transmission of TB infection from an infective to a susceptible per contact per unit time β 2 : Probability of transmission of HIV infection from an infective to a susceptible per contact per unit time

Mathematical Modelling Control (In Indian Context) 9 c 1 : umber of contacts made by a person from active TB class c 2 : umber of contacts made by a person from latent TB class c 3 : umber of contacts made by a person having only HIV and no TB c 4 : umber of contacts made by a person having HIV and TB both α: Rate with which all type of infective develop AIDS. ν 1 : Rate of progression of individuals from the latent TB class to the active class who have only LTB and no HIV. ν 2 : Rate of progression of individuals from the LTB class to the active TB class who have LTB and HIV/AIDS both. γ 1 : treatment rate of latent TB individuals γ 2 : treatment rate of infectious (active TB) individuals. The model equations are given below: ds β1s β2s B c I 1 1 c I 4 2 c A 4 3 c H c E c T 3 4 2 4 2 μs dt de β S βt dt c I c I c A c I c I c A 1 1 1 1 1 1 4 2 4 3 1 1 4 2 4 3 βe 2 1 c 3 H c 4 T 2 c 4 E 2 c 4 I 2 μ ν 1 γ 1 E 1 dh β2s β1h c3h c4e2 c4t2 c1i 1 c4i2 c4a3 μ α1h dt de β H c I c I c A β E c H c T c E c I μ α ν γ E dt 2 1 2 1 1 1 4 2 4 3 3 4 2 4 2 4 2 2 2 1 2

Jyoti Gupta 91 di dt di dt βi ν E c H c T c E c I μ δ γ I 1 2 1 1 1 3 4 2 4 2 4 2 T 2 1 βi ν E c H c T c E c I μ δ α γ I 2 2 1 2 2 3 4 2 4 2 4 2 T 2 2 2 dt γ E γ I βt c I c I c A β T c H c E c T μt dt dt dt da dt da dt 1 1 1 2 1 1 1 2 1 1 1 4 2 4 3 3 4 2 4 2 1 βt γ E γ I c H c E c T μ α T 2 2 1 1 2 2 2 3 4 2 4 2 2 2 βa α H c I c I c A μ δ A 1 1 1 1 1 1 4 2 4 3 A 1 βa α E c I c I c A μ δ ν γ A 2 1 1 2 2 1 1 4 2 4 3 A 2 1 2 da dt 3 α I ν A μ δ δ γ A 2 2 2 2 A T 2 3 da dt 4 γ A γ A α T μ δ A 1 2 2 3 2 2 A 4 We name the above set of twelve equations as system (3.4.1.1). With S E1 E2 H I1 I2 T1 T2 A1 A2 A3 A4 d B μ δ A A A A A δ I I A dt 1 2 3 4 T 1 2 3 d B B μ limt sup dt μ

Mathematical Modelling Control (In Indian Context) 92 So, the feasible region for the system is S, E1, E2, H, I1, I2, T1, T2, A1, A2, A3, A4 : B Λ S E1 E2 H I1 I2 T1 T2 A1 A2 A3 A4, S, E1, μ I1, H, E2, I2, T1, T2, A1, A2, A3, A4 Let, 1, 2,, 1, 2, 1, 2, 1, 2, 3, 4 E S E E H I I T T A A A A be the equilibrium point of the model given above. Since, the recruitment term B can never be zero and population cannot vanish, therefore there is no trivial equilibrium point like,,,,,,,,,,, 1 2 1 2 1 2 1 2 3 4,,,,,,,,,,, E S E E H I I T T A A A A. So, let,,,,,,,,,,, 1 2 1 2 1 2 1 2 3 4 S,,,,,,,,,,, E S E E H I I T T A A A A. Then system of equations at this point gives B S μ So, we can see that there is a disease free equilibrium at E B S, E, E, H, I, I, T, T, A, A, A, A,,,,,,,,,,, μ 1 2 1 2 1 2 1 2 3 4. Let X E, E, H, I, I, T, T, A, A, A, A, S 1 2 1 2 1 2 1 2 3 4 Therefore, dx X = F( X) - V( X) dt where F( X) and V( X) are column matrices given by

Jyoti Gupta 93 F( X ) β H β S c I c I c A c I c I c A 1 1 1 1 1 4 2 4 3 1 1 4 2 4 3 βs 2 c 3 H c 4 E 2 c 4 T 2 c I c I c A c H c T c E c I 1 2 1 1 1 4 2 4 3 3 4 2 4 2 4 2 βi 21 β E 3 4 2 4 2 4 2 βt 2 1 c 3 H c 4 E 2 c 4 T 2 βa 1 1 c 1 I 1 c 4 I 2 c 4 A 3 β T c H c T c E c I and

Mathematical Modelling Control (In Indian Context) 94 V( X ) = βe 2 1 c 3 H c 4 T 2 c 4 E 2 c 4 I 2 μ ν 1 γ 1 E 1 βh 1 c 1 I 1 c 4 I 2 c 4 A 3 μ α 1 H μ α2 ν2 γ1e 2 βi 21 ν1e 1 c3h c4t2 c4e2 c4i2 + μ δt γ 2 I1 ν 2E2 μ δt α2 γ2 I 2 β1t 1 β2t 1 γ1e 1 γ2i1 c1i 1 c4i2 c4a3 c3h c4e2 c4t2 μt1 γ1e 2 γ2i2 μ α2 T βa 1 1 α1h c1i 1 c4i2 c4a3 μ δa A1 α2e2 μ δa ν2 γ1 A 2 α2i2 ν2a2 μ δa δt γ 2 A 3 γ1a2 γ2a3 α2t2 μ δa A4 β1s β2s B c1i 1 c4i2 c4a3 c3h c4e2 c4t2 μs The derivatives DF( E ) and DV( E ) at disease free equilibrium point E, are partitioned as D F F( E ) = and D E V( ) = V J J 1 2 where F and V are 1 1 matrices given by

Jyoti Gupta 95 F β1c 1B β1c 4B β1c 4B μ μ μ β2c3b β2c4b β2c4b μ μ μ

Mathematical Modelling Control (In Indian Context) 96 V μ ν1γ1 μ α1 μ α2 ν2 γ1 ν1 μ δt γ2 ν 2 μ δt α2 γ2 γ1 γ 2 μ γ 2 γ 2 μ α2 α1 μ δa α2 μ δa ν2 γ1 α ν μ δ δ γ 2 2 A T 2

Jyoti Gupta 97 1 Therefore, basic reproduction number R ρfv = spectral radius of FV 1 β c ν B β c B 1 1 1 2 3 R max, μ δt γ2 μ ν1 γ1 μ μ α1 μ R β1c 1ν 1B μ δ γ μ ν γ μ T 2 1 1 (3.4.1.2)

Mathematical Modelling Control (In Indian Context) 98 3.4.2 Stability of Disease Free Equilibrium The Jacobian of model equations at the disease free equilibrium point can be written as β2c3b β2c4b β1c 1B β1c 4B β2c4b β1c 4B μ μ μ μ μ μ μ β1c 1B β1c 4B β1c 4B μ ν1 γ1 μ μ μ β2c4b β2c4b μα1 μ μ μ α2 ν2 γ1 ν1 μ δt γ2 J ν2 μ δt α2 γ2 γ1 γ2 μ γ1 γ2 μ α2 α1 μ δa α2 μ δa ν2 γ1 α2 ν2 μ δa δt γ2 α2 γ1 γ2 μ δ A

Jyoti Gupta 99 The disease free equilibrium is stable if all the eigen values of the Jacobian matrix of the system under consideration have negative real parts. Here, trace (J) is clearly negative. ow for det (J), on solving we reach the relation 2 2 det( J) μ μ δa μ α1 μ α2 μ α2 ν2 γ1 μ α2 δt γ2 β1c 1ν 1B μ δa ν2 γ1μ δa δt γ2 μ δt γ2 μ ν1 γ1 μ For det(j) >, we have 2 2 μ μ δa μ α1 μ α2 μ α2 ν2 γ1 μ α2 δt γ2 β c ν B μ δ ν γ μ δ δ γ μ δ γ μ ν γ μ 1 1 1 A 2 1 A T 2 T 2 1 1 β c ν B μ δt γ μ ν γ μ or 1 1 1 2 1 1 β c ν B 1 1 1 R μ δt γ2 μ ν1 γ1 μ 1 1 (using (3.4.1.2)) This shows that the disease free equilibrium is locally asymptotically stable if R 1 otherwise unstable. 3.4.3 Sensitivity Analysis Sensitivity indices of R to all the different parameters tell us that how crucial each parameter is to the disease spread. This helps us choose the right parameter(s) responsible for making the scenario endemic.

Mathematical Modelling Control (In Indian Context) 1 The values of different parameters used in model formulation are given in Table 3.4.1. Table 3.4.1 Values of Different Parameters used in Model Parameter Value B 2 µ.2 δ T.24 δ A 1. α 1.1 α 2.5 β 1 2. β 2.47 γ 1 2. γ 2 1.5 ν 1.9 ν 2.19 c 1 1 c 2 1 c 3 2. c 4.5 We calculate the sensitivity indices for those parameters on which the value of basic reproduction depends. The results are given in Table 3.4.2. Table 3.4.2 Sensitivity Indices of R to the Parameters Parameter Sign Value - 1. B + 1. µ - 1.85 δ T -.1373 c 1 + 1. ν 1 +.9955 γ 1 -.9916 γ 2 -.8581 β 1 + 1.

Jyoti Gupta 11 These results show that the total population size, new recruitments as susceptible, contact rate with tuberculosis patients and probability of transmission of TB have a constant effect on the scenario. The most important parameters reflected here and need to be addressed are rate of progression of LTB patients to ATB class and their treatment rates. So, if we try to control tuberculosis cases and treat the active TB cases more promptly then we can reduce the level of intensity of this lethal co-epidemic. 3.4.4 umerical Simulation This technique helps us to foresee the future trends of an epidemic and thus help us be ready for future or take safety measures today for making a better future. Here, we take a sample population of 35. The results of simulation in different compartments are shown in following figures 3.4.2 3.4.5. Fig 3.4.2 Simulation of Susceptible, LTB, HIV and LTB with HIV Fig 3.4.2 shows that population with latent TB with HIV infection develop active tuberculosis faster than those having only latent TB.

Mathematical Modelling Control (In Indian Context) 12 Fig 3.4.3 Simulation of ATB, ATB with HIV, TTB and TTB with HIV From fig 3.4.3, we can see the impact of treatment on both compartments namely active TB (I 1 ) and active TB with HIV (I 2 ). Fig 3.4.4 Simulation of AIDS, AIDS with LTB, AIDS with ATB and AIDS with TTB

21 212 214 216 218 22 222 224 226 228 23 232 234 236 238 24 242 244 246 248 25 Jyoti Gupta 13 Figure 3.4.4 shows that people in AIDS with active TB (A 3 ) compartment moves out faster (by means of death) as compared to other three compartments. 25 2 15 1 5 S E1 H E2 I1 I2 T1 T2 A1 A2 A3 A4 Fig 3.4.5 Population dynamics in various compartments in next 4 years This shows the combined dynamics of all compartments. 3.4.5 Results and Discussion Here, we formulated a mathematical model using ordinary differential equations for HIV-TB co-infection scenario. We divided the entire population into twelve compartments. Then a relation for basic reproduction number R in established. Steady state conditions are derived which show that the disease free equilibrium is locally asymptotically stable only if R 1. Sensitivity analysis results tell us that we need to work more rigorously in order to control this co-epidemic. umerical simulation is done using

Mathematical Modelling Control (In Indian Context) 14 MATLAB. Figure 3.4.5 shows the trends of population in different compartments in next 4 years. 3.5 HIV/AIDS with Malaria HIV/AIDS infects and destroys CD4+T helper cells which are responsible for some specific immune responses. With time, as the infection progresses, it goes on damaging human immune system and makes infected human more vulnerable for catching other infections. Tuberculosis and malaria are two most opportunistic diseases affecting an HIV patient. A person infected with HIV develops symptomatic malaria more easily because of low CD4+ T-cell counts. HIV and malaria are the biggest global concern in the present scenario. As per WHO report, by the end of 211, there were 34.2 million people living with HIV infection and in the same year 1.7 million died of AIDS. Also, every year, more than one million people die due to malaria. So, when a person is co-infected with both the diseases, it becomes fatal for him. Also, antimalarial therapy does not respond as nicely in case of HIV Malaria co-infection as in a person with only malaria. Onyenekwe et al. (27) proved that malaria prevalence is tripled in the presence of HIV infection. Xiao et al. (21) modelled interaction of the immune system with HIV virus and malaria parasite in the same body. Shankarkumar et al. (211) presented that the two infections together accelerates malaria cases. Barley et al. (212) designed a model for HIV class and HIV with malaria class not taking AIDS under consideration. They analysed the model for Sub-Sahara Africa. Shah and Gupta (213) gave a

Jyoti Gupta 15 general model for vector-borne diseases and fitted that to malaria applying for both human population as well as mosquito population. Shah and Gupta (213) formulated a model for HIV/AIDS considering the population into different compartments depending on their activity levels. Subramanian (213) presented the effect of P. falciparum on the human immune system. In this section 3.5, a mathematical model is formulated using system of differential equations to understand the co dynamics of two diseases - HIV/AIDS and Malaria. The entire human population (age 16-45 years) is divided into six compartments and mosquito population into two. 3.5.1 Mathematical Model Here, we formulate the model with the objective to see the effect of malaria co-infection on a person infected with HIV. For this we consider a population group of age between 16-45 years. We divide this entire population into six groups. We also consider mosquito population and divide it into two groups. The six groups of human population we formed are: Susceptible (S h ) i.e. persons with no disease, Symptomatic malaria patients (M), Persons infected with HIV only (H), Persons having HIV infection as well as malaria (HM), Persons with AIDS only (A), and Persons having AIDS and malaria both (AM). The two mosquito classes are: Susceptible (S v ) and Infectious (I). The flow of population in these classes is demonstrated by fig 3.5.1 given below:

Mathematical Modelling Control (In Indian Context) 16 S M H HM A AM Fig 3.5.1 Flow of Population in Various Classes The variables and parameters used in model formulation are given below: S h : Susceptible human population i.e. humans of age between 16-45 years with no disease M : Clinical malaria population (i.e. only malaria and no HIV) H : People with HIV infection only HM : Population having HIV and malaria both

Jyoti Gupta 17 A : umber of persons having AIDS and no malaria AM : umber of persons having AIDS and malaria both S v : Susceptible mosquito population I : Infectious mosquito population B h : Recruitment rate in susceptible class for humans B v : Recruitment rate in susceptible class for mosquitoes β m : Probability of transmission of malaria infection from a infectious mosquito to a human per contact per unit time β v : Probability of transmission of malaria infection from an infectious human to a susceptible mosquito per contact per unit time β H : Probability of transmission of HIV infection from an infectious human to the uninfected one per contact per unit time ν : Rate of progression of humans from malaria class to again susceptible class c : umber of contacts made by an HIV infected person per unit time σ : Mosquito biting rate µ h : atural death rate for humans µ v : atural death rate for mosquitoes δ M : Death rate due to malaria

Mathematical Modelling Control (In Indian Context) 18 δ A : Death rate due to AIDS h : Total human population v : Total mosquito population Model equations for human population are as follows: ds H HM h I B νm β σs β cs μ S dt h m h H h h h h h dm I βmσsh νm μh δm I dt h dh H HM I β cs β σh α μ H dt dhm dt H h m 1 h h h I β σh α μ δ HM m 2 h M h da I α1h β σa μ δ A dt dam dt 2 m h A h I α HM β σa μ δ δ AM m h A M h (3.5.1.1) with h Sh M H HM A AM Equations for mosquito population are: ds dt v M HM AM B β σs μ S di β σs dt v v v v v h M HM AM v v v h μ I (3.5.1.2) with v Sv I Adding all the equations of the system (3.5.1.1), we get

Jyoti Gupta 19 d dt h B μ δ ( M HM AM) δ A AM h h h M A or h Bh μ h h Then lim sup t h Sh M H HM A AM μ h B So, the feasible region for the system is B Sh, M, H, HM, A, AM : Sh M H HM A AM, Λ μ Sh, M, H, HM, A, AM ow we use the approach given by Drissche and Watmough (22) for establishing a relation for basic reproduction number R. Let E S,,,,, h M H HM A AM be the equilibrium point of the system (3.5.1.1). It is easy to see that the system has a disease free equilibrium at B h E Sh, M, H, HM, A, AM,,,,,. μh Let E M H HM A AM S,,,,, h de E F( E) V( E) dt where

Mathematical Modelling Control (In Indian Context) 11 F( E) = I βmσsh h H HM βhcs h h I βmσh h I βmσa h and V( E) = νm μh δm I I βmσh α1 μhh h α2 μh δmhm I α1h βmσa μh δa A h α2hm μh δa δm AM I H HM Bh νm βmσsh βhcsh μhs h h h Since, we defined E to be disease free equilibrium, the derivatives DF( E) and DV( E) are partitioned as F DF( E) = and DV( E) = V J J 1 2 where F and V are 3 3 matrices given by F βhcbh βhcbh μhh μh h

Jyoti Gupta 111 and ν μh δm V α1 μh α2 μh δ M B β c B β c 1 h H h H FV = μhh α1 μh μhh α2 μh δ M Therefore, the relation for basic reproduction number for human population turns out to be R h Bhβc H μ α μ h h 1 h (3.5.1.3) So, using approach of Shah and Gupta (213), we can write a relation for basic reproduction number for mosquito population as R B βσ (3.5.1.4) v v v 2 μ v v Here, in order to find the resultant basic reproduction number, we use the next generation operator approach as described by Jones (27). Here, the next generation matrix is R v R h. Therefore, R R R β β cσ v H h v 2 μhμv α1 μh (3.5.1.5)

Mathematical Modelling Control (In Indian Context) 112 3.5.2 Stability of Disease Free Equilibrium In order to check this, we write the Jacobian of the system (3.5.1.1) at B h E Sh, M, H, HM, A, AM,,,,,. μh βhcbh βhcbh μh ν μhh μh h μh ν δm βhcbh βhcbh J μh α1 μhh μhh μh α2 δm α1 μh δa α2 μ δ δ h A M Clearly, trace(j) < and β HcBh det( J) μh μh δa δm μh μ δm μh δa μh α2 δm μh α1 μ h h In order to get a condition for det(j) >, μ μ δ δ μ μ δ μ δ μ α δ μ α βhcbh μh α1 μ β cb H h h h A M h M h A h 2 M h 1 μ h h h h βhcbh 1 Rh 1 μ μ α h h h 1 [by (3.5.1.3)] This shows that the disease free equilibrium will be stable for human population if R 1 otherwise unstable. h

Jyoti Gupta 113 3.5.3 Sensitivity Analysis Here we analyse that how sensitive R is to all the different parameters it depends on. We call these indices as sensitivity indices of R. These indices tell us how crucial each parameter is to disease spread and so help us find the right parameter to be taken care of. We have taken parameter values from published studies and government websites. For some parameters we picked realistically feasible values. We use parameter values as ν =.35, β H =.47, β v =.24, c = 3, σ =.25, µ h =.8, µ h =.33, α 1 =.1, α 2 =.5. Analysis results are given in table (3.5.1). Table 3.5.1 Sensitivity Indices of R to different parameters Parameter Sign Value β H +.5 β v +.5 c +.5 σ +.5 µ h -.537 µ v - 1. α 1 -.463 These results show that the most important parameter for controlling the coinfection scenario is mosquito death rate. This means that we need to keep a control on mosquito population so that the chances of infection spread will reduce.

Mathematical Modelling Control (In Indian Context) 114 3.5.4 umerical Simulation To understand the disease dynamics in future, we simulated the data with sample human population of 25, and proportionally mosquito population of 1,,. The results are shown in figure 3.5.2. Fig 3.5.2 Population trends in various compartments of HIV-Malaria co-infection model The simulation results, as given in fig 3.5.2, show that as soon as a person catches symptomatic HIV infection, it starts getting co-infected with some other opportunistic disease(s) as malaria in this case. Also, it has been shown by last two graphs that people co-infected with AIDS and malaria have shorter life span as compared to those having AIDS only. 3.5.5 Results and Discussion Here, we formulated a model for HIV and Malaria co-infection. The entire human population is divided into six compartments with the objective to compare the trends between HIV only class and HIV with Malaria class, and