Mathematical Modelling & Control of Vector Borne Diseases in India

This report explores the mathematical modelling of infectious diseases in India, with a specific focus on vector-borne diseases. It begins by introducing the importance of mathematical models in predicting and controlling the spread of these diseases, particularly in the context of India's high population density and the need to reduce infant and adult mortality. The report details a basic susceptible-infective-removed (SIR) model, defining variables and demonstrating the transmission dynamics. It includes calculations for threshold densities, herd immunity, and reproduction ratios, along with a practical example using malaria infection data. The analysis provides a graphical representation of the population dynamics, showcasing the interplay between susceptible, infected, and recovered individuals over time. The purpose of this is to understand how such models can be used to inform public health interventions and manage infectious disease outbreaks.

1
Modelling infectious disease in India –
vector borne diseases
I have decided to consider the mathematical approach around the topic of modelling infectious
disease in India. With a greater rise of this vector borne disease and tracing back the origin of
mathematical models of mosquito-borne pathogens transmission that originated back in earlier
twentieth century, with a preference of how to manage and control it. In my thinking I presumed that a
mathematical modelling can be derived out in order to be able to predict its occurrence in most areas in
India and also be able to control and manage it as fast as possible with a greater hope on my side to
reduce with large extend the infant deaths and adult death too.
Mathematical model
Considering population densities
Let
The susceptible individuals be represented by X
The infective individual be represented by Y
The removal individual being either immune or dead be represented by Z
The closed population will therefore be equal to (X+Y+Z = N)
The direct transmission and mass action mixing transferred from X to Y be represented by 𝛽𝑋𝑌
𝛽 = contact rate
The removal of infective transferred from Y to Z be represented by 𝛾Y
Demonstration of the above assumption
𝜸𝒀
𝜷𝑿𝒀
Susceptible infectives removals
𝑑𝑥
𝑑𝑡 = 𝛽𝑋𝑌 𝑑𝑦
𝑑𝑡 = 𝛽𝑋𝑌 − 𝛾𝑌 𝑑𝑧
𝑑𝑡 = 𝛾𝑌
X Y Z

2
If the density of susceptible exceeds threshold then a single infective will results into an
epidemic to a susceptible population
That is;
At t = 0
𝑑𝑦
𝑑𝑡 = (𝛽𝑋 − 𝛾)𝑌 > 0
This will happen if
X > 𝛾
𝛽
And always note that
N ≡ 𝑋
For susceptive rate to be infective (𝛽𝑋𝑌)it must exceed the rate at which the infectives are
removed (𝛾𝑌)
At the end of an epidemic the end population will contain either;
• Removals
• No effective
• Susceptible at point below threshold density
Calculations of threshold
Considering the following
A closed population = N = 8700 people per meter square
𝛽 = 0.001 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑒𝑡𝑟𝑒 𝑝𝑒𝑟 𝑑𝑎𝑦(0.4 probability of transmission per contact)
𝛾 = 0.5 𝑝𝑒𝑟 𝑑𝑎𝑦
Hence
𝛾
𝛽 = 1250 𝑝𝑒𝑜𝑝𝑙𝑒 𝑝𝑒𝑟 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑒𝑡𝑟𝑒
N > 𝛾
𝛽
8700 > 1250
The first secondary case
𝛽𝑁
𝛾 > 1
6.96 > 1

3
In a low population if epidemics begins it becomes non-sustainable without an influx of susceptible, and
major this epidemic does majorly begin in very low-density population
The eradication of an infection by mass immunization is well determine through reducing the density of
susceptible below the the threshold and the process will be known as Herd immunity.
The population being protected from the outbreak usually becomes theoretically possible with less than
100% immunization.
Average age of infection recorded in India
Reproduction ration of the vectors R
R is the total number of the secondary cases produced from single infective case that is introduced into
the susceptible population by the mosquitoes.
At R > 1 the infection persists and there is steady influx of susceptible
R ≅ 𝛽𝑁
The larger value of R is associated with greater contact rate, greater duration of infectiousness or
probability of transmission per contact
At endemic equilibrium 𝑋
𝑁= 1
𝑅 meaning that the susceptible fraction decreases with larger R
Mass immunization
The eradication will be defined by the following formula = R”
R” ≅ 𝑹(𝟏 − 𝒗)
If R” < 1 the immunization level v > 1- 1
𝑅
maleria, 10
whooping cough,
6.5
0
2
4
6
8
10
12
urban rural
Age (years)
Chart Title

4
And if 1 <R” < R the infection will persist in the population with reduced incident and higher mean age.
Example
The malaria infection data was collected within 21 days in a community in India
People susceptible (X) Infectious people(Y)
10 0
15 1
20 3
25 4
30 7
35 9
40 11
in a population of 31 people in a community, 30 were susceptible, 1 was infected and 0 was immune.
D = 1
𝛾
Where D = duration of infection
𝛾 = the recovery rate
𝛾 = 1
𝐷 = 1
21
If the value of R0 = 𝛽
𝛾 = 120
R0 = is the number of people the infectious person will pass on their infection to in a totally susceptible
population
𝛽 = 𝛾𝑅0
𝛽 = 1
21 ∗ 121 = 5.76
𝑑𝑥
𝑑𝑡 = 5.76𝑋𝑌
𝑑𝑦
𝑑𝑡 = 5.76𝑋𝑌 −
1
21𝑌
𝑑𝑧
𝑑𝑡 = 1
21𝑌
Substitution to the formula

5
𝑑𝑥
𝑑𝑡 = 5.76𝑋𝑌
Case 1
𝑑𝑥
𝑑𝑡 = 5.76 ∗ 10*0 = 0
Case2
𝑑𝑥
𝑑𝑡 = 5.76 ∗ 15 ∗ 1 = 86.4
case 3
𝑑𝑥
𝑑𝑡 = 5.76 ∗ 20 ∗ 3 = 345.6
Case 4
𝑑𝑥
𝑑𝑡 = 5.76 ∗ 25 ∗ 4 = 576
Case 5
𝑑𝑥
𝑑𝑡 = 5.76 ∗ 30 ∗ 7 = 1209.6
Case 6
𝑑𝑥
𝑑𝑡 = 5.76 ∗ 35 ∗ 9 = 1814.4
Case 7
𝑑𝑥
𝑑𝑡 = 5.76 ∗ 40 ∗ 11 = 2534.4
𝑑𝑦
𝑑𝑡 = 5.76𝑋𝑌 −
1
21𝑌
Case 1
𝑑𝑦
𝑑𝑡 = 5.76 ∗ 10 ∗ 0 −
1
21 ∗ 0 = −0.0476
Case2
𝑑𝑦
𝑑𝑡 = 5.76 ∗ 15 ∗ 1 −
1
21 ∗ 1= 86.35
Case 3
𝑑𝑦
𝑑𝑡 = 5.76 ∗ 20 ∗ 3 −
1
21 ∗ 3= 345.46
Case 4
𝑑𝑦
𝑑𝑡 = 5.76 ∗ 25 ∗ 4 −
1
21 ∗ 4 = 575.81

6
Case 5
𝑑𝑦
𝑑𝑡 = 5.76 ∗ 30 ∗ 7 −
1
21 ∗ 7 = 1209.27
Case 6
𝑑𝑦
𝑑𝑡 = 5.76 ∗ 35 ∗ 9 −
1
21 ∗ 9 = 1813.97
Case 7
𝑑𝑦
𝑑𝑡 = 5.76 ∗ 40 ∗ 11 −
1
21 ∗ 11 = 2533.88
𝑑𝑧
𝑑𝑡 = 1
21𝑌
Case 1
𝑑𝑧
𝑑𝑡 = 1
21 ∗ 0 = 0
Case 2
𝑑𝑧
𝑑𝑡 = 1
21 ∗ 1 = 0.0476
Case 3
𝑑𝑧
𝑑𝑡 = 1
21 ∗ 3 = 0.143
Case 4
𝑑𝑧
𝑑𝑡 = 1
21 ∗ 4 = 0.1905
Case 5
𝑑𝑧
𝑑𝑡 = 1
21 ∗ 7 = 0.333
Case 6
𝑑𝑧
𝑑𝑡 = 1
21 ∗ 9 = 0.429
Case 7
𝑑𝑧
𝑑𝑡 = 1
21 ∗ 11 = 0.523
Graphical presentation

7
-100
0
100
200
300
400
500
600
700
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
population
time (day)
X
Y
R