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^{*}Corresponding author: Deepak; E-mail:

Hemorrhagic septicemia (HS) is an acute septicemic endemic disease of buffalo and cattle in India with a case fatality rate of up to 80%. This disease causes an estimated economic loss of USD 792 million per year in India. Vaccination and control programs of HS can be understood by mathematical models. The main objective of our study was to design a mathematical model to analyze the effect of vaccination in controlling outbreaks of HS in India. We used posteriori model building approach to create and run the model for HS with the help of an outbreak data from Murshidabad district, West Bengal, India. The best possible transmission coefficient (β) to imitate the outbreak was found to be 0.335 and the best possible basic reproduction number (_{0})^{th} to the 20^{th} day of the HS outbreak reduced the proportion of the susceptible animals from 0.99 to 0.40 on the last day of the campaign. We concluded that animal vaccination modelling for eradication of HS by mass regional or nationwide vaccination campaigns can be understood by simple mathematical models.

Hemorrhagic septicemia (HS) is an immensely deadly septicemic disease of acute nature in buffalo and cattle (Shome

During 1974-1986, HS caused maximum mortalities in buffaloes and cattle among all infectious diseases in India (Dutta

How to cite this article: Deepak, Gulia, D. and Aly, S.S. (2020). Modeling Vaccination programs in outbreaks of hemorrhagic septicemia in India.

Mathematical models of outbreaks are helpful in understanding the impact of vaccination programs from a long time. Target vaccination coverage for eradicating a disease can be understood by simpler one- line models but for other complex questions, complex stochastic mathematical models are required (Schrer and McLean, 2002). The epidemiology of HS and its vaccine characteristics such as proportion of initially vaccinated animals, the duration of protection by the vaccine and the total animal population covered by the vaccination program are important things to understand to model the vaccination programs of HS during an outbreak (Woolhouse _{g}) for HS on the basis of the data available for various outbreaks in Kazakhstan in May, 2016 (“Promed Post - ProMED-mail,” n.d.), in Israel in March, 2017 (“PRO/AH> Hemorrhagic septicemia - Israel: (HZ) bovine, OIE ~ Outbreak Watch,” n.d.), in Malaysia from 29^{th} September to 10^{th} October, 2003 (Bisht

To reduce the number of new infectious cases and hence, to decline the overall morbidity of the disease, each infectious case should generate less than one new infectious case. The total number of secondary infectious cases generated by one infectious case is called as the effective reproductive number (R). Outbreaks start declining when the _{g}). The total number of secondary cases generated by one primary case introduced into a population in which a proportion _{0v}_{0v}_{:}]. For eradication of a disease from the population _{g}_{c)}_{:}) (Schrer and McLean, 2002).

Mathematical models are useful in predicting the dynamics of the disease outbreak with or without the introduction of a vaccination program during the outbreak. A population with a vaccination program contains four different group of animals: susceptible (S), vaccinated (V), infected (I), and recovered (R). The immunity of commercially available and commonly used alum-precipitated HS vaccines is from 4 to 6 months and that for the oil-adjuvant HS vaccines is up to 1 year (Verma and Jaiswal, 1998). Since, the duration of a HS outbreak is generally less than the immunity time period of any HS vaccine, the equations of our

Where

Where

Where

Where

Different groups of animals in the animal population during vaccination: susceptible (S), vaccinated (V), infected (I), and recovered (R). The arrows in the figure show the entry or exit of animals in various afore-mentioned animal groups

The model parameters used in the above transitions’ equations are calculated from a HS outbreak data from Murshidabad district, West Bengal, India (Mitra

Numerical values and interpretations of model parameters

MATLAB and Simulink were used to create and run the model. Since, outbreaks of HS mainly occurred in the monsoon months from July to September (Gowrakkal _{g} parameter (which is equal to 1.011) by comparing the total infectious fraction of animals at the end of the outbreak for different transmission parameter

Best possible _{0} value on the basis of total fraction of animals infectious at the end of the outbreak

The population of cattle and buffalo at risk was 7120. The best possible transmission coefficient (ft) to imitate the outbreak was found to be 0.335 (Kermack and McKendrick, 1927). The efficacy of the HS vaccines in some of the experimental studies was found to be nearly 100% (Mohammad

We used _{0}^{th} September to 10^{th} October, 2003 (Bisht

Finally, we introduced a vaccination campaign by using a constant vaccination signal in the Simulink model starting in the first week the outbreak (Mitra

The best possible transmission coefficient _{0}_{0}

_{0}_{0}_{0}^{th} September to 10th October 2003 (Bisht _{0}

Change in _{0}

R0 values calculated for all the 26 HS outbreaks from July 1995 to June 1999 in Haryana state of India

During the 90-day course of the outbreak, the proportion of the susceptible animals decreased from 0.99 to 0.97 and the proportion of the infectious decreased from 0.0005 to 0.0003. 0.01 proportion of the animals recovered during this duration of the outbreak. The critical proportion of the animals to be vaccinated (1- 1/R_{g}) to stop the outbreak was found to be 0.005.

After the introduction of the vaccination campaign from the 5^{th} to the 20^{th} day of the outbreak, the proportion of the susceptible animals declined from 0.99 to 0.4 on the last day of the campaign. The proportion of the infectious declined from 0.0005 to 0 with in a time span of first 33 days of the outbreak.

The delineation of effectual control programs for HS is often impeded by the lack of the basic sero-epidemiological information. In this study of mathematical model designing for HS, we used an outbreak data from Murshidabad district, West Bengal, India (Mitra

Our results show that the R_{g} calculated for the data of various HS outbreaks in Kazakhstan in May, 2016 (“Promed Post - ProMED-mail,” n.d.), in Israel in March, 2017 (“PRO/AH> Hemorrhagic septicemia - Israel: (HZ) bovine, OIE ~ Outbreak Watch,” n.d.), in Malaysia from 29^{th} September to 10^{th} October, 2003 (Bisht _{g} for HS are may be due to the quicker treatment and vaccination interventions in its outbreaks. Hence, these small R_{0} values also point towards the fact that on time treatment and vaccination campaigns during the outbreaks are always beneficial for the control of this deadly disease. The disease dynamics also vary with the use of different types of HS vaccines based on the efficacy and duration of immunity provided by the vaccine and this is a matter of further research. Using the results for efficacy and duration of immunity of the HS vaccines from the vaccination studies in natural environment instead of the small experimental vaccination studies, can be more reliable. Further analysis of the contact patterns between animals can shed more light on the epidemiology and spread of HS in a specific area. More precise demographic, biological, and epidemiological information and real-time data about HS can be helpful in designing more robust and informative mathematical models for this disease. A combination of factual and theoretical research is needed for better understanding of HS and for animal health policy planning. More complex stochastic models can be used to analyze both endemic cycle and outbreak episodes of HS in endemic animal populations of India.

The success of introduction of a vaccination campaign in controlling an HS outbreak is variable if vaccination is introduced at early, mid or late phases of the outbreak. We also conclude that animal vaccination modeling for eradication of HS by mass regional or nationwide vaccination campaigns can be understood even by simpler mathematical models and these models can be useful in assessing alternate strategies for control of HS and furthermore, in guiding HS related epidemiological research.